Fatigue Reliability of Gas Turbine Engine Structures - NASA Technical ...

NASA/CRD97-206215

Fatigue

Reliability

Engine

Structures

Thomas

A. Cruse,

Vanderbilt

Sankaran

University,

Prepared

under

National

Aeronautics

Space

Lewis

Grant

Administration

Research

October

1997

Center

Mahadevan,

Nashville,

NGT-51053

and

of Gas Turbine

Tennessee

and

Robert

G. Tryon

Available NASA Center for Aerospace Information 800 Elkridge Landing Road Linthicum Heights, MD 21090-2934 Price Code: A04

from National

Technical

Information

Service

5287 Port Royal Road Springfield, VA 22100 Price Code: A04

FATIGUE

RELIABILITY

Thomas

OF GAS TURBINE

A. Cruse,

Sankaran

Vanderbilt

Mahadevan,

University,

ENGINE

STRUCTURES

and Robert

Nashville,

G. Tryon

Tennessee

Abstract The results

of an investigation

description

consists

study.

are described

of two parts.

In Part I, the essential

the gas turbine

industry

with regard techniques

Part I is for method

concepts

are summarized.

safety certification requirements. these methods is also reviewed. to both crack

reliability

reliability

in engine

development.

structures.

approaches

to damage

These

evolved

over the years

have

theory

tolerance

of non-destructive evaluation methods based on probabilistic

and crack growth,

are outlined.

are shown

The

Part II is a specific

and practical

The effect Assessment

initiation

from structural

for fatigue

Limit

to be appropriate

case

design

in response

in

to flight

(NDE) methods on fracture mechanics, state modeling

for application

to this

problem, for both individual failure mode and system-level assessment. In Part H, the results case study for the high pressure turbine of a turboprop engine are described. The response surface approach is used to construct is used with the first order reliability the sensitivity

of the fatigue

stage turbine.

A hybrid

incorporate

time dependent

probability of failure, high pressure turbine. uncertainty

a fatigue performance function. This performance function method (FORM) to determine the probability of failure and

life to the engine

combination random

parameters

of regression variables.

for the first stage

and Monte

System

Carlo

reliability

disk rim of the two

simulation

is to use

is used to determine

the system

and the sensitivity of the system fatigue life to the engine parameters The variation in the primary hot gas and secondary cooling air, the

of the complex

mission

of a

loading,

and the scatter

in the material

of the

data are considered.

Part I - METHODOLOGIES

Summary Part I is a review of the developmentof reliability assessment methodologiesfor gasturbine enginestructures.The essentialconceptsandpracticalapproaches to damagetolerancedesignin the gasturbineindustryaresummarized.Thesehaveevolvedoverthe yearsin responseto flight safetycertification requirements.The effect of non-destructiveevaluation(NDE) methodson thesemethodsis alsoreviewed. Assessmentmethodsbasedon probabilisticfracturemechanics, with regard to both crack initiation and crack growth, are outlined. Limit statemodeling techniquesfrom structuralreliability theory are shown to be appropriatefor applicationto this problem,for both individual failure modeand system-levelassessment.Theseareobservedto provide useful sensitivity information on the influence of various engine parameterson the reliability.

Introduction The reliability (speed,

temperature

conditions design

of a gas turbine

etc.).

safety

recognized

etc.)

design

However,

methods

is affected

generally

the statistical

and given increasingly

requirement

structure

as well as in the structural

Traditional

margins.

engine

nature

appropriate

is that the probability

of crack

properties account

(material

in the operating properties,

and crack

For example,

be less than

growth

a common

1/1000,

environment

geometries,

for these uncertainties

of fatigue

treatment. initiation

by the uncertainties

boundary

using experience-based properties

have long been

commercial

recognizing

certification

the statistical

basis

of this problem. Several

methodologies

fatigue

and

growth

models,

development including paper

fracture

reliability

been

developed

of gas

turbine

non-destructive

of limit

advances

summari_es

have

state

evaluation,

modeling

in response

engine

the past several structures.

probabilistic

approaches

surface,

the development

during

ettieient

of the various

risk

to estimating simulation, methods

These

decades

to address

include

crack

assessment

etc.

Recent

component

and

system

and analytical for fatigue

initiation years have structural

approximation

reliability

the problem

assessment,

and

of

crack

seen the reliability,

techniques.

This

and the advances

made by the gas turbine engine industry in applying these concepts. presents

a case study in which the limit state modeling methodology

as system-level

1

The companion

paper (Part II) [43]

is applied for individual crack as well

assessment.

Damage

Tolerant

Design

in the

Aeropropulsion

Industry

For the most part, engine design today is driven by safe life design philosophy flight cycles before cracking

can be expected.

-- maximize the number of

This has been largely successful over the years, especially

the commercial engine market, due to a very heavy emphasis placed on high performance quality control,

and processing

had 90% of the commercial on the part of suppliers

materials,

in

material

control. The lead organization for much of this was Pratt & Whitney, as they

market share in the 1960's and 1970's, when much of the production

standards

was established.

The FAA life-certifies critical rotating structures Some work is ongoing for life certifying static

in engines,

engine structures

such as disks, shafts,

spacers,

and hubs.

such as diffuser cases which see the same

stress levels as many rotating parts and failure of which is every bit as critical. Two engine manufacturers certify the rotating parts parts,

a statistically

on a minimum life basis, meaning

that

at the mandatory

very small sample of the parts can be expected

Air Force refers to this safe life as the "economic life" of the part. limit results in a rapidly increasing number

of cracked parts.

is different as Pratt & Whitney uses an empirical

retirement life of the

to have initiated a crack. The U.S.

Exposure to more flight cycles than this

The design basis of these two life methods

crack initiation

model and Rolls Royce uses a fracture

mechanics model for initiation. Around program

(ENSIP),

development engine.

1980, the U.S. Air Force Systems Command began to develop the engine structural patterned

of the ENSIP handbook

Non-destructive

the processes

in appropriate

evaluation

and process

mechanics analysis

ways after the successful ASIP standard.

[3] was validated through application

(NDE) standards

for NDE-sized

Much of the work in

to the Pratt & Whitney F-100

were reviewed, improved, and extended,

controls, as a result of that effort. flaws to determine

integrity

All components

as well as were

received a detailed fracture

the depot inspection intervals.

As a result of the durability review of the F-100 engine, it was found that the operating stress level and material crack growth rates in engine structures resulting depot level inspections NDE capabilities.

would be required at uneconomically

That is, the

short intervals, based on the existing

automated

eddy current

component

inspection methods

in

highly sensitive NDE capabilities [26].

Very high strength turbine superalloys,

damage tolerance.

As a result, the San Antonio Air Logistics Center implemented cryogenic proof spin testing

of the F-100 first stage fan, as well as elaborate, order to establish

did not readily support

wheel designs in the 1970's started

using powder metallurgy

such as the Pratt & Whitney IN100 and the General

commercial experience

(PM) nickel base

Electric Rend 95 alloys.

with PM disk design, Pratt & Whitney established

very stringent

Due to prior

powder

handling

andprocessing standards. materials,

for example

Considerable emphasis was placed on characterizing the intrinsic defects in these

[19], in order to define more durable standards

mechanics design basis for the materials.

as well as to establish

Again, this effort followed that in the commercial

& Whitney, but that commercial work was proprietary

and still has not been adequately

The U.S. Air Force initiated an effort to increase the damage tolerance superalloys.

It was found that a coarsening

The trade-off

was that the crack initiation

life limits of the parts.

of the grain structure

a fracture

arena at Pratt

published.

of powder metallurgy, nickel base

resulged in slower crack growth rates.

life of the material decreased, thereby decreasing the economic

Damage tolerance was selected as the more important

design concern

in the final

material process selection. Engine materials have high intrinsic strength with a price. For example,

and resulting fatigue capacity.

a single nickel superalloy

This quality has always come

disk may cost well in excess of $50K. The U.S. Air Force

observed that there was much wasted money in the old method of retiring disks at an economic life limit, because so few might be expected of disk lifing was investigated

to be cracked. As a result, the "retirement

[44], [22]. The RFC approach

developed by the damage tolerance part

requirements

service.

Significant

the exposure

probabilistic

fracture mechanics

and liability associated

was to combine the inspection

with an indefinite

stayed in service until it was found to be cracked;

for cause" (RFC) philosophy processes being

life limit of engine parts,

when cracked,

such that a

the part was to be removed from

issues were uncovered in the RFC approach

owing to

with missed cracks [44].

During this time period, the commercial engine fleets were setting the pace in terms of applied fracture mechanics of engine parts.

Air Force and Navy engines were too few and acquiring

cycles at too slow a rate

to challenge the life envelope the way the commercial engines were. For example, significant numbers of midsize commercial

turbofan

engines were first designed in the late 1950's and are still being sold (in updated

versions, of course) today. There are thousands of such engines, fan disk, a high stressed

nickel turbine

each of which has a high stressed titanium

disk, and multiple high stressed

steel high pressure

compressor

(HPC) disks. The certified life limits on the various designs range from 15,000 flights to 20,000 flights. Some operators might accumulate

in excess of 2,000 fatigue cycles a year.

Some design successes were clearly established & Whitney were retired in such numbers in the minimum turbine

by the commercial fleets. The titanium

by the early 1980's that one had very high statistical

fatigue life of the material.

Equally high reliability

was demonstrated

disk designs as well. However, there were also some design problems

It was found, for example, design periods,

prior to the use of the finite element method.

confidence

for a number of

in the late 1970's and 1980's.

that the stresses in engine disks had not been adequately

calculated

in the early

As a result, several engine models had disks

which would not reach their initial design life. Damage tolerance assessments fracture

fan disks at Pratt

were made using probabilistic

mechanics to determine what inspection intervals could be imposed without

undue risk to the total

fleet operations of the affected engines. The FAA accepted

a probabilistic

risk assessment

(PRA)

4

basis for an in-service,

RFC program

in order

"Monte Cede" Simulations •No Inspection

Establishes Part

RISK ASSESSMENT

Exposure_ J

let inspectionlimit o Roinspectionlimit , Inspectionsensitivity o inspectionreliability

Histogram Program; "Drawdown" Inspection Limits

"L__ Cycles

"INITIATION" Scatter Cumulative % Parts Cracked

t

Propagation • Scatter

-}-

RELIABILITY OF CRACK DETECTION

Size _'_m

No

Inspect,

Crack

to 1/32" Inspect.) Limits

(!Disk

1.1

I

J(

+

Cycles

Figure

to minimize program

1: Probabilistic

the number

of aircraft

was so successful

disk in-service

cracking

for crack

initiation,

obtained

by summing

Cycles,

that

the

desired

level of risk of disk failure

cracking

problems.

in multiple tie-rod

holes,

fatigue

origins.

each

Figure

the probabilistic of the

for the total

in Figure

fleet

(POD). of engine

are then adjusted

were found to be important

a notion

of which has high stresses

as the basis for all engine

1 include

of detection

The

probabilistic

models

The

total

fleet

disks

using

the

risk is known,

in such a way to achieve

the

fleet.

occurs in cadmium 2 gives

due to field cracking.

risk assessment

risks over the entire intervals

for Disk Cracking

from service

PRA shown

probability

The inspection

damage

Flow Diagram

had to be removed

NDE

element

cyclic histogram.

The corrosion

crack

and

individual

and future

and multisite

that

The elements

propagation,

current

Corrosion

engines

the FAA adopted

problems.

crack

Risk Assessment

T

plated,

steel

HPC disks.

of the multisite

on the inside

elements

origins

of some

The multisite

problem

of the hole along

of these

in-service

corrosion

for a disk with

the circumferential

results multiple

diameter

of

the hole. The original sensitivity in effect, model upwards,

PRA

for this problem

for the magnetic statistical

reviews

was non-conservative meaning

was based on the usual

fluropenetrant

NDE

of the in-service and

that

the

that there was a significant

procedure

assumption

then

in use for the part.

crack size data strongly computed

of a single

suggested

risk for the inspection

expectation

of further

field events

that program

crack and While

the

the fracture had

of uncontained

a nominal PRA

was

mechanics

to be adjusted ruptures.

A

Disk

HPC

ole

Sec±ion

(typ) Crack

______Tie

rod

hole

give rise to reduced fracture Several cracking scenarios work. The hardest

from several incidents revealed that the multisite mechanics

(±yp)

(typ)

Figure 2: Typical HPC Disk with Multiple Corrosion-induced

review of the failed hardware

sites

Fatigue Origins

crack initiation process could

lives.

are shown for this problem

in Figure 3, taken from that original unpublished

part of the effort was to combine the factors of multisite

cracking with the physical nature

of the NDE method

used, in that two small cracks close together

engineering

was used for the problem by the first author and a critical combination

approach

spacings was used to define a new fracture mechanics

were identified as one larger crack. An of multiple crack

model. The revised model was used with the Monte

Carlo method to successfully simulate the in-service cracking data.

Based on this success, the inspection

intervals for the airline operators was reduced and no further incidents of disk failures were recorded. In-service turbofan tie-rod

failure history

involving the uncontained

failure of more than a dozen mid-size

engines showed that the final failure event liberated a rim segment

commercial

that usually contained three

holes and, less frequently, two tie-rod holes, as illustrated in Figure 4. While many holes (out of the

23 on each disk) had cracks, the failure occurred when multiple cracks linked up in the patterns

indicated.

Thus, multiple crack origins at multiple holes, crack growth from multiple origins, multiple crack link-up of two-hole and three-hole cracking pattern

patterns,

and growth of the final cracks to the rim defined the probabilistic

for this in-service problem.

Later designs of large-size commercial

turbofan engines made use of rotor designs that eliminated

tie-rods holding the stack of engine disks together. another,

field

Instead, the rotors were assembled

the

by bolting disks one to

near the outer diameter of the rotors. This lighter design concept resulted in much more shell-type

bending of the disks.

However, a new multisite

fatigue origin problem

rotor disks causes high local stresses to occur around locations. A simple diagram is included

was discovered.

Shell bending of

the full circumference of the disk, at the peak stress

to try and clarify the locations of these origins in Figure 5.

Today's engine designs make use of welded rotor construction

with the welds placed in low stress locations

_ 000

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3ooo

Zooo

I000

0 •0.._0 •Og'O

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Figure 3: Various Residual Life Models for multisite Tie-Rod Hole Cracks

3

Hole

Frac±ure

2

HoLe

Frc_cture

Figure 4: Multi-hole Fatigue Cracks Associated with In-service Failures

7

Engine

Cen±er(ine

Figure

for damage bending

5: Typical

tolerance.

However,

these

stress-induced

multi-site

damage.

In all of the commercial analysis

plays

engines,

the FAA manages

repetitive

sized crack.

is untenable.

The

probabilistic

fracture

problems,

destructive

related, which

accepted

of the (NDE)

field reliability

components

based

damage

to inspect,

problems,

cracking

all suspect that

risk assessment.

is detectable

tolerance

this

analysis,

approach

(NDI).

The

measures

field cracking

sensitivity

the nominal

sensitivity

occurs

NDE

is, what

of detection is the largest

as NDE reliability

often

NDE

methods.

level

driven

of the NDE

is adequate

to define

a reality

using

for these

of crack detection. use of nonof the method

(usually

in terms

crack that

given can

an operator-specific

of operational

with one notable

For problems

the inspection

with

intervals.

of the

be missed?

on the subject,

by the number

method.

be managed

the sensitivity

is often really

Past experience,

parts are available

is the effective

For a recent review

are most

which mandates

engine is flying with a

is therefore

include

tolerance

affects multiple

directive

but on the technology

issues

to this same

damage that

problem

to life management

the use of automation.

by standard

probabilistic

until replacement

Retirement-for-cause

mechanics

and life extension

are susceptible

problem

engines

crack size), and the reliability

through

on the nominal

cracking

and

in order to assure that no commercial

now requires

for NDE

Concentrations

the legal control of an airworthiness

of grounding

or inspection

Stress

light-weight

an in-service

issue for NDE is automation,

a crack that

has been

and

question

can only be controlled

generate

through

and

detectable

A key reliability

Engine

When

not only on fracture

nominal

supporting

in-service

of cracked components

mechanics

evaluation

(the minimum,

role.

the problem

FAA

part

are even more

engine

The alternative

and depends

An intrinsic

crucial

Disk with Local Bending

rotors

gas turbine

an absolutely

field inspections

significant

POD).

HPC

A issue

see [24]. cycles

to

exception,

relatively

few

However, for problems with many characteristic hardest

for the

and least

management. sample

NDE

defects

after a field problem is the largest

2

crack

initiation

structure most

of the material.

successful

functions

approaches

of damage data.

and the crack

initiation

relatively

is a defined

performing

material

provides

a cheap

machining

Research and

nickel

fatigue

extraneous

features stress,

alloys.

machined

The

initiation

cracking

with a realistic

the NDE specialist

induced

fatigue

is faced

crack or corrosion

of an in-service

metallurgical

and

and

is therefore

is loosely

fatigue

Rather,

crack[35].

It is only

The question

through

related

influenced

of what

are empirical

the microstructural

The micro-

and strains.

in that

summation

issues. by the

to macro-stresses

some damage

between

defect

greatly

correlated

crack initiation

process

the

testing

for various

various

algorithm

features

was not tied to a well-defined

cycles only

fatigue

to specimen

and

not

failure

developing

lives for various

materials

crack initiation goal

was

were

materials.

is not characterized.

The

algebraic and fitted

of the material

of damage

then,

which,

So long as one is

database,

Even

Again,

state

reported.

a design

to arrive

for the U.S. Air Force by Cruse

to define

to a specified

crack

at the best

all surface

such as finished

holes,

were taken

the size.

preparations and

role of the The

mechanical

replication

(KT = 1), strain-controlled

specimens

in the life model.

specialist

real cracks can be examined.

many

volume

initiation

damage

relative

materials,

unnotched

to in-service

is the

specimen

however,

the generalization

failure

the

role of

of the results

compromised.

factors

two selected

size.

way to assess

into fatigue

crack

involving

give no relationship

characterization

is thereby

Too often,

method

after the failure.

are combined

the fatigue

crack

the specimens

to design

approach

the POD

process.

recently,

generally,

crack

models

for an NDE

the NDE

the true state

--

Modeling

in a very small

parameters

These

tolerance

repetitions

Mechanics

to deterministic

driving

to experimental

Until

determined

process

Typically,

damage

if ever, represent

Initiation

occurs

characteristic

with reliability.

for some time that

is a complex

process

POD

or many

limits on the basis of a laboratory

rarely,

Crack

components,

issue is to provide

are to be detected

Fracture

crack initiation

true

in the overall

crack is often

Probabilistic

Fatigue

that

has been active

undetectable

The

and most difficult

defects

Probabilistic

2.1

element

field inspection

such laboratory

m many

is critical.

The most important

with the task of setting conditionl

method

well developed

of the critical

inspections

mechanical

goal could model.

were closely

to be in an altered

factors

only

be met

Specimens

state

titanium

in notches

governed

the cycles

from

or minimizing

single

specifications to a given

the baseline

whose

that

by eliminating

to design

were considered

mechanical

[14, 15] included

were taken

controlled

was used to define specimens

and Meyer

parameters

heats

of the

for machined crack

material had

size.

specimen;

Zero all

to be included

The results

of the test

was much less than altered

program

the design

by the machining

specimens

to that

conditions

(each of these

not similarly

confirmed

that the scatter

basis for scatter

operation.

specimens

it did appear

through failed

of the material.

to fatigue

due to surface

the life model

in the simple

are two statistical is to construct

3_r). Such

approach

power

approaches

that

was used included

an approach

(Note:

for the treatment

is not particularly

a probability

model

For simplicity,

of fatigue

probability

butions.

There

on the basis

models

of a mechanical

model

of the Weibull

has a volume

• the model

has

The original

three

link.

[27, 28]. A more However,

mathematical

defects,

rather

defects

and

strain

are obviously

model

fatigue

data on a log-log

fatigue

distribution

life capability.

to fatigue

models.

(fatigue)

and

lifetime.

linked

with

and the lognormal distribution [50].

distri-

was developed There

are

three

lives

strength

was defined Weibull

the fracture

the simple

distribution

distribution to have three

by Wirsching

10

and [7].)

modeling:

to characterize

of this fairly standard

is the work

available

by the strength

size effects is available

issue is given in [33].

the lognormal

also has the ability

therein

of materials

have lower

The fact that

the statistical

fatigue

life can be included

methods

physically

makes plot,

The Weibull

was that of a chain whose

most

of probabilistic

is a rich literature

the Weibull

yielding

the probabilistic

wear-out

larger specimens

of various

An example

crack initiation

to both

minimum

There

A or B; mean

The more proper

through

see [42] and references

make it useful in fatigue

that

basis (e. g., Type

on the simplest

design,

most often used

life modeling.

deterministically.

rate indicating

than with fatigue plot

range A¢ and mean

1. The approach

the life scatter

will focus

to the size effect

is really

When

The lognormal

that

such

approach

strain

strength

(1)

for example

for fracture

discussion

life lines on a log-log

distribution

stress

value of the fatigue

cyclic

such as in Eq.

used for fatigue

by Weibull

leads to linear plotting

widely

hazard

parameters

An extensive

the Weibull

process,

effect such that

size effect postulated

of the weakest

can be described

distribution

• the model

both

for a probabilistic

herein

and disadvantages

has an increasing

minimum

sub-surface

A based on a design

good

the discussion

are most

• the model

Ganssian.

local

c°M

for life and characterize

as a random

are advantages

characteristics

to existing

was significantly

law form

for the parameter

those for which the load history

Two

of the

seeks to define the expected

to using a life model

a lower bound

is to define

parameters. models,

behavior

LCF tests

map the data for unnotched

model

crack initiation;

N = AAc_10

minus

one could

a macromechanical

life characterization

For example,

stress _rM parameters

in design

that

fatigue

controlled

affected).

Such an approach

There

lives for these

and that sub-component

However,

for sub-components

in fatigue

of brittle

form of most

the most

materials fatigue

convenient

due

models

of the two.

of the life data would be normal parameters,

application

thereby

also allowing

of the lognormal

and his co-workers,

e.g.,

[32].

probability

or a

The second of fracture

approach,

mechanics

crack initiation

process

structures

is becoming

as the empirical

use the terminology aircraft

which

popular

for modeling

form of the life model.

is characterized

by a single

most often applied. by Yang

more

and his co-workers.

It is assumed

crack from

This approach

fatigue

exists

initiation,

in this approach

an "equivalent

has been widely

There

crack

initial

that the fatigue

flaw size (EIFS),"

and effectively

now an extensive

is the use

applied

literature

to

to aluminum

stemming

form this

work done in the 1980's. An overview

of the EIFS approach

is to use probabilistic

fracture

back-calculate

what

correlate

the known

EIFS

with

(which

life results approach

mechanics

distribution data.

than does

that

to take a given crack

using

a direct

as predicted

load histories

the EIFS would

the EIFS

obtained

allow

in reverse,

If multiple then

his co-workers,

is given in [51, 53]. The basic notion

of initial flaws, growing

does not happen),

by Yang and

for aircraft

calculation

or multiple

engineering

empirical

fatigue

of size effects

cycles

by the fracture types

not be "equivalent"

is an effective

purely

size --

tool

of the EIFS approach distribution,

mechanics

of geometries but would

which

life models.

may

would

all gave the same

produce

the

model,

be intrinsic.

Certainly,

if one can determine

and to

As stated

more

a fracture

volumetric

consistent mechanics

distribution

of

the EIFS. However,

it is important

or to small flaw behavior. model

that specifically

Circulars MSD

It is also important focuses

on the subject

to point out that there

on the nature

state

that

Probabilistic subject

reliability.

of multi-site

a two-lifetime

Fracture

is the single

An example

of probabilistic

most

well defined

fatigue

As discussed test

crack initiation

in [4], the FAA Advisory

is to assure

that

the probability

of

suggests

models

almost

use the lognormal in [36].

The

Paris-form

usual

all probabilistic form,

application

lognormal

micromechanics model

difference

implementations

for crack growth

for probabilistic

between

crack growth

many

form of the probabilistic

others

system

[36]. A review

in the first chapter modeling

because

the various probability

them.

of the standard

[4, 53, 34], and

form of fracture including

crack growth

most

modeling

mechanics

models

of the studies

is illustrated

do

reported

by using

the

as follows da/dN

taken to be deterministic

be taken to be random,

in all of mechanical

et al. [35] cite various works evaluating

statistical

model,

methods

work is found in the book on the subject

and probabilistic

Palmberg

for example

of the crack growth

n is usually

of probability

the use of the lognormal

and that lack of a significant

In fact,

Modeling

of the previous

modeling

it is simple to use and conservative.

may

fatigue

are not tied to microstructure

is no probabilistic

damage.

full-scale

Mechanics

of the scope

crack growth

by the book's editor

where

in the literature

is low.

2.2 This

to point out that the EIFS models

the consensus

-- CAK

n

and

C is taken to be lognormally

is that

one is able to reconstruct

11

(2) distributed the original

(Note: database

While

n

of crack

growth data with excellent we are now describing).

correlation by taking n to be deterministic;

this is certainly far easier for what

Taking the log of Eq. 2, we obtain log{da/dN}

- log C + n log AK

(3)

Each of the terms in Eq. 3 is now a normally distributed variable. The equation if we take the initial flaw size to be quite small, the toughness be constant.

distribution

to be deterministic,

When these are true, then the fatigue life is lognormally

rarely useful and appropriate

approximations

for the fracture mechanics

fracture mechanics Probabilistic

can be integrated

analytically,

and the cyclic stress to

distributed.

Of course, these are

and other methods must be used to determine the probability

fatigue life. The most common method for computing the random

life in the general case is the use of Monte Carlo simulation,

fracture mechanics has been used in the design of titanium,

e.g., [5, 8, 2, 34]. steel, and nickel disks with

internal defects since the early 1970's [16]. Figure 6 shows a set of design curves for a defined reliability of 0.999, where the curves represent differing statistical material toughness. scatter

parameters

for the subsurface defect, defect size, and

The material crack growth rate is taken to be lognormal with the usual level of material

for these alloys. The use of this type of design system has been effective in preventing

titanium fan disks due to intrinsic, type-I hard-alpha to the sensitivity

defects. Such a design system can be directly couple

and reliability of the NDE methods used for detecting the subsurface defects.

A philosophically

different approach

tations of the Markov chain approach approach

to probabilistic

fracture mechanics

to simulating random processes,

is based on various implemen-

e.g. [42]. The thinking

is that if you look at the very local processes of crack growth,

ing whether

the failure of

they appear

behind this

to be random regard-

or not the crack grows in a given cycle and which way it grows. The Markov chain modeling

approach

allows the current event to be random in a manner that is variable tied to the past history.

example,

the current increment

dent on the current

state.

in crack extension

While the Markov chain approach

of crack growth, it has no significant crack growth is microstructural turai in nature; mechanics.

to be beyond the current state-of-the-art

and some non-physical

solutions

(and generally

randomness in to be microstruc-

of microstructural

unavailable)

which the mathematical

of the Markov chain model with a microstructural

The EIFS analysis

The start-stop

in nature and any simulation of this process must attempt

such capability appears

depen-

seems at one level to mimic the observations

relation to the physics of the process.

The Markov chain models need extensive

crack jumping, application

could be totally random or it could he somewhat

For

statistical

formulations

allow.

fracture

data on the An example

flavor is given in [9].

described earlier also treats the crack growth process as a random process, [52], but

in quite a different meaning.

A summary

of the method is found in [52]. In this approach,

the crack growth

rate is given as da(t)/dt where X(t)

is a non-stationary

usual deterministic

fracture

= X(OL(ZXK , K,,,,,,

random process with an expectation

mechanics

R, S, a) E(X)

(4) = 1. The function

model with a general retardation capability. 12

L 0 defines the

Yang

et al. take X(t)

relation

function

to be a lognormal

for the random

process

random

in log-space R,:(r)

Since

the autocorrelation

function

transform

of the autocorrelation

lognormal

random

then

the random

becomes

process. process

the lognormal

in gq.

X(t)

random

= E[Z(t)Z(t

in Eq.

the random

becomes variable

process

a random

of the

various

random

gives

the least.

Based

configurations, the mean function

the

approach

over the fact

[35, pp. 64-65].

that

and,

of Monte today

the two means

in theory,

Carlo though

(FORM)

began

order refers

means

then

(e)

model

where

variance

damage

data

for various

approach.

results

some

process

model,

aircraft

structures

dispersion

Eq.

4, and

approach,

is predicted

as a

at one time point.

assessment

on the sheet

seem to provide

dispersion

In this lognormal

or crack growth

tolerance

depends

in Eq. 7 gives the greatest

the fully random

fractography

to the fractography

for probahilistic While

provides

fracture

for fastener

thickness

and

holes [40, 51, 54].

hole sizes

is given in

quantitative

means

analysis

were direct

to simple

formulations,

the latter

formulation.

The practical

mechanics

the former is strictly

the exact

solution

limited

for any mathematical

is for reasonable

numbers

of probabilistic

for calculating

probability

integrals

in the 1970's for the purpose

to the use of a first-order

variables.

A performance

of limits on the physical where

process

for assessing

advanced

variables,

analytical

as the computer

solutions is totally

time --

limit even

-- is significant.

Approximate

random

crack growth

given by

fully correlated

to aircraft

does

at any two time instants,

Modeling

and Monte Carlo simulations. general

using

distribution

the method

of the

[29].

State

recently,

the EIFS

The random

density

ao

result;

the model

correlated

spectral

The Fourier

[1 - Xeta_)]ll"

models,

the lognormal,

has been applied

However,

alloy developments

Limit

on fitting

model,

=

mechanics

is a deterministic

is the power

is completely X.

is stationary.

= XQa (t)

evaluations

selected

rate

of time, based

Concern

Until

authors

the process

_z,(w),

that the fully correlated

fracture

on extensive

crack growth

The EIFS

2.3

process

state

then the autocor-

(5)

of r only,

variable

crack growth

a(t) In [53], the authors

= logX(t),

+ r)]

5, denoted

da(t)/dt

where e = b-1.

If we take Z(t)

is given as

5 is a function

function

When

process.

X refers

corresponds

system

to the input

to failure,

positive

Taylor series

function response

random

of predicting

g(X)

variables.

value corresponds

(e.g., The

corresponding stress,

to safety,

the first-order

and

g(X)

system

frequency,

is defined

fracture

such

= 0 is referred

that

method

structures.

model

to a performance

natural g(X)

reliability

of civil building

of the physical

function

13

on

the reliability

expansion

is defined

variables

based

First-

in terms of the

criterion

in terms

mechanics

life),

a negative

value

to as the limit

state.

COEFF VAR DEFECT DIST

Stress (KSI)

..........

150

MAX FLAW DIAMETER (in.)

_ s_

0.04 o.o25

100

50%

0.025

100

100

1o0

140 _

__

130

'_

3O%

0.015

1oo

................

30%

0.075 0.025

2o

--

3o%

0.025

50

,oo so _- .,., ,, ,,., m,-,,,'-'_'"--_-._.

8o 678910'

2

3

4 5 8 78910 s

Numberof cycles

'

1/1000 failwo probabili|y Figure

The first-order

Taylor

6: Fracture

Mechanics

series expansion

of g(X)

Design

about

Curves

the mean

for Subsurface

values (pl)

Defects

of the random

variables

is given

by

09 9(x) _ #(x) = ao+ ,=N _ K_

(x,

- _')

(_)

i=1

A safety

index

the above accurate

equation. estimate

probabilistic distributed served

was first defined

= pg/o" I to quantify

For uncorrelated, of the failure

fracture

mechanics,

normal

probabiity

The probability function

point

for effective

of structural

fx(z)

failure

of all the random

variables

the reliability, and linear

as p! = @(-/_).

the function

and may not be uncorrelated.

as the starting

density

as/_

9(X)

Nonetheless,

the above

approximation

algorithms

is mathematically variables

performance

In general,

is not linear

pg and % are easily obtained functions,

however,

and certainly

and the random first-order,

defined

over the region

developed

variables

second-moment

analytical

methods

such as FORM.

14

in the case of

are not normally representation

the limit

of the joint state

probability

is violated,

P/ -- _(x)