Service Life Analysis:

 A Prescription for Fatigue p. 2

A Helicopter with

 Extra Sensory Perception p. 32

What’s Eating Us? Corrosion!

p. 14

aaaa aaaaaaaaa

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Volume 50, Number 4/1998

and more . . .
A.K. Vasudevan

Office of Naval Research

Margery E. Hoffman

Naval Air Systems Command

From the Guest Editors . . .

S

tructures and their components almost

invariably acquire defects at some point

in their life cycle, either during their

manufacture, fabr ication, assembly, or

maintenance. Under a variety of loading

and environmental conditions, these defects initiate

cracks that can grow with time and ultimately lead

to the failure of the component. Depending on the

function of the component, the effects of this accumulating fatigue damage can range from a minor

degradation in system performance to a catastrophic

failure of the entire structure.

Fatigue damage is one of the major life-limiting factors for most structural components. An

analysis of past fatigue failures in aircraft components shows that many cracks initiate at (1) stress

concentration sites such as changes in radius, corners, rivet holes, or other design features in the structure, (2) initial manufacturing defects, or (3) defects

that were inadvertently introduced during assembly, structural inspections, or repairs. Recognition

of these causes has had a profound effect upon aircraft design philosophy, spurred the development

of materials with improved defect quality, and led

to better inspection and maintenance practices.

Even with better design techniques, better

materials, and inspections, however, components

still fail prematurely and unpredictably. The inability to make accurate predictions of fatigue life for a

particular structure or component stems from many

factors. These include — but are not limited to —

variations in loads experienced by components

during service; exposure to different chemical

environments that affect crack growth rates; differing crack growth behavior for short cracks and long

cracks; uncertainties in the size, location, and type

of material defects; and the unpredictability of how

long it takes to initiate a crack.

A better understanding of these factors needs

to be supplemented by accurate, more reliable methods of detecting fatigue damage during routine maintenance or scheduled inspections. This combination

will give operators and maintainers the tools to make

better decisions regarding the inspection, repair, or

replacement of structural components.

The application of damage tolerant approaches

to controlling fatigue damage in aging aircraft fleets

is unquestionably becoming more important. Ideally, these methodologies must comprehensively

model the entire realm of damage by describing

crack growth behavior from the crack nucleation

stage through the stages of short and long crack

growth to the point of final component failure.

The field of fracture and fatigue includes many

topics, but this issue of Naval Research Reviews focuses primarily on two of them: loads and environment. The selected articles give a bird’s-eye view of

some of the developments of the past several decades in the area of fatigue damage analysis. These

articles discuss current practices in fatigue life prediction, the measurement and analysis of loads, new

work in the understanding of the physics of material damage, and the effects of the physical environment upon the rates of corrosion and crack growth.

Research into these topics, and others, can ultimately provide a vastly improved baseline of information for current and future life prediction methods. These data, when used with enhanced predictive models, could have a significant impact on life

cycle costs by reducing maintenance costs. An increased use of health and usage monitoring systems

for aircraft and other systems is leading to more efficient determination of service life for flight-critical components based on the actual severity of their

operational usage. Research to advance life prediction methods needs to keep pace.
Volume 50, Number 4 1

Service Life Analysis: A Prescription for

Fatigue

Current and Future Fatigue Life Prediction Methods for

Aircraft Structures

by Dr. Margery E. Hoffman and Dr. Paul C. Hoffman

What’s Eating Us? Corrosion!

Fatigue and Corrosion: Aircraft Concerns

by Dr. Vinod S. Agarwala

You Can Count on Rain

Cycle Counting and Beyond in Variable Amplitude Fatigue

by Dr. Michael S. Duesbery

A Helicopter with Extra Sensory Perception

Neural Network Based Virtual Strain Gauges for

Monitoring Fatigue Critical Loads

by Dr. David J. Haas, Ms. Kelly M. McCool, and Mr. Lance A. Flitter

Clamping Down on Crack Growth

The Role of Crack Closure and Kmax on Fatigue Crack

Growth Rate Thresholds

by Mr. J. Keith Donald

Milestones in Modeling Fatigue Damage

A Unified Framework for Fatigue Damage Analysis

by Dr. K. Sadananda and Dr. A.K. Vasudevan

Profiles in Science

Dr. K. Sadananda

2

32

14

25

55

44

69
2 Naval Research Reviews

atigue” is more than something you feel after a long day at the office. In

metals, fatigue is the tendency for cracks to form and grow under the

repeated application of stress.

In aircraft, just as in people, fatigue can be a killer: If left undetected,

fatigue cracks in flight-critical components can grow to a point where the parts

can catastrophically fail, causing the loss of an aircraft and possibly resulting in

the death or injury of its aircrew and others. Unexpected fatigue damage also

can require costly maintenance and adversely affect readiness. Fortunately,

however, the formation and growth of fatigue damage is predictable. The expected service life of aircraft components can be calculated and then verified by

full-scale tests.

In the following article, two structural reliability experts from Naval Air

Systems Command explain the naval aviation approach to predicting fatigue life

for aircraft components and discuss how the Navy tracks individual aircraft

according to the severity of their usage. They also review ongoing Science and

Technology work to extend the current way of estimating fatigue life for the

Navy’s older aircraft.

– S.P.
Volume 50, Number 4 3
4 Naval Research Reviews

Margery E. Hoffman and Paul C. Hoffman

Structures Division – Naval Air Systems Command

Patuxent River, Maryland

Current and Future

Fatigue Life Prediction

Methods for Aircraft

Structures

Introduction

The austere operational realities of the Navy’s mission require Navy

aircraft to withstand random fatigue loading during their design

service life without cracking. Carrier-based aircraft operate from

“floating air bases” that have runways only one-tenth the length of

land-based runways, they often perform missions at altitudes barely

clearing the waves, and they are on deployment away from maintenance hangars for months on end. The fundamental constraint for

Navy aircraft, with respect to service life management, is that the

broad theater of operations of aircraft carriers and the limited space

onboard for aircraft maintenance facilities restrict routine inspections

for fatigue cracks.

Readiness is the priority of the Navy airborne fleet. The need for

readiness demands that structural integrity be incorporated into the

design of an aircraft: Cracking should not be an issue when it’s time

for the aircraft to perform its mission. With a crack-resistant design,

aircraft maintenance demands are minimized and safety does not rely

upon routine inspections of the aircraft structure. The challenge is to

design airframes that will last a lifetime, to predict fatigue damage

before it happens, and to recognize it when it does.

Designers of Navy aircraft use a “Safe-Life” approach to structural

design that allows the aircraft to withstand fatigue loading. Furthermore, Navy personnel use structural data recorded onboard the

aircraft to predict the occurrence of crack initiation on every airframe

individually.

In this paper, we will:

• Define and discuss the Safe-Life approach to aircraft structural

design from the Navy perspective

ABOUT THE PHOTO ON THE

PRECEDING PAGE:

A full-scale fatigue test of a Grumman

EA-6B “Prowler” wing, which was conducted

in the late 1960s and early 1970s, determined

that the wing fatigue life limited the aircraft’s

service life. That test was terminated due to

fatigue failure of the lower wing skin in the

center wing panels after flight spectrum loading

equivalent to 9,000 flight hours. The EA-6B is

currently projected to stay in the inventory

until the year 2013, which will result in usage

that is well beyond the service life determined

by the full-scale fatigue test.

To enable the EA-6B to continue flying

until 2013, the EA-6B’s center wing panels will

be replaced. However, this raises questions

about the structural integrity of the outer wing

panels, which did not fail during the full-scale

fatigue test but also have not been subjected

to fatigue testing that would verify and validate

an extended service life.

Will the outer wing panels make it to 2013,

too, or will they have to be replaced as well?

To answer this question, metallurgists at the

Naval Research Laboratory are analyzing

samples from the outer wing panels of the

EA-6B to help determine the aircraft’s

remaining fatigue life.

The photograph on the preceding page

shows a fatigue crack that is growing from an

imperfection in the inside of a fastener hole in

one of those EA-6B outer wing panels, which

was tested more than 25 years ago. Each of

the diagonal bands that can be clearly seen at

the center of page 3 is fatigue crack growth

that resulted from one cycle of stress.

The narrow bands at left center are

approximately 320 nanometers wide. In other

words, it would take nearly 80,000 cycles of

stress to grow the crack an inch. The wide

bands at the center of the page resulted from

cycles of higher stress and are 740 nanometers

wide (about 35,000 per inch).

– S.P.

Photograph courtesy of the Naval Research Laboratory.
Volume 50, Number 4 5

• Show the mathematical basis for using the

number “2” as a factor of uncertainty

• Describe the Navy’s aircraft structural life

surveillance program

• Discuss the future of aircraft life predictions

(with respect to metal fatigue in fixed-wing air

vehicles)

Background

There are two basic approaches to fatigue design. The

first approach, known as “safe-life,” predicts the

amount of time until crack initiation. The second

approach, known as “damage tolerance,” predicts

crack growth.

The Navy has adopted a safe-life approach for a

number of reasons, but especially because carrierbased aircraft can spend months at sea, away from the

hangars and other facilities where routine maintenance

and repairs take place. In contrast, the US Air Force

uses a damage tolerance approach to manage its

aircraft. Unless the aircraft are deployed to the field for

extended periods of time, the US Air Force has ready

access to their fleet; therefore, they can more safely

rely on structural inspections to attain and extend the

life of their aircraft.

Both the safe-life and the damage tolerance methods

have served their respective services well.

The Naval Aviation Approach

to Fatigue Life Prediction

There are two major activities associated with fatigue

life predictions. The first is analytically determining

the service life of an aircraft structure and then validating that life through a full-scale fatigue test. The

second is setting up an individual fatigue-tracking

program that collects aircraft usage data and performs

fatigue predictions quarterly for every fatigue critical

component of over 3,500 airframes in the Navy and

Marine Corps inventory.

Service Life

Service life limits are established for an aircraft based

on operational requirements, design criteria, technical

analysis and fatigue test results. In the safe-life approach used by the Navy, a piece of primary aircraft

structure is considered to have reached the end of its

fatigue life (i.e., it has failed) when a crack reaches

0.01′′ in length.

Engineers can use detailed computer models to predict

the service life of an aircraft’s structure if they know

the details of its design (such as the materials from

which the aircraft is made) and the types of missions

that the aircraft will fly.

The analytically predicted service life is then validated

in the laboratory by subjecting a full-scale aircraft

(usually one of the first to roll off the production line)

to loading that mimics the severe load conditions that

the aircraft might experience during flight. Large sets

of computer-controlled hydraulic rams push and pull

on the aircraft either until part of the structure fails or

until cracks proliferate to the point where further

testing is not warranted.

Critical parts of the aircraft structure are monitored

during the test for evidence of cracking. After the test

has been completed, components that have cracked are

removed from the test article for further study. The

surfaces of the cracks are inspected under a microscope, and the number of flight hours needed for the

crack to grow to a length of 0.01′′ (i.e., the fatigue life

of that component) is noted. Incidentally, cracks

always form in the full-scale fatigue test article; no

aircraft was ever so well designed that no cracking

occurred during the rigorous full-scale test.

The Test Spectrum

The most influential factor in determining service life

from test data is the test spectrum, which is constructed from aircraft usage curves. The test spectrum

is designed to simulate the full set of flight loads that

an aircraft will experience during its entire service life.

In the test spectrum, engineers try to account for every

type of maneuver and mission that the aircraft will be

expected to perform: takeoffs, landings, rapid climbs

to high altitude, in-flight refuelings, high speed turns,

etc.

One way to measure aircraft usage is to look at the

profile of Nz

 exceedances per thousand hours of flight.

“Nz” is measured in g’s and is the acceleration in the z

direction (i.e., downward through the floor of the
6 Naval Research Reviews

actually retard damage formation and crack

growth. Removal of these high loads from the

test spectrum causes damage to accumulate in

the test article more quickly, resulting in a

conservative value for service life for the

aircraft in the fleet.

• Low loads are omitted from the test spectrum if

they do not contribute to crack growth or other

damage. Removal of these loads, which are

often extremely frequent, saves a great deal of

time in testing.

• Loads are introduced to provide “marker

bands” on the surfaces of any cracks present in

the test article. Because these marker cycles are

introduced at known times in the test spectrum

(which correlate to specific numbers of flight

hours), the resulting marker bands help metallurgists analyze the crack growth rate and

determine when cracks reached 0.01′′ in length.

aircraft). An exceedance curve is a cumulative sum of

occurrences reported at several acceleration levels. In

other words, the exceedance curves tell the engineers

how often the aircraft exceeds 2 g’s, 3 g’s, 4 g’s, etc.,

during a certain period of time.

During the aircraft design phase, exceedance curves

are constructed from historical data, including data

that were obtained from accelerometer readings of past

and current aircraft that are flying in the same role

planned for the new aircraft. The design requirements

for the new aircraft include the roles, missions, and

approximate flight time per month, among other

factors.

Military specifications provide guidance for formulating conservative flight-by-flight spectra from Nz data

[1-5]. The current guidance, which is relevant for the

Joint Strike Fighter (JSF) aircraft, is simply that the

test spectrum shall represent the service life and usage

adjusted for historical data, potential weight growth

and future aircraft performance. For aircraft now being

designed, specifications call for 90 percent of the

aircraft in the fleet to meet the service life if they fly

according to the test spectrum. Aircraft designers meet

this guidance by developing a spectrum that represents

at least 90% of the expected fleet usage during the

operational service life.

The ninetieth percentile of a normal distribution is the

mean + 1.32 times the standard deviation. Figure 1

shows the exceedance curves for two operational

aircraft; it also shows the exceedance curve that

matches a test spectrum designed from the fleet

average usage. Aircraft that are now operational, such

as the F/A-18C/D, were tested to a more rigorous

spectrum that represented the mean + 3.0 times the

standard deviation of the usage distribution. The

current joint performance specification is less restrictive due to better notions of aircraft usage.

As mentioned previously, engineers try to include in

the test spectrum loads that represent all of the maneuvers that an aircraft will be expected to make during its

service life. In all full-scale tests, however, the test

spectrum is modified for several practical reasons:

The marker cycles are made by rearranging the

sequence of simulated flights, regrouping sequences of

loads, or substituting cycles into the test spectrum.

Marker cycles are designed so that they neither add nor

retard damage in the test article.

The actual test spectrum is negotiated when the

contract is written to buy the aircraft. For aircraft that

were designed and tested before the F/A-18E/F, the

criteria for test spectra were more severe. This severity

was necessary because there were many unknowns

• Infrequent high loads are omitted from the test

spectrum, because extremely high loads can

introduce residual stresses in metals that can

Fig. 1 - Exceedance curves for two operational aircraft, and an exceedance curve that matches a test

spectrum derived from the fleet average usage EXCEEDANCES/1000 HRS

VERTICAL ACCELERATION, Nz

Test

Mean+1.32*Std. Dev. = 90%

Aircraft 1

Aircraft 2
Volume 50, Number 4 7

about the use and performance of the aircraft design.

The Navy’s newest acquisitions, the F/A-18E/F and

JSF, are incrementally less different in performance

and anticipated usage than the aircraft that they

replace, relative to the differences between F-14 and

F/A-18 A/B/C/D aircraft and the planes that they

replaced.

The Factor of 2

For each critical cracking location in the test article,

the time to crack initiation (i.e., the equivalent flight

time needed for cracks to reach 0.01′′ in length) is

divided by two to obtain the component service life.

For critical locations that have not cracked, the number

of equivalent flight hours at the end of the test is

divided by two. This factor of two is an engineering

factor of uncertainty [1] and is used to calculate the

desired service life by:

 Desired Service Life = factor of uncertainty

time to failure (1)

where service life is a unit of time such as equivalent

flight hours. Factors of uncertainty are used to provide

some degree of assurance that variations in materials,

fabrication, and aircraft usage do not result in the

failure of a structural component. You may ask,

however, “Why the number ‘2’?”

Full-scale fatigue tests take a lot of time to complete

(from two to five years), and therefore they are very

expensive. For this reason, only one test is completed

and the results are assumed to be the median of a

lognormal distribution of fatigue lives, such as the

curve shown in Fig. 2. Next, the acceptable probability

of crack initiation (Pf

) for an aircraft over its lifetime is

required by Navy management to be 0.0013 or less.

This level of lifetime Pf

 is consistent with a Pf of 1 ×

10-7 per flight for an aircraft with average flight

duration of one hour and 10,000 flight hours at retirement. Finally, we assume that the standard normal

deviate, σ, is 0.1, based on multiple tests reported in

References 6 and 7. With these four assumptions in

place, the factor of uncertainty — which has been

reported to be a function of sample size [6] — can be

evaluated according to the following equation:

Factor of Uncertainty = 10(zspec)σ√(1/n+1) , (2)

where σ is 0.1, n is the number of samples, and zspec is

the value from the standard normal table corresponding to the specified Pf

. (For a Pf = 0.0013, zspec = 3.02.)

Figure 3 shows this relationship for several different

combinations of σ and Pf

. The middle curve represents

a Pf = 0.001 and σ = 0.10, which are the values directed by current Navy policy. Notice that the factor of

uncertainty varies from 2.7 for a sample size of 1 (at

the left side of the curve) and that it approaches 2.0 for

a sample size of 10 or greater. The number 2 is selected as the factor of uncertainty, based on the presumption that engineering experience allows us to

assume an infinite sample size.

The top curve in Fig. 3 shows the relationship of factor

of uncertainty and sample size for a slightly larger

standard deviation of σ = 0.12. Note that this slight

increase in standard deviation increases the factor of

uncertainty to 3.2 for a sample size of 1 (and to 2.5 for

Figure 3 - Factor of safety as a function of

sample size

0 1 2 3 4 5 6 7 8 9 10

Sample Size

0

1

2

3

4

Factor of Safety 

Pf = 0.001 σ = 0.10

Pf = 0.001 σ = 0.12

Pf = 0.01 σ = 0.10

Figure 2 - A representative lognormal distribution

of fatigue lives for a fleet of aircraft. (The median

value represents the average fatigue life: Half of

the fatigue lives will be longer, and the other half

will be shorter.)

f(T)

T (hrs) 

Median, p=0.5

p(t≤T)= 0.5
8 Naval Research Reviews

larger sample sizes) to maintain a probability of failure

of 0.001.

The bottom curve in Fig. 3 shows the relationship with

Pf = 0.01 and σ = 0.10. This curve illustrates that

reducing the current factor of uncertainty from 2.0 to

1.8 increases the probability of failure by an order of

magnitude, from 0.001 to 0.01.

The Fatigue Algorithm: Tracking Individual

Aircraft

Very few aircraft will fly missions according to the

service life simulated in the test spectrum. Some

aircraft will fly sets of missions that are more severe

than the test spectrum, while most aircraft will accumulate exceedances at a slower rate. Therefore, the

Navy uses techniques — known as fatigue algorithms

— that can help predict the fatigue damage and crack

formation that may occur in a particular aircraft based

upon the manner in which the aircraft is actually

flown.

Most Navy and Marine Corps fixed-wing aircraft have

equipment that record flight data that indicate aircraft

usage, which is used to calculate crack formation.

Fatigue damage is reported as fatigue life expended

(FLE), which is an index relative to the test flight

hours it took to form 0.01′′ cracks. FLEs are calculated

at 5 to 9 locations for fighter/attack aircraft and at 20

to 30 locations for patrol and support aircraft. FLEs

are used to schedule routine maintenance and structural inspections, life assessment programs, service life

extension programs, and retirements of aircraft from

the active fleet.

fluctuating local plastic deformation in a material to

predict fatigue damage.

Every material has a particular strain-life curve, which

is determined by subjecting smooth, unnotched,

hourglass-shaped specimens of the material to fullreversed constant amplitude loading until they fail.

(Full-reversed, constant amplitude loading means that

the loading cycles between a peak tensile stress and a

peak compressive stress of the same value.) The time

needed to fail the smooth specimens is highly correlated with the time to form a 0.01′′ crack at a fastener

hole under identical loading [8]. The assumption of

equivalency in damage accumulation between the two

cases (i.e., the test article and actual aircraft components) is considered to be slightly conservative and is

therefore useful for calculating fatigue damage and

service life.

Figure 4 shows a typical strain-life curve, which

exhibits two distinct portions: low-cycle fatigue, where

strains are high and fatigue life is relatively short, and

high-cycle fatigue, where strains are low and fatigue

life is relatively long. For most materials, the transition

in behavior between high-cycle and low-cycle fatigue

occurs in the region of the curve between 1,000 and

10,000 cycles to failure. The strain-life curve is

defined by adding the effects of two mechanisms

together. Total strain amplitude, ∆ε/2, is the sum of the

elastic and plastic strains:

 ∆ε

2

 = 

σ ′f

E (2Nf)

b

 + ε′f (2Nf)

c

 (3)

The strain-life approach was first used to predict crack

initiation on the F/A-18 aircraft in the 1980s, and it is

now the standard of all FLE calculations. The strainlife method models the response of a material near

notches or other portions of a component where stress

and strain are elevated. When loads applied to a

structure are low, stresses in the material are elastic

(i.e., stresses and strains are linearly related, or proportional to one another). As load levels rise, however,

stresses near a notch or hole in the structure can easily

exceed the stress that a material can withstand; in such

a case, the material responds plastically to relieve the

local stress field. The strain-life method uses the

where Nf

 is the number of cycles to failure. The terms

b and σ′f are the slope and intercept respectively of the

elastic strain-life relationship, and the terms c and ε′f

are the slope and intercept respectively of the plastic

strain-life relationship, which are obtained when strain

amplitude vs number of cycles to failure are plotted on

a log-log scale, as shown in Fig. 4. Service life is

determined by using the strain-life curve to make

calculations of total strain amplitude. Additional terms

are added to account for cycles of variable amplitude

loading that have mean stresses greater than zero.

The advantage of using the strain-life method over

stress-life methods is the relative simplicity of the

testing required to establish the materials database.
Volume 50, Number 4 9

The strain-life method employs a single strain-life

curve that is used along with the stress-strain curve to

generate predictions for any geometry in the structure.

In contrast, stress-life methods need several stress-life

curves representing a range of stress concentrations

(geometries) present in the structure along with the

stress-strain curve to generate predictions.

But the prime advantage of the strain-life

method is that it can account for localized

plastic behavior of materials near stress

concentrations, whereas the stress-life

method cannot. For these reasons, the

strain-life approach is attractive to the Navy

and manufacturers of aircraft alike.

The major limitation of the strain-life

method is that it is valid only while a crack

is forming in a material. After a crack has

formed, stresses are relieved in the material

through crack-tip plasticity and crack

propagation. At that stage, crack growth

models are needed to predict service life.

conservative stress history assures more efficient use

of service life for individual aircraft. Individual stress

histories are determined from data collected from the

aircraft as they perform their missions. Summary data

are used to schedule structural inspections and/or

modifications, to modify operational usage to extend

service life, and to retire aircraft. Additionally, aircraft

usage data collected now will be used to design new

aircraft in the future.

Fatigue damage accumulation on an airframe (as

measured by Fatigue Life Expended, or FLE) is

reported in a monthly or quarterly SAFE report. The

FLE data is used to project aircraft service life in terms

of flight hours. Type Commanders use the SAFE

report to project aircraft readiness and schedule

maintenance.

Figure 5 summarizes the first two of the four elements

of the FLE tracking process. The four main elements

of tracking are data collection, data reduction, damage

calculation, and information dissemination, which will

be described next.

Data Collection: Data recorders and sensors such as

accelerometers help construct exceedance curves for

the aircraft as it conducts its flight operations. New

aircraft are delivered with manufacturer-supplied

multi-channel data recorders. Older aircraft are being

retrofitted with the Structural Data Recording Set,

SDRS, a generic multi-channel recorder. These

Fig. 5 - The Fatigue Tracking Process: Data are downloaded from the

aircraft and reduced to sequences of stresses at critical locations

0

20

60

-60

-20 Time

S

Stress History at critical locations

SDRS

The Structural Appraisal

of Fatigue Effects Program

The Structural Appraisal of Fatigue Effects (SAFE)

program is the Navy’s tool for individually tracking

critical locations on airframes so that service life is

maximized. Calculating fatigue damage by tracking

aircraft usage rather than using a representative or

Fig. 4 - The strain-life relationship is composed

of the plastic strain-life region plus the elastic

strain-life region and is determined by testing

smooth axial specimens

∆ε

2

2N f

c

b

2N t

σ f′

εf

′ Critical 

Zone
10 Naval Research Reviews

recorders allow sequences of aircraft loading cycles to

be saved in chronological order for input into the

strain-life algorithm. At regular intervals, maintenance

crews perform data recovery from individual aircraft.

Depending on the type of aircraft and recording set,

data storage units fill up weekly. Data are sent monthly

to Naval Air Systems Command, Patuxent River,

Maryland, for processing.

Data Reduction: After the data are subjected to strict

quality control measures, a stress history (such as the

one shown in Fig. 6a) is developed for each monitored

location of aircraft structure. These stress histories are

then used to construct hysteresis loops such as the one

shown in Fig. 6b. Hysteresis loops, which are created

by plotting local stress versus strain, identify damaging

events. The area within a closed hysteresis loop is the

energy per unit volume that is dissipated during a

loading cycle, and it represents a measure of the plastic

deformation work done on the material [6]. After the

ranges and means of the closed hysteresis loops are

determined, equivalent strain amplitude is calculated

and then the strain-life relationship (Figure 6c) is used

to calculate the Fatigue Life Expended.

Damage Calculation: As mentioned earlier, each single

cycle of loading causes a certain amount of fatigue

damage; repeated cycling at that level of loading will

eventually cause the material to fail. Nf

, the number of

cycles needed to cause a material to fail at a particular

level of loading, can be obtained from the strain-life

curve for the material. Correspondingly, the damage

fraction for a single cycle of loading is defined as 1/Nf

.

The incremental damage is calculated by multiplying

each of the damage fractions (1/Nf

’s ) by the number of

loading cycles that have the same equivalent strain

amplitude and summing them, as represented by the

following equation (known as Miner’s Rule):

 Cumulative Damage = n

N

i

fi

nt

i =1

∑ , (4)

where nt is the total number of discrete strain amplitude levels (which correspond to different levels of

loading), ni

 represents the number of cycles that have

occurred at the ith strain amplitude level, and Nfi

represents the number of cycles needed to cause failure

of the material at the ith strain amplitude level. According to Miner’s Rule, when the cumulative damage

equals 1, failure is predicted to have occurred.

In applying Miner’s Rule to naval aviation, failure is

defined as the formation of a 0.01′′ crack, based on data

obtained from the full-scale test article. This linear

damage summation method is widely used in fatigue

algorithms, independent of the sophistication of their

damage equations. Open loops are typically included in

the monthly incremental damage as half the damage of

a closed loop, and are also saved for closing with future

data. The level of residual stress to carryover from

month to month of data receipt is often a topic of

discussion among the engineers responsible for tracking aircraft usage, especially as lapses occur in data

receipt and usage must be assumed.

FLE is calculated by multiplying the cumulative

damage suffered by the aircraft since it began flight

operations by 200%, which is consistent with applying

a factor of uncertainty of 2. (In other words, when an

aircraft component has accumulated fatigue damage

equivalent to the full-scale test article at half of its life,

the FLE has reached 100% and the end of its service

life. As mentioned earlier, the service life of a component is determined by dividing the time to failure — the

time needed to grow a 0.01′′ crack — by 2.) Figure 7

Fig. 6a - Typical stress

history for a critical

structural location

Fig. 6c - The equivalent strain amplitude (obtained

from Equation 3) is used to calculate the Fatigue Life

Expended (FLE)

Fig. 6b - A typical hysteresis loop,

which identifies load cycles that

cause fatigue damage

0

20

60

-20

-60

Time

S

A

F

C,E

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Volume 50, Number 4 11

shows FLE, which represents the percentage of life

used toward failure, plotted versus flight hours.

Aircraft are nominally retired when FLE = 100%.

Information Dissemination: FLE, FLE rates, total

flight hours and the number of aircraft landings are

reported by individual aircraft bureau number and

published monthly on a floppy disk, or quarterly on

paper by aircraft model. Scatter plots of FLE by type,

model, and series of aircraft are used to project maintenance and retirement. These SAFE reports are

provided to the relevant squadron and organizational

level personnel for their planning and fleet management. FLE rates are also used by OPNAV to plan

readiness levels and procurement needs.

These statistics are related to past events, however.

What do they imply about the safety of future flights?

In the current socioeconomic climate, replacement planes are not guaranteed. Legacy

airframes must last 30, 50, or even more years

beyond their FLE limits. As long as an aircraft

is flying within its FLE limits, we know from

the full-scale test experience where to expect

cracking. Inspections are designed for these

specific locations. The probability of crack

initiation is small.

As aircraft fly beyond their limits, however,

we expect that cracks will form at other sites

that were not detected during testing. Consequently, the fidelity of the inspections degrades. To put it simply, it’s infinitely harder

to find cracks if you don’t know where to look. An

inspection technique can be perfect but it will not have

much value if we’re looking in the wrong place.

Service life predictions are complicated by mission

changes for the fleet and by modifications to the power

plant or structure. We need to account for the random

aspects of structural loads in addition to exceeding the

original design life. We need a new tracking approach

to handle intense operational demands in an environment of reduced budgets.

Fig. 7 - FLE is an index that represents the portion of fatigue life

expended, as compared to the fatigue life obtained for the test

article. (When FLE reaches 100%, an aircraft is nominally retired.)

Test Article Actual Flight Hours

FLE

100% Aircraft #1

Aircraft #2

 1/2 Test Life

Each month’s flight data statistics are available at the

end of the next month. Near-term plans include using

secured websites to obtain and disseminate data so that

this turn-around time is reduced.

The Future of Life Prediction for Aircraft

Structural Components

What are the implications of our current approach to

fatigue life estimation? You could say that we are

testing and validating our design every time an aircraft

flies. We’ve had good experiences with our aircraft.

Class A mishaps due to structural failures are extremely rare: A reasonable estimate of occurrence is 1

incident for every 15 million flight hours. The chances

of an aircraft falling out of the sky for structural

reasons is one-tenth the chance that the pilot will be

struck by lightning.

Flying more than the flight hours associated with

100% FLE is venturing into a sky of uncertainty. We

are only beginning to fly aircraft beyond our previous

experiences. The current life prediction approach is not

extensible to FLEs greater than 100%. A current

Science and Technology Initiatives Program (STIP)

that is funded by the Naval Air Systems Command

Science and Technology Office (NAVSTO) is focused

on providing reliabilities resulting from various actions

and allowing the decision maker (operations commander) to heuristically weigh the costs. These

reliabilities go beyond crack initiation; they will be

based on total life. This research and development is a

natural progression in answer to the changing socioeconomic realities.

Normally cracks are modeled as having three distinct

behaviors at three stages of life: nucleation, growth as a

small crack, and growth as a large crack. Of course, the
12 Naval Research Reviews

crack doesn’t know what stage of life it’s in, it simply

responds to the driving forces that are thrust upon it!

Therefore, the Navy’s working definition of total life

will encompass a two-stage model, consisting of crack

initiation and crack growth. Figure 8 compares these

two models of total life.

From our perspective, the Navy has a need to consider

the life of small cracks in the calculation of fatigue life,

because small cracks will develop when aircraft are

required to fly beyond an FLE of 100%. Consequently,

this is an area where the greatest payoff in modeling

can come. Ironically, the Air Force also has a need to

consider the life of small cracks, because that is the

regime that will provide their aircraft with longer

service lives. Both services are working the same

modeling problem but coming at it from the opposite

ends of the life model.

Figure 9 depicts the total life concept. The life of an

aircraft is a random variable: We can not say with

certainty how long an aircraft will fly without suffering

a catastrophic failure due to cracking. Using a probabilistic approach to modeling, we can say with what

reliability life can be obtained. Life is composed of two

stages, as described by our model: crack initiation and

crack growth. The total life probability density function

is a joint probability of times to crack initiation and the

times to reach critical crack size at a critical location.

Concluding Remarks

In this article we have presented an overview of aircraft

fatigue life prediction. We have described our new

focus on a total life approach to predicting life with a

calculated probability of failure. In order to improve

Naval aviation readiness and ensure safety, the following areas of research need to be addressed with funded

programs: small crack modeling, probabilistic methods

and decision theory. Failure to address these areas will

adversely affect readiness and the cost of maintaining

the aircraft in the current inventory. Furthermore,

initiating studies in these areas can possibly lead to

new ways to decrease weight - and therefore decrease

operational costs - in future aircraft.

We hope this article has educated our readers on the

complexity, randomness and uniqueness of naval

aviation and the ways that we currently account for

these uncertainties. Additionally, we hope this article

highlights the need to fund new research that is

directed at formulating a total life approach that will

have the flexibility to quantify safety and readiness for

life extension programs and reduce the cost and weight

of future aircraft.

Acknowledgments

The authors would like to gratefully acknowledge Mr.

Don Polokovics, of the Tactical Strength and Airframe

Technology Branch, for his many valuable discussions

with us; and Mr. Kalmen Leikach, Structures Division

Head, and Mr. Barry Sturgis, of the Aircraft Structural

Life Surveillance Branch, for their continuing support

of our work. Finally, we acknowledge Mr. David Eby

for his thorough historical archive searches and

publication procurement.

Fig. 8 - Two different models of “Total Life”

Fig. 9 - A symbolic representation of the “Total Life”

concept, as applied to a fleet of aircraft

Crack Initiation 

Nucleation

Crack Growth

Small Cracks Large Cracks 3-Stage

Model

2-Stage

Model

Initiation

(hrs.)

Cracking

(hrs.)

Life

(total hrs.)

pc

N.B. P[L<l] = Pf = pi * P[ c < (l -i)|I=i]

I, hrs. C, hrs.

fI(i)

f C

p

(c)

i

L = I + C, hrs. 

Pf fL(l)

l i c

&
Volume 50, Number 4 13

References

1. Joint Service Guide Specification, Aircraft Structures, General

Specifications for, (Draft), ASC/ENFS, Wright-Patterson AFB

OH 45433-6503 or Naval Air Systems Command/4.3.3,

Patuxent River MD 20670-1906, DRAFT 1997.

2. NAVAIR 01-1A-13, Fatigue of Aircraft Structures, Naval Air

Systems Command, Department of the Navy, 1966.

3. MIL-A-8861B, Airplane Strength and Rigidity Flight Loads,

5 Dec 1994.

4. MIL-A-8866C, Airplane Strength and Rigidity, Reliability

Requirements, Repeated Loads, Fatigue, and Damage

Tolerance, 3 Oct 1994.

5. MIL-A-8867C (AS), Airplane Strength and Rigidity Ground

Tests, 3 Oct 1994.

6. A.O. Payne, “Determination of the Fatigue Resistance of

Aircraft Wings by Full-scale Testing,” in Full-scale Testing of

Aircraft Structures, 5-11 June 1959, Amsterdam, Netherlands,

(Pergamon Press, New York NY, 1961), pp. 76-132.

7. E.D. Bouchard, “Evaluation of Fatigue Capabilities on High

Performance Military Aircraft,” (paper No. 24 presented at the

5th Pacific area national meeting of the American Society for

Testing and Materials, Seattle WA, 31 Oct.-5 Nov. 1965).

8. J.A. Bannantine, J.J. Comer, and J.L. Handrock, FundamenThe Authors

Dr. Margery E. Hoffman is a Senior Technical

Investigator in the Aircraft Structural Life Surveillance

Branch of the Structures Division, Naval Air Systems

Command, Patuxent River, MD. She holds B.S. and M.S.

degrees in Structural Engineering from Purdue University,

and a Sc.D. in Structures and Dynamics from George

Washington University. She also has held engineering

research positions at the USAF Wright Laboratories and

the USDA Forest Products Laboratory. Her research

interests are metal fatigue and crack growth, probabilistic

methods, and computational neural networks as

approximators.

Dr. Paul C. Hoffman is a Senior Technical Investigator

in the Tactical Aircraft Strength Branch. He is the

Structures S&T team lead and the Structures/Materials

Aging Aircraft team lead for the Air Vehicle Department,

Naval Air Systems Command, Patuxent River, MD. He

previously held a faculty position at Villanova University.

He holds a B.S. degree in Mathematics, University

of Scranton; a B.C.E. degree and M.C.E. degree, Villanova

University; and a Ph.D. in Structural Engineering from

The Pennsylvania State University. His research interests

include structural reliability, uncertainty factors, structural

repair, corrosion fatigue, and aging aircraft.
14 Naval Research Reviews

What’s

 Eating

 

Us ?

What’s

 Eating

 

Us ?

Corrosio

n

•

Cor or ois n •

Cor

rosion • Corrosio

n

•

Cor or ois n •

Cor

rosion •
Volume 50, Number 4 15

Corrosion!

Corrosion!

Corrosion!

Corrosion!

Corrosion!

Corrosion!

Corrosion!

Corrosion!

utomobile mufflers and old tin cans aren’t the only things that get

rusty. Even high-strength steels, titanium alloys, and other advanced

aerospace materials succumb to the gradual degradation that results

from exposure to the elements.

In the weakened state brought on by corrosion, materials become even

more susceptible to the fatigue damage caused by the repeated cycling of

high stresses. And even though the degradation can be substantial, it often

isn’t easily detectable. As a result, unexpected failures in aircraft components

(such as the in-flight loss of a tail surface from an E-2 aircraft shown in the

photograph at left) can occur.

Most of the damage that comes from corrosion, however, isn’t as dramatic

as pieces falling off of an aircraft. Usually, it’s merely similar to the rust on your

porch swing and your lawn mower. But that doesn’t mean it’s not a problem:

DoD spends nearly $3 billion each year for corrosion repairs and other

maintenance costs for its fleet of aircraft.

In the following article, a scientist in the Aircraft Division at the Naval

Air Warfare Center at Patuxent River, Maryland, discusses some of the

corrosion-related factors that affect the rate of crack growth in aerospace

materials.

– S.P.
16 Naval Research Reviews

Introduction

Unfortunately, many of the high-strength materials used

in load-bearing aircraft structures are also susceptible to

the effects of corrosion. Although materials like 7075-T6

aluminum alloy have higher toughness, they also have

poor resistance to corrosion fatigue and stress corrosion

cracking. Similarly, steels like AISI 4340 or 300M, in

their highest strength temper, are well suited for landing

gear, but these materials exhibit problems in an environment where the effects of corrosion play a dominant

role.

Corrosion and corrosion-related factors undermine the

fatigue and toughness properties of these materials to

such a significant extent that service failures become a

serious concern. For example, corrosion in aging aircraft

can aggravate other problems such as metal fatigue to

the point where service life is reduced. In fact, catastrophic failures of aircraft structure due to corrosion

have occurred. To help prevent such failures, the effects

of corrosion on the fatigue properties of existing aircraft

materials are being reviewed with a renewed perspective. At the same time, the search is on for new materials

that are both strong and tough and resistant to corrosion

fatigue and stress corrosion cracking.

Research into the mechanical, metallurgical, and electrochemical aspects of corrosion fatigue may lead to an indepth understanding that will allow structural analysts to

better assess the remaining life of an aging component.

Sensors that can detect and monitor the initiation and

growth of cracks from sites of corrosion pitting or

crevice corrosion are being developed; if reliable, these

new tools may allow maintenance crews to apply

preventive measures and possibly extend the service life

of affected components.

This paper discusses how various chemical environments interact with in situ properties of a material, as

well as the electrochemical properties of the surface

and tip of a crack, to affect the crack growth rate in the

material [1]. Other topics include the interaction of

corrosion and stress, and how crevices in structures

play a critical role in initiating and accelerating

cracking under low and high amplitude stresses in the

presence of a corrosive environment.

The Concern

The Department of Defense spends nearly $3 billion

each year for corrosion repairs and other maintenance

costs for its fleet of approximately 15,000 aircraft. The

Navy’s share of this annual amount for its fleet of

more than 4,600 aircraft is about $1 billion. Not all of

these maintenance costs are directly associated with

corrosion fatigue, but this phenomenon most certainly

causes a significant portion.

Avoiding corrosion fatigue for any aircraft is a formidable task, but it is especially difficult for those

involved in naval aviation because they operate in one

of the world’s most severe environments. Aircraft

components are exposed to a multitude of corrosive

agents (including sea water and exhaust gases like

sulfur dioxide [SO2], nitrogen oxides [NOX], and

others) under extremely high flight loads and vibrational stresses that often approach and sometimes

exceed the limits that the aircraft’s materials can

handle.

These high-performance naval jets not only experience

damaging stresses when they perform flight maneuvers. Because they operate from aircraft carriers, the

planes also must endure enormous loads and stresses

Vinod S. Agarwala

Naval Air Warfare Center – Aircraft Division

Patuxent River, Maryland

Fatigue and Corrosion:

Aircraft Concerns
Volume 50, Number 4 17

when they are catapulted from the flight deck or when

they land at high speeds and high sink rates to make

sure that their tailhooks engage the carrier’s arresting

cable. In such an operational environment, it is virtually impossible to keep water from intruding into the

aircraft joints, places where hinges overlap, or other

gaps in the structure even when sealant is present. The

most common result of this exposure is a phenomenon

called crevice corrosion and pitting.

Engineers can use detailed analytical techniques (such

as fracture mechanics and finite element analysis) to

design an aircraft component that can operate safely

even though it contains a crack or other defect of a

certain size. However,

these analytical techniques become much less

accurate when the effects

of corrosion are considered, because the precise

effect of corrosion on the

material’s strength and

other properties are

neither easily quantifiable

nor fully understood.

Therefore, the analytical

model that predicts the

rate of fatigue damage in aircraft components fresh off

of the factory floor (i.e., those made of pristine, uncorroded material) is no longer valid for materials that

have been degraded by corrosion.

As discussed below, each of these methodologies used

to design aerospace structures — the “fail-safe”

approach and the “safe crack growth” approach — are

intrinsically inadequate to consider the

effects of corrosion.

“Fail-safe” design: The fail-safe design

approach assumes that multiple, redundant load paths through a component

will reduce the stresses produced by the

applied loads to safe levels (i.e., levels

that produce stress intensities below a

threshold stress intensity [∆Kth] or a

certain value of fracture toughness

[KIC]). This approach to aircraft design

also assumes that the multiple load paths

in a component will provide design

THRESHOLD STRESS INTENSITY

(∆KTH).The value for change in stress

intensity below which, theoretically,

there is no crack growth.

FRACTURE TOUGHNESS. The

parameter that measures a material’s

resistance to fracture (usually

symbolized by “KIC”). When the stress

intensity factor for a component is

equal to the fracture toughness for

the material, the component breaks.

features that arrest a crack before it grows to a length

that will cause the part to catastrophically fail.

The question, then, is whether the ∆Kth and KIC values

assumed for a material are realistic if the effect of

corrosion is included. The KIC value for most materials

depends upon the test conditions under which it was

measured, including factors such as the magnitude of

the stresses, the rate at which the stress cycles occur,

and the environment in which the test was conducted

(temperature, humidity, the presence of corrosive

agents, etc.).

In most cases, a component’s built-in safety margin

allows engineers to develop a conservative estimate of

fatigue life. But as the aircraft ages, the cumulative

damage due to corrosion and plastic deformation

(caused by vibrational stresses and wide spectrum

fatigue) change the material properties, which become

difficult to estimate.

Obviously, analytical models that predict service life

can only be as good as their input data. Therefore,

developing a fair understanding of the effects of corrosion so that we can obtain reasonably good data that can

be applied to an aging aircraft is an important element

in more accurately estimating an aircraft’s fatigue life.

The Aloha Airlines Boeing 737 disaster, where widespread fatigue damage around the fastener holes was

determined to be the cause for failure, is a clear example that fail-safe design criteria do not always offer

protection from catastrophic failure of components.

FRACTURE MECHANICS. The

concepts and equations used to

determine how cracks grow and how

cracks affect the strength of a

structure.

FINITE ELEMENT ANALYSIS. An

analysis technique (usually

computerized) that divides a

complicated structure or component

into smaller, discrete, more

manageable units called finite

elements.

“Slow Crack Growth” design: The U.S. Air Force uses

the slow crack growth design approach for their

aircraft. This analytical approach

presumes that undetectable manufacturing flaws exist in structural

components when the aircraft rolls

off of the production line. However, these flaws are not a matter of

great concern because the aircraft is

designed to ensure that even the

largest undetectable flaw in the

worst possible location will not

grow to critical size during two

lifetimes of service life. For this

reason, the “slow crack growth”
18 Naval Research Reviews

design approach is also known as the “damage tolerance” approach.

The approach depends on the presumption that the

flaws in a structure are smaller than the detectable size

and are therefore not of great concern. What is frequently overlooked is that no inspection can be absolute and deterministic in value, especially for damage in

hidden areas of structure (such as those beneath the

heads of rivets or screws, areas for which there is no

precise minimum detectable flaw size). Therefore, new

and innovative tools and techniques that can rapidly

and effectively detect small cracks that exist in large

areas of structure are needed.

The Navy Approach: The US Navy has adopted the

safe-life design approach for its aircraft. This approach

differs from the damage tolerance approach in that the

aircraft structure is designed to have a low probability

of cracking during the design service life, the so-called

“safe” life. This approach generally accounts for

changes in loads and usage, but makes no

acknowledgement of actual material condition, which

may degrade with time. Large structures will always

have some defects or small flaws that either are introduced during manufacture or develop during operational service. Even if these flaws are not detected due

to limitations in the inspection technique, it is dangerous to conclude that they do not exist.

Another assumption about flaws that should be reconsidered is that small, supposedly subcritical cracks will

not grow. The effects of corrosion can cause even small

cracks under dynamic stress (and even small cracks

under static, or constant, stress) to grow. Therefore, the

relevance of a threshold for cracking under the fail-safe

design approach does not exist. In some instances,

cracks have been observed to grow at stress intensity

factors even below the so-called design KIC values. In

other cases, tests conducted in corrosive environments

under static conditions have found KISCC values that are

lower than those determined for the same material in

non-corrosive environments.

The Role of

Corrosion

A simplistic way to

understand the effects of

corrosion on the low-cycle

fatigue behavior of a high

strength alloy [2] is to

look at Fig. 2, which

shows a plot of crack

growth rate in AISI 300M

steel as a function of the

change in stress intensity

factor (∆K). In the plot,

the rate of crack growth is

displayed along the vertical axis and is expressed in units

of change in crack length per cycle of stress (da/dN).

STRESS INTENSITY FACTOR. A

parameter that mathematically

describes the stresses at a crack tip

in terms of both the magnitude of

the stress applied to a component

and the shape of the component.

Stress intensity factors are usually

symbolized by the letter “K”;

subscripts help distinguish between

different types of stress intensity

factors.

KISSC. The threshold stress intensity

for stress corrosion cracking. If the

stress intensity for a component is

kept below KISSC, then no stress

corrosion cracking should occur.

Fig. 1 - Multiple fatigue cracks found at countersunk

fastener holes during a teardown inspection of a US Air

Force KC-135 tanker aircraft. (The cracks can be seen

extending from each of the circled holes at the 10 o’clock

and 4 o’clock positions.)

The crack growth rate data shown in Fig. 2 were

obtained in tests where the stresses varied between

their maximum and minimum values at a frequency of

0.1Hz (or 1 cycle every 10 seconds), a typical rate for

low-cycle fatigue. Test results show that the crack

growth rates are about an order of magnitude higher

for a wide range of ∆K values when test conditions are

changed from very dry (less than 15% relative humidity, depicted by the open circles) to very moist (MA,

depicted by the open triangles).

As chlorides are added to the moist air (MA + Cl -,

depicted by the X’s in Fig. 2), the crack growth rate

accelerates to a rate several times faster than the crack

growth rate in dry air. Mechanical or stress intensity

aspects of fatigue seem to play a smaller role in this

behavior. In fact, the crack growth rate becomes

almost independent of the stress intensity factor, which

is a clear indication that the effects of corrosion are

beginning to dominate the damage process [2].
Volume 50, Number 4 19

Even for values of stress intensity factor below

30 ksi√in, if moisture is present the fracture mechanics

under pure fatigue behavior seems to be less important. Researchers at Lehigh University have shown that

even at very low vapor pressures of moisture, the crack

growth behavior of aluminum under low-cycle fatigue

is still governed by the fast kinetics of corrosion and

not by pure fatigue [3]. As long as there is some

moisture present, the predominant reaction kinetics at

the bottom of a crack, pit, or other defect is due to

corrosion mechanisms. Other studies of the growth of

short cracks have also indicated that crack velocities

under corrosive conditions are significantly high [4].

Electrochemical Factors: Under cyclic stresses and in

the presence of an ion-conducting liquid, the bottom of

a corrosion pit becomes an anode (i.e., it acquires a

positive charge) and its adjacent side the cathode (i.e., it

acquires a negative charge). Measurements of the

changing potential for a crack growing through a 300M

steel compact tension (CT) specimen in a moist saline

environment show that the electrochemical potential of

the surface of the crack tip can be quite high (nearly

–1.1 volt relative to a Saturated Calomel Electrode

(SCE)). The crack tip can also develop a voltage

difference of up to 300 mV with respect to its adjacent

side.

Figure 3 shows measurements of electrochemical or

electrode potential that were taken with a microprobe

(electrode) located at the bottom of the notch in a CT

specimen. Voltages were measured relative to an SCE,

which served as a reference electrode. As seen in the

figure, the notch-tip (electrochemical) potential drops

very rapidly from approximately –0.1 V (the potential

associated with a passive surface of alloy steel) to

–1.1 V when the tip of the crack reaches the microprobe. This 1.0-V drop in electric potential acts as a

driving force that will further accelerate corrosion in the

material at the crack tip. If described in terms of

galvanic action, this difference in electrochemical

potential is similar to a battery and can produce a

current as high as 10 amperes per square centimeter

(A/cm2) over the exceedingly small area of the crack

tip. Under the influence of such high currents, the

material at the bottom of a notch or the tip of a crack

could dissolve at rates that are orders of magnitude

higher than the rate that a crack would grow under pure

fatigue.

Studies of Agarwala [6], Turnbull [7], Chung and

Macdonald [8], and Pickering [9] were some early

efforts to characterize the crack tip processes during a

transient state; they were also the first studies that

attempted to measure the electrochemical potential of

the crack tip.

Mechanistic studies on fatigue and corrosion have

been very difficult to relate to one other, because most

crack growth rate measurements have been made in a

quasi-transient state condition (i.e., on a time scale that

has extended well beyond the state where transient

phenomena occur, and after an initial stress has been

applied) [5]. Therefore, most analytical prediction

models previously available in the scientific literature

were primarily based on defining the physico-chemical

processes occurring either after the cracking or in a

quasi-steady state condition. These types of models

provide an inaccurate prediction of the remaining

fatigue life of a material.

Fig. 3 - Changes in electric potential as

an advancing fatigue crack passes a

microprobe

Fig. 2 - Effects

of various

environmental

factors on

crack growth

rates during

low-cycle

fatigue
20 Naval Research Reviews

In 7075-T6 aluminum alloy compact tension specimens undergoing corrosion fatigue in a 3.5% sodium

chloride (NaCl) solution, Kendig and Mansfeld [10]

measured a potential that was at least 200 mV more

active than the bulk potential for the material. However, when the crack tip was passivated by an inhibitor

(i.e., when corrosion was reduced to very low rates),

the potential became more positive (called “passive” or

“noble”) as measured with alternating current impedance spectroscopy.

To better understand the effect of corrosion on crack

growth rates for short cracks and for very low values

of stress intensity, crack-tip potentials were monitored

during corrosion fatigue experiments [6,11]. Figure 4

shows a plot of crack growth rate vs. the change in

stress intensity for 300M steel exposed to moist

(relative humidity above 90%), salt water conditions.

Monitoring crack growth rates below 10-6 mm/cycle is

very tedious and time consuming. Nevertheless, the

data show that the rate of crack growth rose very

sharply even for changes in stress intensity below

12 ksi√in, a threshold that is markedly below the

critical flaw size where cracks are not supposed to

grow under pure fatigue. The electric potential measured at the crack tip was approximately –1.1 V vs. an

SCE reference electrode, a value that indicates corrosion is occurring [6]. This data suggests that even short

cracks have a very active crack tip in the presence of a

corrosive environment.

The data in Fig. 4 were obtained in an experiment

where the stress varied at a rate of 1 cycle every

2 seconds. As seen in the upper right portion of the

figure, for large changes in stress intensity in a moist

salty environment, the crack would grow through

300M steel at approximately 45 mm per minute. At

that rate, a crack would take less than 10 hours to grow

about 1 mm. If the conditions were dry, the predicted

time to grow the same distance would have been an

order of magnitude greater under similar fatigue

loading.

The Effect of Chemical Modification on

Crack Growth Rate

Table 1 summarizes the results of experiments of

chemical modification on crack growth rates in 10-in.

three-point loading bend bar specimens made of highstrength 4340 steel. In these studies, the notch area of

the specimen was chemically modified to reflect the

test environment by introducing vapors of various

additives through atomization in a closed chamber

[1,2,12]. The test results show dramatic effects of

chemicals in the specimen’s environment on the rate of

crack growth. In wet conditions, the crack growth rate

increased by nearly a factor of 10 over the rate observed for dry conditions (from 12 to 110 μin/cycle,

respectively). Correspondingly, the fatigue life (i.e.,

the number of cycles need to cause the specimen to

fail) dropped from 17,500 cycles for dry conditions to

1,800 cycles for wet conditions. The presence of

chloride ions further accelerated the rate of crack

growth and reduced the fatigue life of a 1-inch-thick

bend bar to 1,200 cycles.

Rows 4 through 7 of Table 1 show the effects of some

of the individual chemicals in the DNBM formulation

on crack growth rate and fatigue life. The dichromate

is excellent corrosion inhibitor (“passivator”) and

performs exceedingly well in the absence of chloride

(see Row 4). However, dichromate’s passivating

property is compromised when the chloride ion is

present in the environment; the chloride ion breaks

down the passive film as soon as it forms.

Table 1 indicates that chemical formulations such as

DNBM (which contains ions of dichromate [D], nitrite

[N], borate [B], and molybdate [M]) can be used to

effectively retard crack growth. When you compare the

data in Rows 3 and 8 of the table, you see that the

presence of DNBM slowed the crack growth rate in a

wet air and chloride ion environment by a factor of 8.

Fig. 4 - Fatigue crack growth data for 300M

steel in moist, salt water conditions
Volume 50, Number 4 21

Table 1 – Effect of crack-tip chemistry

on low-cycle fatigue behavior of 4340 steel [13]

The test results show high crack growth rates and poor

fatigue lives even when chemicals like borate (a pH

modifier) and nitrite (an oxygen scavenger) were

added (see Row 7). However, when molybdate is

added to the mixture of DNB, the other inhibitors (N

and B) retain their functional properties and the

passive film is reinforced. Therefore, the corrosion

fatigue data for DNBM in the presence of chloride

showed the lowest crack growth rate and the best

fatigue life (see Row 8) for tests conducted in potentially corrosive environments. It must be noted,

however, that the fatigue life obtained by using DNBM

in the wet air and chloride environment does not even

approach that obtained during tests in dry conditions;

indeed, it is less than 40% of the fatigue life in dry-air

conditions. This confirms the hypothesis that as long

as there is some moisture present, the low frequency

fatigue behavior will not show a crack growth rate

equivalent to that obtained for pure fatigue in dry-air

environment, even when stress intensities are very low

or the cracks are short.

material from the corrosion pit (located at the left

center of the picture), grew through the material, and

then continued growing until the component failed.

Therefore, understanding the complex relationship

between corrosion and pitting behavior and fatigue life

is critical. Research that studies the depth, size, and

density of pits that nucleate and grow is relevant to the

fatigue life of materials, because these characteristics

have a significant effect upon the amount of service

life that can be expected from a structure. The frequency with which pitting occurs is also relevant,

because up to 10% of pits can become passive or

dormant and then reactivate. Furthermore, the deepest

corrosion pit is not necessarily the weakest spot in the

structure; under the proper conditions of stress and

chemical environment, any non-critical pit can become

activated, link up with others and grow. Also, it is well

known that the growth rate for corrosion pits can be

several times faster than general corrosion. Therefore,

any prediction of remaining structural life that is based

on the current size and depth of corrosion pits will be

inaccurate and will likely be non-conservative (i.e., an

overestimate). A corrosion pit or short crack that is

now subcritical may not remain so indefinitely; indeed,

it may grow very fast even under little or no stress.

Crack-tip Environment Corrosion

Mechanism/Observations

Crack

Growth

Rate

(μin/cycle)

Stress

Intensity

Factor

(ks√in)

1. Dry air (15% RH)

2. Wet air (100% RH)

3. Wet air + chloride

4. Wet air + dichromate (D)

5. Wet air + borate (B)

6. Wet air + chloride +

molybdate (M)

7. Wet air + chloride +

(D) + (B) + nitrite (N)

8. Wet air + chloride +

DNBM

Almost no corrosion

Severe corrosion

Extremely severe corrosion

Passive film formed at the

crack tip

Adjustment in pH

Some passivation and 

chloride counteraction

Breakdown of passivity

due to chloride

Passivation, pH modification,

chloride counteraction

10

110

200

40

65

60

180

25

70

35

30

50

35

45

34

48

Fatigue

Life

(cycles)

17,500

1,800

1,100

6,800

3,300

4,500

1,200

6,700

Role of Pitting

It is believed that most cracks that lead to corrosion

fatigue failure initiate from portions of the material

where there is pitting or localized corrosion. Figure 5

is the photo of a crack surface in 7075-T6 aluminum

alloy in which the fatigue crack began growing into the

For aging aircraft, one can not simply use the existing

condition of a component (pit size, pit depth, etc.) and

predict its remaining fatigue life. As discussed earlier,
22 Naval Research Reviews

a corrosion pit that is passive or dormant today may

become active tomorrow; if certain chemical ingredients are present, only moisture is needed to activate the

corrosion and further degradation of the material.

8.5 mm in depth. The research further demonstrated

that if the electrochemical potential at the bottom of the

pit increases, the pit growth rate increases and deeper

pits are formed. This phenomenon happens even at

lower stress intensities, where analyses that consider

only pure fatigue may overestimate the remaining life

for a component or material.

For example, a number of F/A-18s deployed on the

USS Midway in the mid-1980s developed corrosion pits

up to 40 mils (750 mm) deep in a 7075-T6 aluminum

longeron after less than 6 months of exposure. In

another example, researchers have produced corrosion

pits up to 80 mm in depth on

a bare sheet of aluminum

(0.08′′ thick, with no

crevices or joints) in the

laboratory environment

by placing the material in

a salt spray chamber [14].

A Possible Solution

The interaction of corrosion and fatigue must be

addressed if we hope to find a solution for widespread

fatigue damage and catastrophic failures in aircraft

structures. We must understand the mechanisms of

crevice corrosion and pitting — as well as the role of

the chemicals in the environment — when the material

is subjected to stress, fretting, and fatigue.

LONGERON.A major load-bearing

structural component that runs along

the length of an aircraft’s fuselage.

Fig. 5 - The fatigue crack that led to the failure of this

component initiated at the active corrosion pit shown at

the left center of the photograph

To be accurate and deterministic, any analytical model

to predict the remaining life of a material under fatigue

must account for the effects of localized corrosion. The

damage should be characterized in terms of pit nucleation and growth, regardless of whether the damage

initiated from existing cracks or flaws in the material,

stress corrosion cracks or crevices, or galvanic corrosion and/or microstructural corrosion. The research of

Macdonald [13] confirms this, and it also suggests that

no crack growth model can be considered deterministic if it does not conform to the laws of physics and

chemistry, including the laws of equilibrium, Faraday’s

Law, the laws of conservation of mass, charge and

momentum, and others. Therefore, accurately determining corrosion behavior (such as the susceptibility

of materials to localized corrosion; the rates of corrosion and pit nucleation; and damage growth parameters (size, density, etc.)) is just as important as an

accurate understanding of fatigue behavior in general.

For aluminum alloys, the rates of pit nucleation and

growth are directly related to electrochemical potential

of the crack or pit surface, the concentration of chloride ion in the environment, and the length of exposure

to the environment [13]. This research also showed

that the pit density grows by a factor of 4 within

5 years, and up to 2% of corrosion pits can be up to

Systematic data on the fatigue life and fracture toughness of materials must be determined in an environment

that is representative of expected operational conditions, not in dry or unspecified conditions. Fatigue data

obtained in a vacuum or in an inert atmosphere can

help provide estimates of the maximum effects that one

can expect due to environment. Even under modest

design engineering limits, a component’s margin of

safety can have limitations, because corrosion can

reduce the actual fatigue life to one-fifth of that reported in the literature or in handbooks.

The rates of crack growth through corroded or

uncorroded areas of a structure must be determined

under realistic conditions. Furthermore, no pit of

corrosion should be assumed to be passive or dormant.

Under certain conditions, any defect can become

active and grow faster than the predicted rate.
Volume 50, Number 4 23

Materials that have a high resistance to localized

corrosion and a greater threshold for corrosion fatigue

and stress corrosion cracking must be used for structural applications. The avoidance of galvanic coupling

(i.e., placing dissimilar metals in contact with each

other) and the use of corrosion-preventive compounds

and sealants in joints and other structural areas susceptible to corrosion will provide greater structural

reliability and will enhance fatigue life. This is especially important for naval applications, because

corrosion-resistant coating systems serve as a barrier

that blocks the entry of moisture and harmful chemicals into hidden areas of aircraft structure. These

measures do not protect an aircraft indefinitely,

however, because coating systems gradually break

down with exposure to vibration, ultraviolet light, and

extremes in temperature. Once developed, intelligent

sensors that can be placed between the metal and its

protective coating will help maintenance crews detect

early signs of corrosion and greatly enhance the

practice of condition-based maintenance.

Concerns about the gradual structural degradation

brought about by corrosion and fatigue require that

aircraft structures be inspected periodically for damage. The wide variety of sizes, shapes, materials, and

levels of access for structural components means that

many different types of non-destructive inspection

(NDI) techniques must be used to locate and characterize damage. When more than one NDI technique is

used to inspect the same area of structure, the probability of detecting damage increases, and the results

provide substantial benefit by increasing the overall

reliability of each set of inspections. NDI techniques

now used with some degree of success include visual

inspections, eddy current, X-ray, ultrasound, neutron

radiography, and thermal imaging.

monitor its own “corrosion health” for hidden corrosion damage. The ability to detect damage early will

be the key to reducing aircraft maintenance and

enhancing the service life of their components.

Concluding Remarks

Even though corrosion can significantly aggravate

fatigue damage — resulting in increased crack growth

rates and reduced fatigue life — interest in the effects

of corrosion and the development of corrosionpreventive technology lags behind the general research

into fatigue, crack growth, and analyses of residual

strength. The influence of corrosion on fatigue life is

complex and is not well understood, because the rates

and the effects of corrosion can vary greatly depending

upon operational and environmental conditions. For

example, the current standard for the allowance of

thickness loss due to corrosion alone used in analytical

models — a value of 10% — is based on tests that

were conducted under static conditions (i.e., unvarying

stress), not under conditions of fatigue (where stresses

vary).

In addition to the mechanical aspects of stress and

fatigue, one must also look at the effects of a material’s

microstructure on corrosion. The corrosion susceptibilities of individual grains and inclusions in a metal,

their compositions, and the active nature of the grain

boundaries have a strong effect on the rate of corrosion; therefore, these properties have a great influence

upon the crack growth rate and the remaining fatigue

life of aircraft components.

The large number of factors that affect the rate of

corrosion — including the characteristics of the

corrosive environment, changes in the rate of cycling

of stresses, the role of corrosion inhibitors on the

crack-tip chemistry, and the influence of electrochemical parameters on pit nucleation and growth — suggest

that no analytical model will be sufficient to accurately

predict fatigue life and address all of the Navy’s

concerns about the structural integrity of the aircraft in

its fleet. Where corrosion is concerned, the present

analytical models will, at best, be non-conservative

and overestimate the fatigue life of aircraft and their

components.

It seems that no crack length is a safe crack length

when it comes to estimating the remaining fatigue life

Although these inspection techniques can be precise,

finding a crack, corrosion, or other defect under two or

three layers of paint or protective coating is a very

time-consuming process. New NDI technologies such

as broad scan visual enhancement, early stage

corrosivity monitoring, guided wave ultrasonic, auto

signal processing, fluorescence imaging spectroscopy,

and superconducting quantum interference devices

(SQUIDs) are making their way into the field. The best

option to increase aircraft and component service life

is to develop a system with intelligent sensors that can
24 Naval Research Reviews

of an aircraft component, unless corrosion is controlled. Research into the growth rates for short cracks

(even those of microstructural defect size) in the

presence of a chemical environment that simulates

actual operational conditions must be pursued. Because very little research of this type has been done, it

is recommended that additional studies that investigate

the effect of corrosion on low-cycle fatigue be conducted.

References

1. V.S. Agarwala and J.J. DeLuccia, Corrosion, 36, 208-212,

1980.

2. V.S. Agarwala, “Multipurpose Corrosion Inhibitors for

Aerospace Alloys in Naval Environments”, in New Materials

and New Processes, Vol. 3 (JEC (Japan) Press Inc., Cleveland,

1985), 178-189.

3. A. Alavi, C.D. Miller, and R.P. Wei, “Electrochemical

Reactions with Bare Metal Surfaces and Corrosion Fatigue

Crack Growth,” Lehigh University Technical Report 19, IFSM

86-143, 1986.

4. R.O. Ritchie and J. Lankford, Small Fatigue Cracks, (The

Metallurgical Society of AIME, Warrendale, PA, 1986).

5. P.S. Pao and R.P. Wei, In Environment Sensitive Fracture of

Engineering Materials, (The Mineral, Metals and Materials

Society, Warrendale, PA, 1979), 566-580.

6. V.S. Agarwala, “An In-Situ Experimental Study of the

Mechanisms of Catastrophic Damage Phenomena, in

Hydrogen Effects on Materials Behavior”, Eds. N.R. Moody

and A.W. Thompson (The Mineral, Metals and Materials

Society, Warrendale, PA, 1990), 1033- 1045.

7. A. Turnbull, Corrosion Science, 23, 833, 1983.

8. H. Chung and D.D. Macdonald, Corrosion, 41, 150, 1985.

9. H.W. Pickering, Corrosion, 42, 125, 1986.

10. M.W. Kendig, F. Mansfeld, and V.S. Agarwala, Evaluation of

Electrochemical Kinetics of the Fatigue Crack Tip of

Aluminum Alloy (The Electrochemical Society, San Diego,

CA, 1986).

11. V.S. Agarwala, “Catastrophic Damage Phenomena: An In-Situ

Crack-Tip Study”, in Proceedings of 6th International

Conference on Mechanical Behavior of Materials, Kyoto,

Japan, Vol.2 (Pergamon Press, 1991), 711-718.

12. V.S. Agarwala, “Modification of Crack-tip Chemistry to

Inhibit Corrosion and Stress Corrosion Cracking in High

Strength Alloys”, in Embrittlement by the Localized Crack

Environment, ed. R.P. Gangloff (The Metallurgical Society of

AIME, Warrendale, PA, 1984), 405-419.

13. D. Macdonald, “The Deterministic Prediction of Corrosion

Damage in Aircraft”, paper given at the Third International

Aircraft Corrosion Workshop, Solomons, MD, August 1998.

14. K.K. Shankaran, R. Perez, R.J. Lederich, et al., “The Effect of

Pitting on the Mechanical Behavior of 7075-T6 Aluminum,”

paper given at the AEROMAT 98 Conference, Abstract

AALE1.2.ASM International, June 1998.

The Author

Dr. Vinod S. Agarwala is a Senior Staff Scientist

in the Aerospace Materials Division of Naval Air

Systems Command, Patuxent River, MD. He was

educated at Massachusetts Institute of Technology and

at Banaras Hindu University in India. He holds a B.Sc.

in Physics; an M.Sc. in Physical and Electrochemistry; an

M.S. in Metallurgy; and a Ph.D. in Metallurgy (Corrosion

Science). He is a Fellow of ASM International and

National Association of Corrosion Engineers (NACE)

International, a registered Professional Engineer, and a

NACE-certified Corrosion Specialist. He has published

more than 120 research papers, edited two books, and

has been awarded 10 patents. Dr. Agarwala’s research

interests include sensors for corrosion detection and

monitoring, “smart” coatings and materials, mechanisms

of corrosion fatigue and inhibition, hydrogen

embrittlement and environment-assisted cracking, alloy

development, electroplating and finishing, and tribology

(wear and lubrication).
Volume 50, Number 4 25

ain makes some things grow and others shrink. In the figurative sense, one

of the things that rain can shrink is the amount of data needed to represent

the sequence of stresses that a component experiences during operation.

In the following article, a scientist from Fairfax Materials Research in

Springfield, Virginia, explains a technique known as the “rainflow” method of

counting stress cycles, which can be used to reduce extremely large stress

histories to a relatively small set of data that can be easily stored and analyzed.

These abbreviated stress sequences also more closely match the type of stress

histories used in crack growth testing in the laboratory.

The article also discusses the development of mathematical techniques

for modeling crack growth that use the basic physics of material behavior to

more closely link the analytical world of cycle counting methods and the real

world of fatigue damage.

– S.P.
26 Naval Research Reviews

Introduction

In the early 1980s, the American Helicopter Society

conducted a bold experiment in which hypothetical

load sequences for a pitch link — a flight-critical

component in a helicopter’s rotor system — were

provided to seven major helicopter manufacturers. The

manufacturers were expected to use their preferred inhouse methods to analytically determine the amount of

the pitch link’s fatigue life that was expended during

the load sequence. The manufacturer’s estimates varied

by more than three orders of magnitude, from 9 hours

to 28,000 hours. A statistical analysis of these results

[1] showed that 85% of the variance in the estimates

was attributable to just two sources: the amount of

safety factor used in the analysis, and the method used

to count the cycles of stress in the load sequence.

The various methods used to count cycles in load or

stress histories have evolved from two basic imperatives.

1. First of all, a variable amplitude, cyclic loading

sequence must be converted into a form that

can be analytically compared to data obtained

from constant amplitude fatigue tests, which is

the only form in which laboratory results are

widely available.

2. The second need — which, by the way, is

becoming increasingly unimportant as the cost

of storing computer data drops — is to distill a

general load sequence into a form that permits

the compact storage of information without the

loss of its relevant physical characteristics.

An unprocessed stress history includes a large number

of data points that provide a comprehensive record of

Cycle Counting and Beyond

in Variable Amplitude Fatigue

Michael S. Duesbery

Fairfax Materials Research, Inc., Springfield, Virginia 22153

stress at some location of interest in an aircraft or other

structure. If stress data is sampled every few seconds

(as it often is in flight test programs that investigate

fatigue crack growth in components that experience a

rapidly changing stress environment), the resulting

stress histories can quickly become very large. For

example, a mere 100 hours of flight data would yield a

stress history that contained 36,000 data points if

measurements were taken once every 10 seconds — a

rate that is not uncommon.

Many cycle counting techniques have been developed

over the years and are thoroughly discussed in several

excellent reviews [2-4]. The technique that has come

to be accepted as the modern standard — in large part

because it implicitly includes the effects of many of

the competing methods of cycle counting — is known

as the “rainflow” method.

Background

A simplified visual depiction of the rainflow method

of counting and reducing the number of cycles in a

stress history is presented in the sidebar at right. This

technique also can be described in procedural and

mathematical terms as follows [5]:

1) A sequential set of maximum and minimum

stresses are extracted from the full stress history

by discarding all data points that lie within

monotonically increasing or decreasing ramps.

The remaining “peaks” and “valleys” of stress

are then digitized to form an abbreviated stress

history.

2) Data points in the abbreviated stress history are

then considered sequentially, in sets of four.
Volume 50, Number 4 27

load

cycles

i+2

i+1

i+3 i

i+2

i+1

i+3

i

load

cycles

i+2

i+1

i+3 i

i+2

i+1

i+3

i

load

cycles

i+1

i+3 i

i+2

i+3

i

load

cycles

i+3 i

i+3

i

The Rainflow Method

of Counting Stress Cycles

The “rainflow” method of counting and reducing

stress cycles takes its name from a visual representation of the process, which is shown at left.

The top diagram (which also appears as Fig. 1 in

the main article) depicts two sets of four stress

cycles. The set shown on the left represents a trend

of generally increasing stress, with a small intermediate drop in stress; the one on the right represents

a trend of generally decreasing stress, with a small

intermediate rise in stress.

If you imagine these diagrams to be the crosssectional, side view of a roof, you can see that the

intermediate peaks in the roof form “gutters” that

will collect rain. When the gutters become completely full of rainwater, the roof will appear as

shown in the second diagram. Any additional rainfall that strikes the upper portion of the roof will

spill into the full gutter and then immediately continue to flow downward as if the gutter weren’t

there.

Now, let’s move from the “roof” analogy back to

our sequence of loads and stresses. If we neglect

the gutter shown in the second diagram, the sequence of four stresses then becomes a singledirection change in stress level that is interrupted

by a short pause at some intermediate level of

stress, as shown in the third diagram.

Because short pauses in stress should have little if

any effect on the rate of fatigue crack growth, these

pauses can be neglected. Therefore, the sequences

of four stresses shown in the top diagram can be

replaced by the simple changes in stress shown in

the bottom diagram.

– S.P.
28 Naval Research Reviews

(Figure 1 shows two examples of variations in

stress for load points i, i+1, i+2, and i+3. Note

that for any particular component, the stress in

the component is directly proportional to the

applied load.)

If conditions on the stress values for the sets of four

load points shown in Fig. 1 satisfy the criteria:

 σσ σσ ii ii ++ ++ 21 32 −≤−

σσ σσ i i ii ++ + 21 1 −≤−

 (1)

(in other words, if the subcycles defined by the load

points i+1 and i+2 [which also can be represented as

subcycles A and B in the load-strain diagram shown in

Fig. 2] are completely contained within the stress

cycles defined by the extrema load points i and i+3),

as is true for both sequences of four load points shown

in Fig. 1, then the range of stress for the intermediate

load points (i.e., ∆σ = σi+2 - σi+1) and the mean stress

(i.e., σmean = 0.5(σi+2 + σi+1)) are recorded in a density

matrix n. Information is stored in the density matrix

(where each element takes the form of njk) according

to the following pattern:

∆σ σ σ k = − i i + + 2 1

σ σσ j ii = + + + 0 5 2 1 . *( )

 (2)

After all of the load points in the stress history have

been processed using this technique, the result can be

expressed as the density matrix n of counted subcycles

with recorded mean stress and stress range and a

Fig. 1 - Four-point subsections of an

abbreviated stress history. (Generally

increasing stress [at left] and generally

decreasing stress [at right].)

load

cycles

i+2

i+1

i+3 i

i+2

i+1

i+3

i

residual load sequence with few features. To complete

the cycle counting exercise, the residual sequence is

duplicated, joined to itself and reprocessed as above

[6].

The accumulation of fatigue damage

By the Palmgren-Miner rule [7], the fatigue damage

Dij that is caused by the matrix element nij is

D

n

N ij

ij

ij

= (3)

where Nij is the total number of fatigue cycles to

failure for a mean stress of σi

 and stress range ∆σj. The

parameter Nij can be estimated from N0j, the number of

cycles to failure at zero mean stress for the stress range

∆σj, by means of the relation

N N ij j

i

a

b = − 0 ( ) 1 σ

σ (4)

in which σa and b are constants that are determined

from the stress-strain curve for the material. Finally,

the fraction of lifetime expended by the load sequence

(also known as Fatigue Life Expended, or FLE) is

given by

f Dij

j

m

i

m

= − −

∑∑1 1

 (5)

where the damage Dij is found using Eq. (3).

Fig. 2 - Load-strain cycles for load

histories shown in Fig. 1. (Generally

increasing stress is shown by the

upper curve; Subcycle A represents

intermediate charges in stress from

load point i+1 to load point i+2.)

The power of the rainflow method is that the original

load sequence can be reproduced in terms of the
Volume 50, Number 4 29

number and magnitude of subcycles by inversion of

the reduction procedure. The rainflow method of

counting stress cycles is complete, because the decomposition matrix of the reconstructed load sequence is

the same as that of the original load sequence. However, the sequential order of the subcycles removed

from the original load sequence cannot be recovered;

therefore, any time correlation in the loading history is

lost.

In effect, fatigue life analyses that use such a reconstructed stress history ignore load-load interactions.

Lifetime predictions that use stress histories reduced

by the rainflow method tend to be conservative [7]. In

other words, these calculations overestimate the

accumulation of fatigue damage in a component and

therefore underestimate its fatigue life.

A Different Way to Calculate Fatigue

Damage

There are several drawbacks with using the PalmgrenMiner rule, discussed in the previous section, to

predict the accumulation of fatigue damage.

Equation 3 assumes that damage increases linearly as

the number of cycles increases, and Eq. 5 assumes that

this damage combines linearly to predict the Fatigue

Lifetime Expended. This is clearly not true; for any

particular change in stress (and with all other factors

being equal), the rate of growth of a long crack is

faster than that for a short crack. Perhaps most importantly, however, the Palmgren-Miner rule includes no

tangible damage parameter. Therefore, the accuracy of

any predictions using this technique can only be

verified when the component ultimately fails.

tion in the Palmgren-Miner rule can be replaced by

the change in stress intensity, which is more closely

related to the rate of crack growth. Incorporating these

two fundamental changes of philosophy into the

techniques for calculating fatigue crack growth require

us to modify both our cycle counting procedure (Eq.

(2)) and our method for calculating the accumulation

of fatigue damage (Eqs. 3 through 5).

Changes in the cycle counting technique will be

addressed first. Figure 3 illustrates the conventional

definition of the driving factors for crack growth

according to the Palmgren-Miner rule — the average

stress and the change in stress — as they were defined

in Eq. (2). [Note, however, that the stress values in

those equations has been replaced by the stress intensity at the crack tip (K).] For subcycles in both increasing and decreasing loading ramps (labeled A and B,

respectively, in Fig. 3), the change in stress intensity

(∆K) is defined by the amplitude of the subcycle, and

the magnitude of the stress intensity K is determined

from the average stress value for the subcycle.

Fig. 3 - Conventional definition of stress

intensity factor (K) and change in stress

intensity factor (∆K)

From the physics of fracture, however, it is known that

Kmax and ∆K play well-defined roles in fatigue crack

propagation. Kmax is a parameter that governs the ease

of emission of dislocations from the crack tip; it is

therefore related to the strength of the interatomic

bonds at the crack tip and is sensitive to corrosion.

Because it is localized to the crack tip, Kmax is unaffected by details of the microstructure of the material.

The parameter ∆K, on the other hand, governs the

behavior of the plastic zone — a region of stressdeformed material that surrounds the crack tip and

extends some distance from it. The plastic zone

usually contains paired dislocation dipoles, which tend

to harden the material without shielding the crack.

However, when the applied Kmax exceeds a certain

Any method of counting cycles — whether it be the

rainflow method or a competing technique — is

completely empirical, and both its performance and its

justification can be improved by considering the basic

physics of fatigue crack propagation. There are two

obvious changes that can reinforce the connection

between the analytical world of cycle counting methods and the real world of fatigue damage accumulation. First, the length of a fatigue crack can be adopted

as the damage parameter that is missing from the

fatigue life calculations that are based on the

Palmgren-Miner rule. Second, changes in load or

stress that serve as the driving force for crack propaga-
30 Naval Research Reviews

threshold value for the material, unpaired positive

dislocations are emitted from the crack tip into the

plastic zone. Conversely, a reduction in stress intensity

of ∆K permits the plastic zone to restore its paired

condition, either by emission of negative dislocations

from the crack tip or the absorption of positive dislocations into the crack. The degree of restoration possible

depends on the magnitude of the drop in ∆K and also

on the microstructure of the material (e.g. the hardening, grain size, second phase content).

This clarification of the physical roles of Kmax and ∆K

enables a more logically intuitive decomposition of the

variable load spectrum. Figure 4 shows the same stress

history and subcycle (B) that were shown in Fig. 3, but

the new definitions of stress intensity parameters are

depicted. In this case, Kmax is defined naturally as the

value of stress intensity that is associated with the peak

stress, rather than the value associated with the mean

stress for the cycle. The parameter ∆K is shown as the

range of stress intensity values associated with the

unloading stroke after the peak of the cycle. The effect

of these changes in definitions for a simple rainflow

decomposition will be to leave ∆K unchanged for

subcycles on a loading ramp (such as subcycle A in

Figs. 3 and 4). For subcycles on an unloading ramp

(such as subcycle B in Figs. 3 and 4), the value of ∆K

will be increased. For both the loading and unloading

types of subcycle, Kmax will be increased slightly, from

the mean to the peak value.

link between cyclic stress and the resulting damage

due to fatigue crack growth.

The rate at which the fatigue crack extends for each

load cycle (da/dn) is usually expressed in the form

da

dn = F KK [ ] ∆ p E , ,, max β β (6)

where F is a mathematical function that depends upon

the parameters enclosed by the brackets. For Eq. (6),

βP and βE are parameters that describe the effect of the

plastic zone and the chemical environment of the crack

tip, respectively. If the detailed dependence of crack

growth on these parameters is known, then the calculation of accumulated fatigue damage is a matter of

numerical integration rather than simply counting the

cycles in the load spectrum.

Because this type of damage calculation does not

necessarily condense stress histories into an abbreviated form (as is done for many classical techniques of

counting cycles of stress), time correlations are

preserved and load-load interactions can be retained in

the damage estimate. The first attempt to quantify

fatigue damage using an equation of the form shown in

Eq. (6) was made by Paris and Erdogan [8], who

suggested a form dependent on ∆K only

da

dn

const K p = .( ) ∆ (7)

where the exponent p is empirically determined from

experimental data. It is now known that the rate of

crack extension depends on Kmax as well as ∆K; this

dependence is usually introduced into the crack growth

calculations either explicitly or by a dependence on R,

the stress ratio (R = Kmin/Kmax).

Fig. 4 - New definition of maximum

stress intensity factor (Kmax) and change

in stress intensity factor (∆K)

As previously mentioned, the prediction of accumulated fatigue damage according to the Palmgren-Miner

rule involves both the linear scaling of damage (Eq.

(3)) and the linear combination of damage (Eqs. 4,5).

Both of these dubious assumptions can be avoided if

we adopt the fatigue crack length a as the damage

parameter and then use fracture mechanics to form the

To account for load-load interactions, it would normally be necessary for crack growth to be dependent

upon both Kmax and ∆K. This can be avoided, however,

by including a dependence on the plastic zone (via

which the loads are time-correlated) by way of the

parameter βP, although it is by no means clear how this

should be done. Lastly, the effect of the environment

on crack growth rate is acknowledged by including a

parameter βE; the detailed nature of this dependence is

similarly unknown but is expected primarily to take the

form of a modification to the Kmax dependence.
Volume 50, Number 4 31

When all these dependences are accounted for, the

right-hand side of Eq. (6) becomes a state function. In

other words, if the values and relative influences of the

parameters enclosed in the brackets on the right side of

the equation remain the same, then the resulting crack

propagation rate will always be predictable. Therefore,

it is not necessary to know the full load history of a

component or test specimen in order to predict its

fatigue life, provided that all the parameters included

on the right side of Eq. (6) are known or can be measured.

The effects of most or all of the parameters in Eq. (6)

are included in the crack growth calculations performed by several advanced computer programs [9].

Figure 5 shows crack growth data collected from a test

specimen made of 7075-T651 aluminum alloy that was

subjected to a loading spectrum from a fighter aircraft

in laboratory air conditions. The solid line, which

closely follows the more conservative of the two sets of

data, is the prediction of the US Air Force AFGROW

computer program [10]. While AFGROW does not

show the same high performance in every case, this

example shows that in some cases the combination of

cycle counting and fracture mechanics can provide an

excellent capability for predicting fatigue life.

2. C. Amzallag, J.P. Gerey, J.L. Robert, and J. Bahuaud,

“Standardization of the Rainflow Counting Method for

Fatigue Analysis,” Fatigue, 16, 287 (1994).

3. R.K. Holman and P.K. Liaw, 1997, “Methodologies for

Predicting Fatigue Life,” J. of Metals, 49, 46 (1997).

4. S. Suresh, Fatigue of Materials (The University Press,

Cambridge, 1998), pp. 264-267.

5. A.K. Khosrovaneh and N.E. Dowling, “Fatigue Loading

History Reconstruction based on the Rainflow Technique,”

Int. J. Fatigue, 12, 99 (1990).

6. K. Sobczyk and B.F. Spencer, Jr., Random Fatigue - from

Data to Theory (Academic Press, New York, 1992), pp. 104-

107

7. N.E. Dowling, Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture and Fatigue (PrenticeHall, Englewood Cliffs, NJ, 1993).

8. P.C. Paris and F. Erdogan, “A Critical Analysis of Crack

Propagation Laws,” J. of Basic Engrg., 85, 528 (1963).

9. K. Walker, “ An Evaluation of Empirical and Analytical

Models for Predicting Fatigue Crack Propagation Load

Interaction Effects” (Paper presented at USAF Structural

Integrity Program Conference, San Antonio, Texas, 2-4 Dec.

1997).

10. J. Harter, (1998). AFGROW, http://fibec.flight.wpafb.af.mil/

fibec/afgrow.html.

The Author

Dr. Michael S. Duesbery is Research Director

at Fairfax Materials Research in Springfield,

Virginia. He was educated at Cambridge

University and holds M.A., Ph.D., and Sc.D.

degrees in Physics. Before joining Fairfax

Materials Research in 1990, he was employed

at Oxford University, the National Research

Council of Canada, and the Naval Research

Laboratory in Washington, DC. Dr. Duesbery

has published more than 80 papers on the

physical properties of materials and was

awarded the annual medal of the Canadian

Metal Physics Conference in 1986. In addition

to his work at Fairfax Materials Research, he

co-edits the Elsevier series of monographs

“Dislocations in Solids” with F.R.N. Nabarro.

His current research interests center on

computer modeling of the deformation, fatigue

and fracture of metals.

Fig. 5 - Plots of actual crack growth data in test specimens (dots) and the prediction of the AFGROW

computer program (solid line)

References

1. C.T. Gunsallus, E. Nagy, P.G. Stennett, and W.G. Flannelly,

“Investigation of Variation in Fatigue Life Calculated using

Damage Fraction,” J. Amer. Helicopter Soc., May, 35 (1990).
32 Naval Research Reviews

a helicopter with

odern aircraft are literally bristling with antennas, sensors, and

instruments. Some of these sensors provide the aircrew with

airspeed, altitude, and other vital real-time information. Other

sensors, such as accelerometers and strain gauges, enable post-flight

analysis and tracking of the strains and stresses that the aircraft

experiences during flight maneuvers.

Some parts of the aircraft, however, are difficult  if not

impossible  to instrument. This is especially true for components

that are embedded in engines or other rotating systems.

Nevertheless, the stresses and strains in such components are of

immense interest because these moving parts can be a significant

source of maintenance costs for the aircraft.
Volume 50, Number 4 33

esearchers at the Carderock Division of the Naval Surface Warfare

Center in Bethesda, Maryland, have designed a system for the SH-60

helicopter that uses easily measured flight parameters to predict the

loads in critical life-limited components in the helicopter’s rotor system.

In the following article the scientists describe the system, which acts as

a “virtual” strain gauge that, in essence, gives the helicopter a sensor

where none actually exists.

This extra “sense” isn’t a crystal ball, but it could be almost as

good as one. These load predictions could help SH-60 maintenance

personnel more effectively schedule their aircraft inspections and

component replacements, thereby avoiding costs associated with the

retirement of a component before it actually reaches the end of its

service life. The technology can also be used as a real-time warning

system that tells pilots when damaging loads occur in the rotor, thereby

helping to conserve the useful life of critical components.

– S.P.
34 Naval Research Reviews

STRAIN GAUGE. An instrument

used to measure the small deformations of components or test

specimens that are caused by tension,

compression, bending, or twisting.

Strain gauges are usually adhesively

bonded to the specimen at or near a

location of interest to structural

analysts.

ALGORITHM.A step-by-step, often

computerized procedure for solving

a mathematical problem.

David J. Haas, Kelly M. McCool, and Lance A. Flitter

Carderock Division – Naval Surface Warfare Center

Bethesda, Maryland

Neural Network Based Virtual

Strain Gauges for Monitoring

Fatigue Critical Loads

Abstract

A neural network approach is utilized to develop

“virtual” strain gauges to monitor rotor system vibratory loads on a Navy SH-60 helicopter. The neural

network model is based on input parameters that can

be readily measured and recorded in the fuselage using

a flight data recorder. The

neural network was

developed using flight test

data from a highly instrumented SH-60B aircraft

which included a slipring

assembly and strain

gauges mounted in the

rotating system. The strain

gauge measurements

provide a reference

vibratory load for training

and validation of the

neural network. Two types

of neural network algorithms are used, a self organizing feature map algorithm for selection of training data

and a modular neural network algorithm for load

predictions. The virtual strain gauges were demonstrated in a flight test evaluation on a second SH-60

aircraft where the algorithms were embedded in an onboard processor for real-time load predictions. Assessment of the flight test results show that the predicted

loads compare quite well with strain gauge measurements on the two separate flight test aircraft. The

results also demonstrate the potential to use the

algorithms for real-time pilot cueing to minimize

operation in fatigue damaging flight conditions.

Introduction

In an effort to improve the reliability and reduce

operating costs of helicopters, an increasing emphasis

is being placed on developing systems that can monitor the severity of aircraft usage and the health of

various subsystems [1,2]. The rotor system, in particular, is a significant source of helicopter maintenance and operational costs. During flight, the

rotor blades encounter complex and variable

aerodynamic loads that subject flight-critical

components to severe vibratory loads. These

loads cause fatigue damage that can accumulate

and ultimately result in the catastrophic failure of

a structural component if the damage remains

undetected.

The rate of accumulation of fatigue damage in

the structure varies according to the magnitude of

the stresses and the frequency of the cycling,

which ultimately depends upon the types of

maneuvers flown. Accurate knowledge of the component loads is the key to making valid predictions of

component service life. Due to the challenges of

instrumenting the components of a rotating system,

however, there is currently no practical way to measure

in-flight rotor system loads on fielded aircraft. Therefore, calculations of accumulated fatigue damage are

typically based on data that is more easily obtained,

such as the number of flight hours logged and an

assumed severity of flight usage. This assumed usage

spectrum must be quite conservative in order to

represent (and therefore protect) the most severely

flown helicopter in the fleet. As a result, some aircraft
Volume 50, Number 4 35

Fig. 1 - Schematic of a feed forward

neural network

INPUT

LAYER

HIDDEN

LAYER

OUTPUT

LAYER

components on helicopters with less severe flight

usage are needlessly retired before they reach the end

of their useful life.

The objective of this work is to develop and demonstrate a method of using an artificial neural network to

monitor dynamic loads in fatigue-critical rotor system

components. Such an approach would use only

parameters that are readily measured in the helicopter

fuselage, and therefore the need to directly instrument

components of the rotor system would be eliminated.

Approach

Background: An artificial neural network (ANN) is a

method of computing that is inspired by the structure

and function of the brain and nervous system, which

is made up of large numbers of individual neurons

(processing elements) that, in concert, can evaluate

inputs and make decisions. ANNs use sets of complex

equations with nonlinear transfer functions to solve

difficult and seemingly intractable problems. (In

engineering terms, they can “map” a given set of

inputs to a set of outputs.)

network. The hidden layer is composed of individual

processing elements, which are represented as circles.

For each set of inputs, the ANN uses the connection

weights, node biases, and the transfer functions for

each node in the network to calculate a set of outputs.

At the same time, the ANN calculates a “prediction

error,” which represents a measure of the difference

between the predicted outputs (from the model that is

built into the network) and the desired outputs (as they

are observed in the real world).

The most distinguishing feature of a neural network is

its ability to “learn.” As an ANN is exposed to more

and more data, it adjusts the connection weights

between the nodes in the network. When the changes

in these connection weights stabilize, the ANN is

considered to be “trained”; at that point, the values for

the connection weights are fixed (i.e., they are no

longer allowed to fluctuate). Once a network has been

successfully trained, it is capable of analyzing sets of

data that are somewhat different from those it was

originally exposed to during its training sessions. In

other words, the ANN is capable of generalization.

Furthermore, neural networks are often capable of

producing desired responses even if they are presented

with “noisy” or confusing data.

An ANN consists of a set of processing elements

(“nodes”) that are connected by links. Each processing

element is linked to at least one other processing

element and is usually linked to several others in the

network. Each of these links has an associated

“weight,” which represents the importance of that link

in computing the output of the network. The outputs of

one set of processing elements may become the inputs

to the processing elements in the next “layer” of the

network. In addition to a set of inputs, processing

elements frequently have a special input called a

“bias” that is set to a constant value (which is usually a

value of 1). A processing element’s contribution to the

final output of the network is usually calculated by

summing the bias and inputs to the processing element, modified by their associated weights, and then

sending the sum through a transfer function to obtain

the output for the processing element [3].

Neural networks vary in physical design, training

model, and functionality [4]. A feed forward neural

network consists of an input layer, one or more hidden

layers, and an output layer. Figure 1 shows a schematic

of a simple three-input, two-output feed forward neural

There are two basic methods for training ANNs:

supervised and unsupervised.

• In supervised training, the neural network is

presented with paired sets of stimuli and
36 Naval Research Reviews

1

2

3

4

5

6

7

8

9

10

11

12

Airspeed (kn)

Main rotor speed (%)

Engine torque (%)

Load factor (g’s)

Pitch attitude (deg)

Roll attitude (deg)

Lateral stick position (%)

Longitudinal stick position (%)

Collective stick position (%)

Pitch rate (deg/sec)

Rate of climb (ft/min)

Gross weight (lb)

INPUT PARAMETER

Table 1 - Neural Network Inputs

responses (inputs and outputs). The network

then attempts to determine the relationship

between the inputs and outputs so that it can

reproduce the appropriate response (output)

when it is presented with a specific stimulus

(input). During the learning phase, the prediction error is communicated through the network

in some manner and is used to adjust the

network weights such that the prediction error

is minimized.

• In unsupervised training, the neural network is

exposed only to stimuli, and it organizes itself

based on characteristics of the input data. The

output of such networks is usually some

categorization of the input space.

A key advantage of ANNs over traditional statistical

methods is that they provide a straightforward means

to include nonlinear effects without having to determine the specific relationships between inputs and

outputs.

Neural Network Development: For this research, the

neural network-based load prediction algorithms were

developed using data collected from an SH-60B

helicopter during a flight loads survey test program.

The helicopter was instrumented with strain gauges

and a slipring assembly to measure loads in the rotor

system. During the flight test program, which included

56 test flights, the helicopter was flown through a wide

variety of maneuvers at several different combinations

of altitude, gross weight, and weight distribution.

Flight data were recorded throughout these 56 flights

and sampled at 8 times each second. The nearly

9 hours of data obtained during this flight loads survey

yielded 255,401 data points, which were then used to

train and validate the neural network.

sets of inputs may occur much less often than others.

In such a case, a random selection of training data

points may result in the inadvertent exclusion of data

that occur infrequently. To ensure that a training data

set fully represents the entire population of possible

inputs (i.e., to ensure that it also includes data that

occur infrequently), a neural network called a SelfOrganizing Feature Map (SOFM) is utilized.

The SOFM is a neural network algorithm that uses

unsupervised learning to develop a map of the input

space. In general, a SOFM merely changes the format

of the sets of inputs. For this research, the SOFM

transformed a 12-dimensional input space (see Table 1

for the 12 inputs measured during the flight loads

survey) into a more manageable 2-dimensional grid

that measured 50 nodes by 50 nodes. Each of the

2,500 nodes in this grid served as a “bin” into which

data could be placed. Each set of inputs collected from

the flight loads survey (more than 255,000 data points

total) was then placed in one of the 2,500 bins, which

were characterized by the 12 measured parameters.

Several data points from each bin were then selected as

training data for the neural network.

Another example of how a SOFM’s “binning” of data

can be useful, which may be easier to understand, is

related to the US postal system. Suppose we wanted to

do a survey of representative letters and packages that

flow through the mail system, and we want to make

sure that a sample of mail is included from all parts of

One of the keys to successfully training a neural

network is the proper selection of a training data set.

The data set used to train the ANN must include data

that are representative of the problem domain. In other

words, the ANN must be trained through exposure to a

wide range of inputs that will be similar to those that

the network will actually encounter while in service.

Great care must be used to select a set of training data

because, in the entire population of data points, some
Volume 50, Number 4 37

the country. We know that a lot of mail goes to and

from New York City (because a lot of people live

there), and quite a bit less goes to relatively rural parts

of the country such as northern Maine and western

Wyoming. In fact, a random survey of 10,000 pieces of

mail may not find many (if any) letters or packages

that are destined for northern Maine. However, if we

sort a very large amount of mail by ZIP code groups

and then take an equal number from each group, we

can ensure that we include examples of mail from ZIP

codes that begin with “047” (northern Maine), as well

as other portions of the country that typically have a

much lower volume of mail.

For the SH-60 helicopter used in our research — and

for helicopters in general — the population of loads

data primarily consists of low loads, because a significant amount of a helicopter’s flight time is spent in

relatively steady flight maneuvers, such as straightand-level flight or hovering. A random selection of

data points from this population would therefore be

skewed toward sets of inputs that contained low loads.

However, data sets that contained high loads were of

the most interest, because high loads result in more

fatigue damage in the components of the rotor system.

To increase the importance of high loads during the

training of the ANN, the training data that contained

high loads were weighted. Using this approach, only

7,500 data points — less than 3% of the baseline flight

data collected in the flight loads survey — were

required to train the neural network.

(Fig. 2). During training, the expert networks learn

how to predict the proper response to inputs, and the

gating layer learns how to partition the data among the

experts.

The MNN used in this research has 3 expert networks

(each with 65 hidden nodes) and a gating network with

9 hidden nodes. The network uses 12 input parameters

(previously listed in Table 1) that were measured by a

flight data recorder unit to predict rotor system component loads.

Network Validation on Baseline Data: After the neural

network was sufficiently trained using the baseline

data, the connection weights were fixed and the

performance of the network was assessed with validation data. Validation data are distinct from training data

and include data to which the network has not previously been exposed. As previously discussed, only

7,500 of the 255,401 baseline data points were used to

train the network used for this research. After training

was complete, the network’s load monitoring algorithms were validated using the remaining 247,901 data

points obtained during the first phase of flight testing.

Because the training data makes up such a small

portion of the overall data (less than 3%), results are

presented for the entire baseline data set.

Table 2 shows the overall statistical performance of the

MNN for load predictions in two components of the

helicopter system: the main rotor pushrod (which

controls the pitch of each rotor blade), and the main

rotor lead-lag damper (which stabilizes and damps the

in-plane motion of the rotor blade). Linear correlation,

R, is displayed in the center column of Table 2. The

value of R indicates the degree of linearity in the

This research evaluated several different neural

network designs for load prediction, including simple

feed forward networks and modular networks. Results

showed that a modular neural network (MNN) was the

best predictor of rotor system loads because it produced a lower average root mean square error and a

higher average correlation. A modular neural network

is a “collection-of-experts” style network that uses

supervised learning. In a collection-of-experts network, several neural networks function in concert with

each other. Each individual network is an “expert” in

solving some portion of the problem, and the collective

network is designed so that input data are “directed” to

the appropriate expert. The MNN design used for this

study is the “Adaptive Mixtures of Local Experts” [5],

which consists of several experts and a gating network

Gating Network

Output Layer

Gating Network

Hidden Layer

Output Layer

Gating Layer

Expert 1

Output Layer

Expert 2

Output Layer

Expert 3

Output Layer

Expert 3

Hidden Layer

Expert 2

Hidden Layer

Expert 1

Hidden Layer

Input Layer

Fig. 2 - Modular neural network architecture
38 Naval Research Reviews

Fig. 3 - Scatter plot of main rotor

pushrod predictions for the

baseline data

relationship between the actual and predicted loads. A

number close to 1.0 indicates that a linear relationship

exists and that the neural network does a good job of

predicting the loads in the components of the rotor

system.

Root mean squared (RMS) error, which is shown in the

right column of Table 2, is normalized as a percentage

of the maximum vibratory load. This maximum load

corresponds to the largest anticipated load and is

determined from an inspection of the baseline data set.

The maximum vibratory load was found to be 5,000 lb

for the pushrod and 7,000 lb for the lead-lag damper.

Because the baseline data consists of a variety of high

load maneuvers that were performed to determine the

boundaries of the flight envelope, these values are

considered to be reasonable estimates for the largest

load that could be anticipated during actual service.

For assessment of this study’s load predictions, the

normalized RMS error is the primary measure of

network accuracy.

Analysis of the load prediction results shows that, for a

given load value, between 94% and 96% of the data

point errors fall within a range equal to twice the RMS

error. For this reason, there is a 95% confidence that

the MNN’s load predictions will be accurate to within

twice the RMS error value. Using this approach and

the results shown in Table 2, there is a 95% confidence

that the pushrod load predictions are accurate to within

±7.3% of the load range for the baseline data.

Results for load prediction in the lead-lag damper were

found to be significantly different when the baseline

data was separated into prediction above and below

60 kn of forward airspeed. For flight test data collected

at airspeeds above 60 kn, the normalized RMS error is

4.1% of the load range. However, for flight test data

collected at airspeeds below 60 kn, the RMS error is

13.5% of the load range. This significant change in

prediction accuracy may be due to the damping

characteristics of the SH-60 lead-lag damper, which

operates in two separate modes depending on blade

lead-lag velocity. At small lead-lag velocities, the slope

of the force/velocity curve is rather high; at higher

lead-lag velocities, the damper enters a “bypass mode”

and the slope of the force/velocity curve is much lower.

This transition into bypass mode occurs at helicopter

airspeeds between 60 and 70 kn. Also, at airspeeds

below 60 kn, the damper load is quite sensitive to the

blade lead-lag velocity, which is not a neural network

input. This characteristic is likely to be one reason for

decreased prediction accuracy for data obtained at

airspeeds below 60 kn. Another factor that may contribute to inaccurate predictions for lead-lag damper

loads in the low-speed flight regime is that blade leadlag velocity is a function of aircraft airspeed, which

cannot be reliably measured when the helicopter is

traveling at airspeeds below 40 kn.

Figure 3 is a scatter plot of the ANN’s load predictions

for the pushrod load; Figures 4a and 4b are scatter

plots of the load predictions for damper loads at

airspeeds above and below 60 kn, respectively. A

scatter plot is useful for identifying specific data points

where the load predictions were particularly inaccurate, but it does not allow for a complete view of how

dense the data points are around the 45° line (where

the predicted load matches the actual load). A comparison of Figs. 4a and 4b clearly shows that load predictions for helicopter airspeeds greater than 60 kn have

much less scatter.

Pushrod load

Damper load

Damper > 60 kn

Damper < 60 kn

0.94

0.78

0.88

0.61

3.65

7.62

4.14

13.51

LOAD R NORMALIZED

RMS ERROR

Table 2 - Statistical summary of load

prediction results for the baseline data
Volume 50, Number 4 39

Fig. 4b - Scatter plot of damper

predictions for airspeeds less

than 60 kn for the baseline data

Fig. 4a - Scatter plot of damper

predictions for airspeeds greater

than 60 kn for the baseline data

10 30 50 70 90 110 130 150 170 190 210 230

TIME (sec) 

0

1000

2000

3000

4000

5000

6000

7000

MAIN ROTOR DAMPER LOAD (lb)

STRAIN GAUGE

NEURAL NETWORK

Fig. 6 - Damper load predictions for a 60o angle of

bank right turn in the baseline data

Fig. 5 - Pushrod load predictions for a 60o angle of

bank right turn in the baseline data

10 30 50 70 90 110 130 150 170 190 210 230

TIME (sec) 

0

1000

2000

3000

4000

5000

MAIN ROTOR PUSHROD (lb)

STRAIN GAUGE

NEURAL NETWORK

While scatter plots, correlation, and RMS error

provide good measures for statistically analyzing load

predictions, these measures do not fully highlight the

extent to which the load predictions of the neural

network follow the trends of the actual measurements

obtained from the strain gauges. Figures 5 and 6 show

time history traces of the predicted vs. actual pushrod

and damper loads encountered during a right turn with

a 60° angle of bank. Even though the RMS error for

the pushrod load prediction was found to be the

highest during this particular maneuver, Fig. 5 indicates that the predictions of the neural network accurately capture the significant changes in pushrod load

level during this highly dynamic maneuver.

Flight Test Evaluation and Results

System Description: During a second phase of validation and flight testing, a flight test evaluation of the

neural network’s load prediction algorithms (which

could also be referred to as “virtual strain gauges”)

was conducted on an SH-60F helicopter that was

instrumented with strain gauges on the main rotor

pushrod (see Fig. 7) and the lead-lag damper. (These

strain gauges were located in the same position that

was used on the SH-60B helicopter during the first

phase of flight testing.) Power was supplied to these

rotor system strain gauges through an existing aircraft

slipring assembly that is typically used for power

during blade folding procedures. (In the test helicopter,

the blade fold capability was disabled during the flight

test evaluation.)

Analog strain gauge measurements were recorded on a

flight data recorder, digitized, and then post-processed

into steady and vibratory components. These strain

gauges provided a reference strain measurement that

could later be compared with the neural network load

predictions.
40 Naval Research Reviews

The load prediction algorithms that had been developed

in the first phase of flight testing were programmed in

the C computer language and compiled, and then the

executable program was embedded in a 486-based

central processing unit (CPU) in an upgraded Structural

Data Recording Set.

During this phase of flight testing, the neural network

load predictions were performed in real time, using

12 parameters of flight data (see Table 1) that were

collected 8 times each second. Each data sample was

time-tagged so that the neural network’s load predictions could be synchronized with actual load measurements during post-flight data analysis.

Flight Test Demonstration Data: The second phase of

flight testing consisted of 20 flights and gathered almost

8 hours of flight data, which yielded 228,704 data

points. There are two significant differences between

the baseline data collected from the SH-60B and the

flight evaluation data collected from the SH-60F in this

phase of the test:

• The baseline data was obtained during a flight

loads survey that was designed to explore the

boundaries of the helicopter’s flight envelope. In

other words, some of the flight conditions

encountered during that phase of testing were not

representative of normal flight operations. For

example, the baseline data contain some data

points that were collected when the helicopter’s

gross weight exceeded 22,000 lb and when

engine torque levels were nominally 120% of the

basic operating torque. Flight operations at high

gross weight and high torque levels can result

in abnormally high vibratory loads.

However, during the flight test demonstration

of the virtual strain gauges (the second phase of

flight testing), the helicopter was limited to a

nominal torque value of 100% of basic operating torque, which is consistent with fleet

operations. Helicopter gross weights during this

phase of testing were also consistent with fleet

operations, with values ranging from 18,500 lb

to 20,600 lb.

These differences in gross weight and torque

levels are among the reasons that the rotor

system loads in the flight test demonstration are

generally lower than the loads in the baseline

data.

• The pullout maneuvers performed during the

second phase of flight testing were not flown as

aggressively as they were in the baseline data.

Peak load factors of 2.7 g’s were achieved

during the flight test, as compared to 3.3 g’s

that occurred in the baseline data collected

during the flight loads survey.

Fig. 7 - Instrumented

main rotor pushrod

While high gross weight and engine torque

levels and severe high-g flight maneuvers are

very useful for neural network training purposes, the data obtained during the flight test

demonstration of the neural network algorithms

are more representative of the way the aircraft

is actually operated in the fleet.

Results: Neural network predictions for loads in

components of the helicopter’s rotor system were

performed on-board the flight test helicopter in real

time. Table 3 shows the overall statistical performance

of the MNN for load predictions during the second

phase of flight testing. Normalized RMS error for the

pushrod load predictions is quite good, at ±2.1% of the

load range. Again, the effect of helicopter airspeed on

loads prediction for the main rotor lead-lag damper

was found to be very significant: Flight test data

collected at airspeeds above 60 kn yielded an RMS

error of ±3.2%, while data collected at airspeeds below

60 kn yielded an RMS error of ±16%. Scatter plots for
Volume 50, Number 4 41

Pushrod load

Damper load

Damper > 60 kn

Damper < 60 kn

0.91

0.84

0.83

0.49

2.11

7.24

3.22

16.06

LOAD R NORMALIZED

RMS ERROR

Table 3 - Statistical summary of load

prediction results on the flight test data

Fig. 9a - Scatter plot of damper

load predictions for airspeeds

greater than 60 kn for the flight

test demonstration

Fig. 9b - Scatter plot of damper

load predictions for airspeeds

less than 60 kn for the flight test

demonstration

Fig. 8 - Scatter plot of main rotor

pushrod load predictions for the

flight test demonstration

this set of flight test data are shown in Figs. 8 and 9; as

was seen in the scatter plots for the baseline data, there

is clearly less scatter in the damper predictions when

airspeed is greater than 60 kn (Fig. 9a).

The results provided in Table 3 show that the RMS

error for load predictions in this phase of flight testing

compares favorably with the results for the baseline

data, which were shown in Table 2. Furthermore, this

indicates that the results obtained from one aircraft can

be applied to a second similar aircraft without loss in

prediction accuracy.

fatigue-critical helicopter rotor system components

have been successfully developed and demonstrated in

flight tests of an instrumented SH-60 helicopter. The

neural network uses easily measured aircraft parameters (such as pilot control positions, engine torque,

rate of climb, etc.) to predict loads on components in

the helicopter’s rotor system. The load prediction

algorithms were developed using data collected during

56 test flights that included a wide range of helicopter

weights, configurations and flight maneuvers. The

ANN-generated load prediction algorithms were then

embedded in an on-board processor and successfully

used to make real-time load predictions during 20 test

flights of a second flight test helicopter.

As a result of this research, the following conclusions

can be drawn:

Figs. 10 and 11 depict time history traces of predicted

vs. actual loads (as measured during a high-g symmetric pullout maneuver) for the pushrod and the lead-lag

damper, respectively. These figures show that the

neural network load predictions compare quite well

with those obtained from the actual strain gauge

measurements. The overall trend, as well as the

increase in load level, is captured in both cases.

Summary and Conclusions

Using an artificial neural network approach, virtual

strain gauge algorithms that monitor vibratory loads on

• The flight test results show that the neural

network load monitoring algorithms that were
42 Naval Research Reviews

100 200 300 400 500 600

TIME (sec)

0

1000

2000

3000

4000

5000

MAIN ROTOR PUSHROD (lb)

STRAIN GAUGE

NEURAL NETWORK

Fig. 10 - Pushrod load predictions for a high-g

symmetric pullout maneuver from the flight

test demonstration

100 200 300 400 500 600

TIME (sec)

7000

6000

5000

4000

3000

2000

1000

MAIN ROTOR DAMPER LOAD (lb)

0

STRAIN GAUGE

NEURAL NETWORK

Fig. 11 - Damper load predictions for a high-g

symmetric pullout maneuver from the flight

test demonstration

developed using flight data from one aircraft

are general enough to successfully predict

vibratory loads on a second aircraft.

• Artificial neural networks, when properly

trained, can be used to consistently predict

loads in helicopter rotor system components.

• For pushrod load, the neural network normalized RMS error was 3.6% of the load range for

the baseline data (obtained from the first flight

test helicopter) and 2.1% for the flight test data

(obtained from the second flight test helicopter).

• The neural network’s accuracy of predictions

for damper loads was highly dependent upon

helicopter airspeed. At airspeeds greater than

60 kn, the neural network normalized RMS

error is 3.2% of the load range for the flight test

data. At airspeeds below 60 kn, the RMS error

increases to 16%. Therefore, while damper load

predictions are quite good for airspeeds greater

than 60 kn, the neural network cannot accurately predict damper loads in low speed flight.

• Artificial neural networks can accurately

monitor the significant changes in load levels

during highly dynamic maneuvers.

• Demonstration of a real-time load prediction

capability illustrates the potential to use virtual

strain gauges as part of a warning system that

indicates to helicopter pilots that they are flying

under conditions that can cause excessive

fatigue damage accumulation in flight-critical

components.

Acknowledgments

This work was supported by the Office of Naval

Research, Dr. P. Abraham and Mr. W. King. The

authors also acknowledge the support and guidance of

Mr. A. Ragghianti and Mr. G. Barndt of the Naval Air

Systems Command and the flight test team at

NAWCAD, headed by Mr. M. Hollins.

References

1. M.J. Augustin, “Specifying HUMS to Meet Safety and

Reduced Operating Cost Requirements,” in Proceedings of the

American Helicopter Society 54th Annual Forum, 20-22 May

1998, Washington DC, pp. 1201-1210.

2. J.D. Cronkhite, “Practical Applications of Health and Usage

Monitoring (HUMS) to Helicopter Rotor, Engine, and Drive

Systems,” in Proceedings of the American Helicopter Society

49th Annual Forum, 19-21 May 1993, St. Louis, MO,

pp. 1445-1455.

3. B. Muller and J. Reinhardt, Neural Networks, An Introduction

(Springer-Verlag, Berlin, 1990).

4. M. Zeidenberg, Neural Network Models In Artificial Intelligence (Ellis Horwood Limited, Chichester, West Sussex,

England, 1990).

5. R.A. Jacobs, M.I. Jordan, S.J. Nowlan, and G.E. Hinton,

“Adaptive mixtures of local experts,” Neural Computation 3,

79-87 (1991).
Volume 50, Number 4 43

The Authors

Dr. David J. Haas is an aerospace engineer in the

Marine and Aviation Department and has worked at the

Carderock Division of the Naval Surface Warfare Center

since 1985. His current research activities involve the

development of advanced diagnostic techniques and

health monitoring technologies for aircraft and shipboard

mechanical systems. He is an Associate Fellow of the

American Institute of Aeronautics and Astronautics.

Ms. Kelly M. McCool is an aerospace engineer in the

Marine and Aviation Department at the Carderock

Division of the Naval Surface Warfare Center. She has

worked at NSWC since 1989 on projects involving

helicopter rotor system health monitoring technologies,

neural network applications, and helicopter low speed

flight issues.

Mr. Lance A. Flitter has been a computer scientist at

the Carderock Division of the Naval Surface Warfare

Center since 1987. Since 1992, he has been in the

Shipbuilding Technologies Department working on the

application of artificial intelligence technologies for

manufacturing and maintenance purposes. His focus has

been on neural network applications for virtual sensing.

Mr. Flitter also chairs and participates in standards

committees related to computer science and

manufacturing.
44 Naval Research Reviews
Volume 50, Number 4 45

lmost 30 years ago, researchers discovered the phenomenon

of crack closure. Simply put, closure occurs when the surfaces

of a crack come into contact with each other before the stress in

the material falls to zero.

This premature contact of the crack surfaces can result from

a number of sources, including the permanent deformation of

material as a crack passes through, which tends to act as a clamp

on the material just behind the crack tip. Whatever the cause,

however, closure prevents the tip of the crack from being exposed

to the full range of stresses that it might otherwise experience.

Theoretically, closure should cause crack growth to slow

down, or even to stop in some cases. Experiments, however, reveal

that closure is a complex phenomenon: some results even show

that crack growth can occur in situations where the surfaces of

the crack never fully lose contact with each other.

In the following article, the director of Fracture Technology

Associates in Bethlehem, Pennsylvania, discusses the phenomenon

of crack closure and compares several of the many analytical models

to predict crack growth that have attempted to incorporate the

effects of crack closure.

– S.P.
46 Naval Research Reviews

Introduction

Fatigue crack growth rate (FCGR) data, when combined with stress analysis of cracks, provides a means

of predicting the useful life of a structure or a component that is subjected to cyclic stress in the presence of

a crack or defect. However, fatigue life prediction

methodology is complicated by a large number of

variables, including — but not limited to — the

frequency of the cyclic stresses, the cyclic stress

intensity range at the crack tip (∆K), the stress ratio R

(the ratio of minimum to maximum stress intensity),

the behavior of small cracks, the effects of load

interactions, and the physical environment of the

component (temperature, humidity, presence of

chemicals, etc.).

In theory, a detailed analysis of laboratory tests should

allow us to determine the material properties that

govern the rate of fatigue crack growth, regardless of

the size and shape of the sample of material being

tested. In practice, however, this is often not the case.

One of the key variables related to fatigue crack

growth is the threshold value of the cyclic stress

intensity (∆Kth), below which fatigue cracks will not

propagate. In general, the cyclic stress intensity (∆K) is

given by:

∆ ∆ KY a =⋅ ⋅ σ π , (1)

interest (such as the presence of holes, notches,

corners, or other changes in shape that tend to increase

stress).

A commonly used laboratory test to determine ∆Kth

subjects a sample of material to cyclic stresses and

gradually decreases the value of ∆K as the crack length

increases. When the fatigue crack either stops growing

or grows at a rate less than 10-7 mm per cycle of stress,

researchers stop the experiment and use Eq. (1) to

calculate ∆Kth. In this type of test, however, ∆Kth is

determined at a crack length that is longer than is

typical of small cracks at similar stress intensities.

Interference in the crack wake, also known as crack

closure, can reduce the range of ∆K and is considered

one reason why the value of ∆Kth determined from a

long crack may be non-conservative for small cracks.

A better understanding of this effect, as well as the

factors causing stress ratio effects on the rates of

fatigue crack growth, should lead to improvements in

fatigue life prediction methodology.

J. Keith Donald

Fracture Technology Associates

Bethlehem, Pennsylvania

The Role of Crack Closure

and Kmax on Fatigue Crack

Growth Rate Thresholds

Background

Cyclic stresses cause fatigue cracks to grow in metals

and other ductile materials. As a general rule, when the

stress in a cracked material increases, the surfaces of

the crack pull away from each another and the crack

has the opportunity to grow. As the stress decreases,

the surfaces of the crack should move back toward

each other. In theory, you’d expect that the crack

surfaces should not touch each other until the stress

falls to zero or becomes compressive.

where a is the length of the crack, ∆σ is the difference

between the maximum and the minimum values of

stress in the cycle, and Y is a stress concentration factor

that is related to the geometry of the component of
Volume 50, Number 4 47

In 1970, however, Elber discovered the phenomenon

of crack closure [1,2]. In simple terms, closure occurs

when the two surfaces of a crack come into contact

with each other before stress in the material falls to

zero. Crack closure can result from several sources,

including permanent deformation of the material that

occurs when the crack tip passes through, irregularities

in the crack surface that do not align precisely as the

surfaces come back together, and the buildup of

corrosion products on the crack surfaces.

The Crack Closure Dilemma

The traditional concept of closure has led to the

assumption that the driving force for crack propagation

— the effective stress intensity, ∆Keff — exists only

when the crack is fully open. In other words, this

concept implies that the material just in advance of the

tip of a crack will not experience the full value of

stress intensity if the surfaces of the crack are touching

each other, because this contact will in effect prop

open the crack and relieve some of the stress in the

material surrounding the crack tip.

However, it will be shown that totally excluding the

driving force that exists below the level of load needed

to fully open the crack can yield misleading values of

∆Keff. This is particularly true in the near-threshold

regime, where the loads needed to open a crack are

typically high. In the past, experiments that have

attempted to consider the effects of crack closure have

been hampered by non-repeatable measurements of

closure and by contradictory results [3-10].

change in compliance — the inverse of the slope of the

load/displacement curve — as the stress intensity

increases and the crack approaches the fully open

state. Also, measurement of the load needed to fully

open the crack is subject to experimental scatter from a

number of sources: signal conditioning noise, sources

of non-linearity from test fixtures or instrumentation,

etc.

American Society for Testing and Materials (ASTM)

standard E647 recommends defining Kop as the stress

intensity associated with the load needed to cause a

2% deviation in the slope of the linear section of a test

specimen’s load-displacement curve [11]. The effective stress intensity factor, ∆Keff , for the “opening

load” method is then calculated as follows:

(∆Kop method) ∆Keff = Kmax - Kop , (2)

where Kmax is the stress intensity calculated for the

maximum stress, and Kop is the stress intensity value

calculated for the crack opening load (when measured

as defined above). The 2% offset value used to establish Kop in this method results in an opening load that

is somewhat lower than the “true” opening load. Also,

measurements of displacement at locations remote

from the crack tip generally yield lower apparent

opening loads (and, therefore, higher values of ∆Keff)

than do measurements of displacement taken near the

crack tip.

The opening load method: A common interpretation of

the closure mechanism visualizes the surfaces of a

crack as “peeling apart” or “unzipping” as stress is

applied. Such an interpretation can lead to the assumption that contributions to ∆Keff are only applied when

the last point of contact is lifted off and the crack is

fully open. This places tremendous significance on the

value of the stress intensity at which the crack is fully

open (Kop), a parameter that is very difficult to measure with any accuracy.

Attempting to measure Kop by comparing the load

placed on a test specimen (which is proportional to

stress in the material) and the displacement (or movement) of the specimen is difficult due to the gradual

Based on opening load, the ∆Keff at threshold for a low

stress ratio (R = 0.1) is often lower than the change in

applied stress intensity (∆Kapp) for cycles of stress with

a high stress ratio (R = 0.98, for example) for which

crack growth can be observed. This seemingly contradictory result can only be explained if crack closure

only partially shields the crack tip from damaging

cyclic strain. Alternative methods of estimating ∆Keff

— ones that account for partial closure or that evaluate

the closure contribution at the minimum load instead

Despite these limitations, however, this method is able

to correlate stress ratio effects very nicely for fatigue

crack growth rates above 10-5 mm per cycle of stress.

Surprisingly, there is a lack of correlation at nearthreshold crack growth rates, where opening levels are

typically much higher. This lack of correlation is not

related to the use of a remote measurement location.
48 Naval Research Reviews

of at the opening load — will be discussed in a later

section.

Sources contributing to crack closure: The most

common types of crack closure are plasticity-induced

closure (which is caused by the permanent deformation

of material in the wake of the crack tip), roughnessinduced closure (which is caused by asperities, or

irregularities, on the surfaces of the crack), and oxideinduced closure. Experimental evidence shows that

plasticity-induced closure can be a very significant

source of closure under plane stress conditions, which

are most commonly associated with relatively thin

specimens of material. However, the effects of plasticity-induced closure are less obvious under conditions of

plane strain (which are usually associated with relatively thick specimens of material or with values of

stress intensity near threshold) and may be overshadowed by the effects of roughness- and oxide-induced

closure.

Three factors are related to the diminishing impact of

plasticity-induced closure in plane stress conditions.

First, the proximity of the free surfaces of a specimen

under plane stress provides a source for the flow of

material that ends up in the crack wake. Under plane

strain, however, any flow of material that contributes to

residual plastic deformation and interference in the

crack wake must also contribute to voids or areas of

non-contact (due to constant volume plastic flow

assumptions).

A second issue related to plasticity-induced closure is

scaling. If we assume that residual plastic wake effects

are proportional to the size of the plastic zone at the

crack tip, the magnitude of this wake effect is proportional to the square of the stress intensity, whereas the

elastic profile of the crack is simply proportional to the

stress intensity. In other words, as the value of the stress

intensity is reduced, the ability for significant plasticityinduced closure to occur beyond the immediate vicinity

of the crack tip is proportionally reduced. Therefore,

whether plasticity-induced closure even occurs under

conditions of plane strain is probably unimportant. The

results of opening load measurements on 2024-T3

aluminum alloy show that Kop is fairly constant at low

Kmax levels (as can seen in the lower curve in Fig. 1),

presumably due to a fairly constant roughness and/or

oxide contribution to crack closure.

A third issue related to plasticity-induced closure is the

effect of the initial level of Kmax in decreasing K tests

for determining threshold. If a decreasing K test is

initiated at a level of Kmax that is sufficiently high to be

partially or predominantly plane stress, the resulting

∆Kth is elevated above the “true” long crack threshold

and is sensitive to the magnitude of the decreasing Kgradient. The K-gradient refers to the rate of change of

K or ∆K with respect to change in crack length. For a

high starting Kmax level of 22.0 MPa√m, the measured

Kop remains high as ∆K is decreased, although the

interference is clearly not near the crack tip (as can be

seen in the upper curve in Fig. 1).

Fig. 1 - For a standard fatigue crack growth rate test

in 2024-T3 aluminum alloy, the Kop is fairly constant at

low Kmax levels. At higher Kmax levels, the

contribution of plasticity-induced closure significantly

elevates the Kop level. The Kop level remains high for

decreasing K from a high starting Kmax level.

Although the data are not shown in Fig. 1, the test that

generated the upper curve of data was continued until

crack growth rates near threshold were achieved. The

opening load at threshold was found to be about 50%

higher than the cyclic maximum load at threshold.

Using the traditional definition of ∆Keff (i.e., the

calculation shown in Eq. (2)) would not be meaningful, because Kmax is less than Kop; under these conditions, Eq. (2) would yield a negative value for ∆Keff.

Therefore, the results of this test are clearly strong

evidence for the occurrence of cyclic crack-tip strain at

values of stress intensity below Kop.

Figure 2 shows that the influence of the starting Kmax

level on the near-threshold behavior in 2024-T3

aluminum alloy occurs above a starting Kmax of
Volume 50, Number 4 49

15 MPa√m. However, if a decreasing K test is initiated

at a Kmax level that produces predominantly plane

strain conditions, neither the K-gradient nor the

starting Kmax level appears to have any impact on the

resulting value of ∆Kth. Figure 3 shows seven decreasing K tests using different K-gradients and crack

lengths on three separate test samples. The variation in

∆Kth for these seven tests is only about 3%. Although

the ∆Kth determined from these tests would still be

considered a “long crack” threshold, the crack closure

effects observed appear to be related to the presence of

roughness and oxide, not plasticity.

The influence of near crack-tip vs. crack wake influence on crack closure: Perhaps the biggest controversy

surrounding closure is the relative importance of crack

surface contact near the crack tip versus contact over

the entire crack wake. The notion that small cracks and

long cracks behave differently because of wake effects

also suggests that the forces exerted over the entire

crack wake may be important. The magnitude and

distribution of these forces depend on the location,

height and distribution of asperities or other interference mechanisms. However, solving for crack wake

forces is not straightforward, and therefore closure

measurement has traditionally relied on contact to

estimate ∆Keff .

There is strong evidence to support additional crack tip

cyclic strain below the contact or opening load (i.e.,

even after contact between the crack surfaces occurs),

although in such cases the effective change in stress

intensity (∆Keff) would no longer be directly proportional to the applied change in stress intensity (∆Kapp).

It is this evidence that supports the partial crack closure

and crack wake force distribution models (discussed

below) as alternatives to traditional opening load

concepts.

Alternative Procedures for Estimating ∆Keff

Several methods of estimating ∆Keff have been attempted [12] and will be briefly summarized here. Each

of these methods analyzes the same raw load/displacement data; the difference in the various methods lies in

the interpretation of that data. Four of these alternative

methods are briefly summarized below, and then

analytical results using each of the techniques is

presented. Fig. 2 – Fatigue crack growth rate plot that shows the

effects of different values for starting Kmax

Fig. 3 - Fatigue crack growth rate plot that shows the

effects of decreasing K-gradient

The 2/π correction to the opening load method: The

partial crack closure model proposes to account for the

crack-tip strain that occurs at values of applied load

below the opening load. This “partial closure” interference is modeled as a thin rigid wedge of uniform height

with a small gap at the crack tip [13]; the effective

change in stress intensity factor (∆Keff) is calculated as

follows:

 (∆K2/π method) ∆K K KK eff = − ⋅− ∆ ∆∆ app







 ( ) op + op 1 2

π , (3)

where ∆Kapp is the change in stress intensity between

the maximum and the minimum values of stress intensity for the loads applied to the test specimen, and ∆Kop

is the stress intensity value calculated for the crack

opening load.
50 Naval Research Reviews

The adjusted compliance ratio method: Another

means of calculating ∆Keff is based on interpreting

measurements of strain near the crack tip [14]. This

technique, which is called the compliance ratio

method, assumes that the crack driving force should be

proportional to the magnitude of the elastic strain field

near the tip of the crack. Therefore, ∆Keff is based on

the ratio of the measured strain magnitude (which is

proportional to the secant compliance, Cs

) to that

which would have occurred in the absence of closure

(which is proportional to the fully open compliance C).

(Figure 4 graphically depicts each of these parameters.)

An alternate form of this technique, which is referred

to as the adjusted compliance ratio (ACR) method, has

been developed to estimate ∆Keff by using remote

measurements of compliance (i.e., those taken at

locations far from the crack surface or crack tip). This

technique estimates ∆Keff by subtracting the initial

compliance (Ci

, measured prior to the initiation of a

crack), from the compliance ratio as follows [15,16]:

 (∆KACR method) ∆K = ∆ C - C

C - C eff

s i 

i

⋅ Kapp , (4)

where ∆Kapp is the change in stress intensity between

the maximum and the minimum values of stress

intensity for the loads applied to the test specimen. The

compliance parameters shown in Eq. (4) are defined as

shown in Fig. 4.

The crack wake influence method: The previously

discussed methods of estimating ∆Keff are based

primarily on the direct analysis of experimental data.

The crack wake influence (CWI) method, however,

attempts to mathematically describe the effect of

interference in the wake of the crack (i.e., crack

closure) on measurements of crack surface displacement and on the stress intensity at the crack tip [17].

Solving the complex mathematical equations that

describe the stress and strain fields surrounding a crack

provides a method of relating measurements of compliance far from the crack tip to the reduction in elastic

strain at the crack tip, which has a direct impact on the

magnitude of the cyclic plastic crack tip strains that

cause fatigue damage.

The CWI method calculates ∆Keff as follows:

Fig. 4 - Load-displacement trace that shows

the influence of opening load, compliance

ratio and crack wake

When the amount of crack growth is small compared

to the overall length of the crack, the ACR method is

equivalent to the CWI method for a distributed force

that tapers linearly from some value at the point of

crack initiation to a value of zero at the crack tip. This

equivalence allows us to use the mathematical results

from the CWI method to improve the empirically

determined estimates obtained from the ACR method

of estimating ∆Keff .

The modified adjusted compliance ratio method (n2):

The equivalence of the mathematical CWI method and

the empirical ACR method for certain crack growth

conditions allows the development of an improved

ACR method for estimating ∆Keff . A slight modification of the ACR formula (Eq. (4)) replaces the initial

compliance (Ci

) with the compliance of the load

displacement curve at minimum load, Cn2, as follows:

(∆KCWI method) ∆ ∆ K K V

V IFR eff app

cwi = ⋅− ⋅







 1 (5)

where ∆Kapp is the change in applied stress intensity,

Vcwi and V are displacement measurements that are

obtained from load/displacement curves (these parameters are shown and defined in Fig. 4), and IFR are

non-dimensional (i.e., solely numerical) functions that

depend only upon the position of the crack wake

interference in relation to the crack tip [17].
Volume 50, Number 4 51

 (∆ ∆ KACRn2 method) K K eff app = ∆ C - C

C - C

s n2 

n2

⋅ (6)

Experimental results: To evaluate these various methods of estimating ∆Keff , fatigue crack growth was

measured in laboratory tests of center crack tension

samples (also known as M(T) geometry specimens)

fabricated from 2024-T3 aluminum alloy. The crack

growth data were generated at stress ratios of 0.1 and

0.7 using decreasing K (below 1 x 10-5 mm/cycle) and

increasing K (above 1 x 10-5 mm/cycle) test procedures.

Figure 5 shows crack growth data obtained at a stress

ratio of 0.1 for the 2024-T3 alloy as a function of

∆Kapp (small open circle symbol) and of the ∆Keff

calculated from the five estimation procedures (including the traditional opening load concept, which is

shown as a large circle symbol). For comparative

purposes, a second sample of 2024-T3 aluminum alloy

was used to generate crack growth rate data at a stress

ratio of 0.7. Analysis of the load-displacement data

obtained at a stress ratio of 0.7 revealed no evidence of

non-linearity associated with crack closure. Therefore,

the data at this stress ratio could serve as a baseline for

evaluating the various methods of estimating ∆Keff .

Note that at near-threshold growth rates, the opening

load (OP) method gives a distinctly lower value for

∆Keff than is obtained from the other methods. Also

note that all of the estimation methods that account for

crack tip cyclic strain below the opening load are in

good agreement with the data obtained at high stress

ratio (R = 0.7). Finally, note that for crack growth rates

above 5 x 10-6 mm per cycle of stress, the ∆Keff data

using the opening load method agrees very well with

the crack growth data obtained at high stress ratio.

The Role of Kmax

The observations noted above do not account for

second-order effects that are due to differences in Kmax

at a stress ratio of 0.1 and 0.7. Previous work has

shown that properly correlating stress ratio effects

requires the correct closure methodology as well as an

evaluation of the Kmax contribution to stress ratio

effects [12,15,16,18]. Research has shown that this

Kmax dependence can be fitted by a power-law function, with the magnitude of the exponent being a

measure of Kmax sensitivity. The combined effects of

∆Keff and Kmax can be normalized using the following

relationship [18]:

∆ ∆ Knorm = Keff

( 1 - n ) 

⋅ K max

n

 (7)

where n is a normalized Kmax sensitivity exponent

determined from experimental data.

To investigate the sensitivity of the 2024-T3 alloy to

changes in Kmax , specimens were tested at a constant

Kmax of 6.6, 9.9, 14.8, 22.0, 33.0, and 49.5 MPa√m and

a decreasing ∆K. Figure 6 shows the data obtained

from these tests. At near-threshold crack growth rates,

the value of stress ratio for the constant Kmax test varies

from 0.78 to 0.98; therefore, none of these data should

be affected by crack closure. These results suggest that

Fig. 5 - Fatigue crack growth rate plot that shows data

collected at stress ratios R = 0.1 and R = 0.7 as a function of ∆Kapp and five estimation methods for ∆Keff

Fig. 6 - Fatigue crack growth rate plot that shows the

effect of Kmax at a stress ratio above the influence of

crack closure
52 Naval Research Reviews

the difference between the curves in Fig. 6 is real and

is actually related to differences in Kmax. To test this

hypothesis, the data are replotted in Fig. 7 using a Kmax

sensitivity exponent (n) of 0.100 in Eq. (7). As can be

seen in Fig. 7, the separate crack growth data curves

from Fig. 6 collapse into a single curve that provides a

unique crack growth rate curve that is intrinsic to that

material. Although unproven, it is reasonable to

assume that Kmax sensitivity may play a role at low

stress ratios as well. If so, then the Kmax sensitivity

exponent may be universal over a wide range of Kmax

and ∆K, although crack closure is expected to play a

more dominant role at low stress ratios.

Figure 8, which is a plot of ∆K versus Kmax at threshold, shows how the data obtained from the constant

Kmax tests (shown in Fig. 6) converge with the data

obtained at low stress ratio for all ∆Keff estimation

methods (∆Kother) except opening load (∆Kop). As

noted in Fig. 8, when the experimental crack growth

data and the empirically determined Kmax sensitivity

exponent (n) of 0.100 are plugged into Eq. (7), ∆Knorm

for 2024-T3 aluminum is determined to be 1.69.

Figure 8 also shows a sharp rise in ∆Kapp at low stress

ratio due to crack closure, as well as a drop in ∆Kop

due to the ignored elastic crack-tip cyclic strain below

the opening load.

Summary and Conclusions

The experimental determination of ∆Keff using the

opening load concept is sensitive to the location where

the displacement of the material is measured (whether

close to or far from the crack surface), the arbitrary

nature of the percent offset, and the variability associated

with selecting a specific percent offset due to noise and

non-linearity. However, it appears that an opening load

method may be appropriate under the right circumstances. At growth rates above 10-5 mm/cycle, the ASTM

opening load method appears to provide satisfactory

results.

The 2/π method of making a correction to the opening

load has a physical model and is simple to apply. Although the model may only approximate typical mechanisms of crack closure, it does provide a means by which

cyclic crack tip strain that occurs below the opening load

may be accounted for. One disadvantage, however, is

that the opening load must be either measured or predicted to utilize this method.

Fig. 7 - Fatigue crack growth rate plot that shows a

unique intrinsic curve obtained by plotting the crack

growth rate as a function of ∆K0.9 * Kmax0.1

Fig. 8 - A plot of ∆K vs. Kmax at threshold reveals a

trend at high stress ratios and a convergence at low

stress ratio with all estimation methods (∆Kother)

except opening load

The adjusted compliance ratio (ACR) method is also

simple to apply and appears less sensitive to variability

than the opening load method. Experiments that measure

displacement of the material at multiple locations have

shown the ACR results to be insensitive to the measurement location. The measurements are also unique (i.e.,

no arbitrary percent offset is required) and repeatable,

because full range load and displacement (rather than the

deviation in linearity between load and displacement/

strain) are measured. However, results of the comparison

of the crack wake influence and the ACR methods

suggest that the ACR method only partially accounts for

crack-tip interference due to closure.
Volume 50, Number 4 53

The modified adjusted compliance ratio (ACRn2)

method attempts to account for limitations in the ACR

method through refinements in the initial assumptions

about the initial compliance. At present, however,

these refined assumptions lack a solid theoretical basis

and warrant further investigation.

The crack wake influence (CWI) method is mathematically complex and requires knowledge of the

crack wake force distribution, which limits its usefulness. Like the ACR method, however, the CWI method

is based on the change in crack surface displacement at

minimum load due to closure, rather than the load at

which contact in the crack wake first occurs.

The overwhelming difficulty associated with experimental estimations of ∆Keff arises both from the

challenges of obtaining quality experimental data and

from the broad range of assumptions required to

interpret the measurements. By nature, estimating

∆Keff from a single load-displacement curve is

indeterminant. Furthermore, the estimation of ∆Keff is

complicated by evidence that Kmax also influences

stress ratio effects. Because the procedures used to

estimate ∆Keff are not exact, it is difficult to separate

the precise roles or effects of crack closure and Kmax.

Additional research should provide a better understanding of these phenomena and should lead to

improvements in fatigue life prediction methodology.

Acknowledgments

The author wishes to acknowledge the support of

Fracture Technology Associates, Alcoa Technical

Center (Alcoa Center, Pennsylvania), and the Office of

Naval Research in the above endeavors. Many thanks

for the contributions of Paul C. Paris and Hiroshi Tada

as well.

4. E.P. Phillips, “Results of the Round Robin on Opening-Load

Measurement,” NASA Technical Memorandum 101601,

NASA Langley Research Center, Hampton, VA, 1989.

5. E.P. Phillips, “Results of the Second Round Robin on

Opening-Load Measurement,” NASA Technical Memorandum

109032, NASA Langley Research Center, Hampton, VA,

1993.

6. K. Sadananda and A.K. Vasudevan, “Analysis of Fatigue

Crack Closure and Thresholds,” Fracture Mechanics: 25th

Volume, ASTM STP 1220, edited by Erdogan and Hartranft

(American Society for Testing and Materials, Philadelphia,

PA, 1995).

7. K. Sadananda and A.K. Vasudevan, “Short Crack Growth

Behavior,” Fatigue and Fracture Mechanics: 27th Volume,

ASTM STP 1296, edited by R. Piascik, N. Dowling, and J.

Newman, (American Society for Testing and Materials, West

Conshohocken, PA, 1997).

8. D.L. Chen, B. Weiss, and R. Stickler, “Contribution of the

Cyclic Loading Portion Below the Opening Load to Fatigue

Crack Growth,” Materials Science and Engineering, A208,

1996, pp. 181-187.

9. D.E. Macha, D.M. Corbly, and J.W. Jones, “On the Variation

of Fatigue-Crack-Opening Load with Measurement Location,”

Experimental Mechanics, June 1979, pp. 207-213.

10. J.E. Allison, and C.P. You, “Problems Associated with the

Quantification of Fatigue Crack Closure,” in Fatigue ’90, The

Proceedings of the 4th International Conference of Fatigue

and Fatigue Thresholds, edited by H. Kitigawa and T. Tanaka

(Materials and Components Engineering Publications, Ltd.,

United Kingdom, 1990), pp. 1249-1255.

11. J.K. Donald, “A Procedure for Standardizing Crack Closure

Levels,” Mechanics of Fatigue Crack Closure, ASTM STP

982, edited by J.C. Newman, Jr. and W. Elber (American

Society for Testing and Materials, Philadelphia, PA, 1988),

pp. 222-229.

12. J.K. Donald, and P.C. Paris, “An Evaluation of ∆Keff Estimation Procedures on 6061-T6 and 2024-T3 Aluminum Alloys,”

Fatigue Damage of Structural Materials II, Engineering

Foundation Conference, September 7-11, 1998, Hyannis, MA.

To be published in International Journal of Fatigue, Elsevier

Science Ltd.

13. P.C. Paris, H. Tada, and J.K. Donald, “Service Load Fatigue

Damage — A Historical Perspective,” Fatigue Damage of

Structural Materials II, Engineering Foundation Conference,

September 7-11, 1998, Hyannis, MA. To be published in

International Journal of Fatigue.

References

1. W. Elber, “Fatigue Crack Closure Under Cyclic Tension,”

Engineering Fracture Mechanics, Vol. 2, 1970, pp. 37-45.

2. W. Elber, “The Significance of Fatigue Crack Closure,”

Damage Tolerance in Aircraft Structures, ASTM Special

Technical Publication (STP) 486 (American Society for

Testing and Materials, Philadelphia, PA, 1971), pp. 230-242.

3. R.W. Hertzberg, C.H. Newton, and R. Jaccard, “Crack

Closure: Correlation and Confusion,” Mechanics of Fatigue

Crack Closure, ASTM STP 982 (American Society for Testing

and Materials, Philadelphia, PA, 1988), pp. 139-148.

14. J.K. Donald, “Introducing the Compliance Ratio Concept for

Determining Effective Stress Intensity,” International Journal

of Fatigue, Vol. 19, Supp. No. 1, 1997, pp. 191-195.

15. J.K. Donald, G.H. Bray, and R.W. Bush, “An Evaluation of

the Adjusted Compliance Ratio Technique for Determining

the Effective Stress Intensity Factor,” 29th National Sympo-
54 Naval Research Reviews

sium on Fatigue and Fracture Mechanics, ASTM STP 1332,

edited by T.L. Panontin, and S.D. Sheppard (American

Society for Testing and Materials, West Conshohocken, PA,

1998).

16. G.H. Bray and J.K. Donald, “Separating the Influence of Kmax

from Closure-Related Stress Ratio Effects Using the Adjusted

Compliance Ratio Technique,” Advances in Fatigue Crack

Closure Measurement and Analysis: Second Volume, ASTM

STP 1343, edited by R.C. McClung and J.C. Newman, Jr.

(American Society for Testing and Materials, West

Conshohocken, PA, 1998).

17. J.K. Donald, G.M. Connelly, P.C. Paris, and H. Tada, “Crack

Wake Influence Theory and Crack Closure Measurement,”

National Symposium on Fatigue and Fracture Mechanics:

30th Volume, ASTM STP 1360, edited by K.L. Jerina and P.C.

Paris (American Society for Testing and Materials, West

Conshohocken, PA, 1999).

18. J.K. Donald, G.H. Bray, and R.W. Bush, “Introducing the

Kmax Sensitivity Concept for Correlating Fatigue Crack

Growth Data,” High Cycle Fatigue of Structural Materials

(Professor Paul C. Paris Symposium), edited by W. Soboyejo

and T.S. Srivatsan (The Minerals, Metals & Materials Society,

West Conshohocken, PA, 1997), pp. 123-141.

The Author

Mr. J. Keith Donald received his B.S. degree in

Mechanical Engineering from Lehigh University in 1972

and is currently the director of Fracture Technology

Associates. FTA was formed in 1984 to provide software

and testing services for fatigue crack growth and fracture

toughness testing. Mr. Donald has been active in

developing automated test techniques for fatigue crack

growth rate testing since the 1970s and has authored

numerous papers on the subject. His past activities within

the American Society for Testing and Materials (ASTM)

have included chairing the organization’s task group on

Near-Threshold Fatigue Crack Growth and co-chairing

the task group on Non-Visual Methods of Crack Length

Determination. Mr. Donald was co-editor for ASTM

Special Technical Publication (STP) 1251 on “Special

Applications and Advanced Techniques for Crack Size

Determination.” He is currently co-chairman of ASTM

task group E08.06.03 on Crack Closure Measurement

and Analysis.
Volume 50, Number 4 55

The first successful methods of

modeling the nucleation of fatigue cracks

in materials were developed in the 1920s.

Today’s multitude of empirical models to predict

the growth of longer cracks — many of them

computerized — have been developed and fine-tuned since

the 1960s.

In the following article, Navy scientists review some of this

century’s significant advances in crack growth modeling. They also discuss

a new unified model that incorporates two main crack driving forces for long

fatigue cracks. In this model, the presence of internal stresses helps to explain

deviations from long crack behavior, such as the significant differences between

the crack growth rates for short cracks and long cracks.

This comprehensive new model may, for the first time, allow researchers to

model fatigue damage behavior from the point of crack nucleation at small

material defects to a component’s final failure.

– S.P.

1940

1960

1980

2000

Milestones 

in Modelingn Modeling

Fatigue 

Damage

Milestones 

in Modeling

Fatigue 

Damage
56 Naval Research Reviews

Background

Fatigue damage can be divided into several stages:

crack nucleation, short crack growth, long crack

growth, and final failure. The rate of damage accumulation in each of these four stages depends upon a

variety of factors: loads, environment, alloy chemistry,

the presence of stress concentrations, specimen

geometry, etc. In the past several decades, empirical

approaches have been used to predict the life at each

stage of fatigue damage.

K. Sadananda

Naval Research Laboratory – Materials Science and Technology Division

Washington, DC

A.K. Vasudevan

Office of Naval Research – Materials Science Division

Arlington, VA

A Unified Framework for

Fatigue Damage Analysis

The most important drawback of these linear damage

models is that they do not account for the effects of

prior loading history. In other words, the damage

caused by each cycle is not affected by the size or the

sequence of the previous cycles of stress. Another

important consideration: if an unknown defect or flaw

were present, the total life of the component would be

significantly less than that predicted by the PalmgrenMiner model. Finally, these models are specifically

suited to address predictions of total component life

when the crack propagation stage of fatigue damage is

significantly small compared to the crack nucleation.

The Palmgren-Miner model of fatigue damage accumulation has undergone several modifications [3,4].

Most of the models that describe the behavior of short

cracks are concerned with the prediction of high crack

growth rates at lower stress intensities (below the long

crack threshold). These approaches encompass semiEarly methods to predict fatigue life included determining the relationship between stress or strain and the

number of load cycles to needed to produce component failure. One of the most common life prediction

calculations used for components subjected to variable

loads was based on the Palmgren-Miner linear damage

law [1,2]. This law proposes that damage sustained by

the component at different stress levels is linearly

additive; in such a case, the component’s total life can

be estimated by summing the incremental damage

n/Nf

, where n is the number of cycles of stress experienced at a certain level of stress, and Nf

 is the number

of cycles needed to produce a component failure at

that stress level. (This number is usually obtained from

laboratory tests of specially prepared specimens.)

When Σ [n/Nf

] = 1, the component is considered to

have failed.

In the strain-life approach, the material in the component is assumed to be highly strained near areas of

stress concentration (such as notches), and this ligament of material is assumed to behave the same as

virgin material (without the stress concentration)

having the same stress-strain history. Therefore, for

such fatigue damage models, the only material property data needed to estimate the component life are the

cyclic stress-strain and strain-life curves. In comparison to the stress-life model, the strain-life approach

accounts for any local plasticity effects at regions of

stress concentration. There are over 70 different crack

initiation models [3], and these models are described in

detail by Dowling [5,6] and Polak [7].
Volume 50, Number 4 57

empirical formulations of Hobson [8], subsequent

extensions of this model [9,10], and dislocation-based

models [11-14] in terms of the effects from material

inhomogenities like grain boundaries and local plasticity. These models predict the existence of a threshold

for a short crack and its variation from deceleration to

acceleration. Recently, Navarro and de los Rios [15]

have modified these models to predict the short crack

growth lives under constant and variable amplitude

loading [16].

(a), stress ratio (R), and applied stress is used. With some

modifications of the ∆Keff at near-threshold regions, the

model was fit to the small crack data and endurance

limits. It was then used to predict the fatigue lives of

aluminum alloys and titanium alloys for notched samples

under both constant and variable amplitude loadings. An

equivalent initial flaw size concept was used to bound

the fatigue lives in agreement with test results. Their

model does predict the ∆Kth decrease with load ratio for

constant amplitude loading, which is commonly observed under environment. They are able to predict to a

reasonable extent the life of short and long cracks under

constant and variable amplitude loading and also under

some service loading conditions. The analytical predictions from these models of crack growth have been

incorporated into several other computer programs [21].

Lazzeri et al. [22] have compared the life predictions

from various such empirical models for an aircraft

spectrum under a flight-by-flight load history at a mean

flight load of 75 MPa. The results are shown in Fig.1.

All of the models underpredicted the flights to failure.

The MODGRO (1988) model has been improved recently and renamed AFGROW, which now seems to be

comparable to the FASTRAN model [23]. Improvements

in these models that will incorporate the effects of

material slip behavior and of the environment are in

progress. Commonly, inadequacies in the prediction

methods are compensated by using several adjustable

parameters that are correlated using component test data.

In practice, vehicle safety is guarded by the use of safety

factors in design, the selected use of component test data,

periodic non-destructive inspections, the use of statistics

to assign data scatter, material quality control, etc.

Research shows that the constraint factor α is not very

sensitive to constant amplitude loadings. As a result,

long crack spike overload data (which is taken from

plots of crack length vs. number of cycles) is used to

fine tune the α values. For small cracks, a plane strain

condition is assumed and therefore the value of α is

selected between 2 and 3. The estimated value of α is

then used to correlate the long crack growth rates as a

function of ∆Keff for constant amplitude loading

conditions; for which the analytical functional relations for crack opening stress in terms of crack length

There are about 40 empirical models proposed for the

prediction of long crack growth [17]. Among these

models, a modified Forman equation [18] is considered the best approximation for the engineering

application. Newman and co-workers [19,20] have

done extensive work in the past 20 years to predict

fatigue life of commercial aluminum and titanium

alloys. They modified the Dugdale strip yield model to

include plasticity-induced closure and introduced a life

prediction code called FASTRAN to predict the lives

of alloys using small crack theory. The plastic zone

and the wake zone are modeled for rigid-perfectlyplastic elemental strips with an average stress between

the yield stress and the ultimate stress for the material.

A crack tip constraint factor α was assumed to account

for the three-dimensional state of stress at the crack

tip. This constraint factor α varies from a value of 1

for plane stress (which is usually encountered in thin

sheets of material) to a value of 3 for plane strain

(usually encountered in thick materials), and it is

determined by a trial-and-error method by fitting the

crack growth curves at high and low values of stress

ratio R. Furthermore, α varies from material to material; for instance, for aluminum alloys α is 2 and for

steel it is 2.5.

Fig. 1 - Comparison of experimental data

with crack growth predictions from various

computer models
58 Naval Research Reviews

The drawbacks in the prediction of fatigue life by the

abovementioned empirical models stem from several

sources, including: (1) the assumption of plasticity

induced closure, (2) the lack of terms in the model

that address environmental effects and slip behavior,

and (3) several adjustable parameters to fit the experimental data. In order to develop a more reliable life

prediction model, one must first recognize the correct

physics of the problem. We find that a better approach

is to treat the overall damage as an intrinsic behavior

instead of an extrinsic factor such as crack closure.

First, we must recognize that the true material behavior is represented in the long crack growth properties.

Second, fatigue damage has to be described by two

driving force parameters instead of one; these two

driving forces are ∆K and Kmax. Third, we must

recognize that deviations from the long crack growth

behavior arise from internal stresses that are present

ahead of the crack. For example, these internal

stresses are responsible for the accelerated crack

growth in the short crack region and the decelerated

growth due to overloads. Finally, the overall damage

must be described from the initiation stage to final

failure.

The following is a brief description of the analysis of

damage that bridges the gap between stages of fatigue

in a unified manner. Due to the restriction of space,

we discuss only damage in the long and short fatigue

crack growth regions. The analysis has also been

extended to include overloads and underloads.

Premature crack closure is believed to be a result of:

(1) plasticity, (2) asperities from oxides or other

corrosion products, (3) surface roughness, (4) viscous

fluids, and (5) phase-transformed material ahead of the

crack tip. While this interpretation has been used for

more than two decades, questions have been raised in a

number of areas, including experimental uncertainties,

interpretation of the closure data and basic physical

concepts. For detailed discussions on the topic of crack

closure, see Suresh [24] and ASTM proceedings [25].

From recently published articles on crack closure [26-

30], however, one can summarize that: (1) plasticityinduced closure is not possible (which complies with

the experimentally observed vacuum results where

∆Kth is independent of stress ratio), and (2) roughnessand oxide-induced closure are not significant enough

in most common aqueous environments (like lab/moist

air) to account for the large variations in the dependence of ∆Kth on stress ratio (i.e., factors of 3 to 7) that

have been experimentally observed.

Fig. 2 - Schematic diagram of various stress

intensity (K) factors

Long Fatigue Cracks

It has been established that for long fatigue cracks

there is a threshold stress intensity [∆Kth] below which

crack propagation does not occur. The threshold values

have been experimentally observed to depend primarily

upon stress ratio [R = Kmin/Kmax], microstructure of the

material, and environment. These stress ratio effects

have been rationalized on the basis of premature crack

closure in the wake of a crack that reduces the applied

crack tip stress intensity [∆Kapp = Kmax - Kmin] by an

amount due to closure [Kcl]. Therefore, the effective

driving force [∆Keff = Kmax - Kcl] that is less than the

applied force [∆Kapp], which tends to retard the crack

growth process. All of these parameters are depicted

schematically in Fig. 2.

Fatigue crack growth results are traditionally presented

in terms of variation in ∆K with stress ratio. Figure 3

shows such a plot, which compares crack growth data

for 7075-T7351 aluminum alloy obtained in a vacuum

with that obtained for the alloy in laboratory air. The

data in Fig. 3 is taken from different sources [31-33].

Most materials show a linear reduction in ∆Kth with

increase in stress ratio for lab air conditions; however,

Fig. 3 seems to show no such dependence for data

obtained in a vacuum. The former condition is interpreted as being due to plasticity-induced crack closure,

and the latter due to lack of it. Crack closure is considered to be prevalent only at low stress ratio, with
Volume 50, Number 4 59

closure levels decreasing towards higher stress ratios.

The relative magnitude of the reduction in ∆Kth with

stress ratio in air depends on the alloy and the environmental conditions in which the test was conducted.

Attempts to correlate the closure values to variations in

alloy microstructure and environmental conditions

have led to conflicting conclusions [28,29]. Closure

interpretations are considered as extrinsic since the

effects are due to the effects in the wake of a crack

(i.e., behind the crack tip). However, one can interpret

the results shown in Fig. 3 without invoking closure

explanations. This view accepts the fatigue behavior of

the material to be intrinsic to the conditions of the

applied loading and environment. The implications of

this explanation are interesting and are basic to the

behavioral properties of the material.

growth. As the stress ratio approaches a value of 1,

Kmax would increase to a level that is dictated by the

toughness of the material. On the other hand, in the

region where stress ratio is less than 0.4, Kmax approaches a constant critical value [K*max] with an

attendant decrease in ∆Kth. This portion of the plot is

labeled as a Kmax-controlled region, which means that

∆Kth must be varied to maintain the critical static stress

intensity K*max constant for crack growth. In the entire

range of stress ratios (i.e., from -10 to +0.8), while

both ∆K*

th and K*max critical values are being satisfied,

either one parameter or the other is controlling the

fatigue damage process. Thus, we postulate that the

crack advancement will occur if these two critical

stress intensities are simultaneously satisfied. These

two limiting values are the minimum conditions

necessary for crack growth to occur independent of

closure. The magnitude of these critical stress intensities for a given alloy can vary, and they also depend on

the load history and environmental conditions.

Because stress ratio is not a driving force for crack

growth, the data from Fig. 4a can be replotted in terms

Fig. 3 - Plot of threshold change in stress

intensity vs. stress ratio for an aluminum

alloy in lab air and in vacuum

Fig. 4 - [a] A plot of Kmax and ∆Kth vs. stress ratio,

and [b] the same data plotted as ∆Kth vs. Kmax

This new interpretation can be described using an

example from the behavior of a 2024 aluminum alloy,

for which larger amounts of data are available. Figures

4a and 4b show data obtained by Kemper et al. [34] in

laboratory air conditions for a 2024-T3 alloy over a

wide range of stress ratio (from –10 to +0.8). In Fig.

4a, values for both ∆Kth and Kmax are plotted against

stress ratio. The data show that ∆Kth decreased linearly

as R increased until the stress ratio reached approximately 0.4 and then leveled off. We have designated a

critical cyclic stress intensity [∆K*

th] when stress ratio

is greater than 0.4, a region where Kmax increases nonproportionally and ∆Kth is constant. This portion of the

plot is labeled as a ∆K-controlled region, where Kmax

must increase to maintain the minimum ∆Kth for crack
60 Naval Research Reviews

of ∆Kth vs Kmax. The result, Fig. 4b, shows an Lshaped curve that defines the region where crack

growth can occur. Critical values of ∆K*

th and K*max

are marked by the two arrows along the axes of the

plot. For 2024 alloy, the critical values of ∆K*

th and

K*max are approximately 1 MPa√m and 3 MPa√m,

respectively, for the given environmental and loading

condition. Such an interpretation is independent of

testing methods [28,29] and is applicable to many

types of materials. Figures 5a and 5b show a welldefined values for both ∆K*

th and K*max at low and

high temperatures for a variety of materials ranging

from monolithic alloys to composites. Plots such as

these can be developed from any constant crack

growth rate data that allow the extraction of ∆K and

Kmax values for the material.

Using these concepts, one can extend the analysis to

classify the fatigue crack growth behavior of a variety

of materials (including metals, ceramics, and polymers) that fall into five different types, where each is

distinguished by the relative contributions from the

environment and cyclic plasticity [35,36]. For a given

material, the classification scheme applies over the entire

crack growth rate and identifies the transitions of the

behavior from one class to another [35,36]. The analysis

presented above would move the fatigue analysis more

towards the intrinsic material behavior which is needed

for reliable life prediction analysis.

A literature survey indicates that several key studies

[37-42] have directly or indirectly noted the role of two

driving force parameters, ∆K and Kmax, in fatigue. (See

Table 1.) Wilkinson and Roberts [43] were able to

predict the ∆Kth vs. Kmax curve for copper using a

computer simulation of dislocation motion near a crack

tip. This is the first attempted theoretical work that

confirms the need for two parameters to model the

driving force for crack advance, without invoking crack

closure.

We have rationalized these observations of early investigators and proposed the following physical basisfor

crack growth:

• The two driving forces ∆K and Kmax are intrinsic to

the material behavior and are prerequisites to

uniquely defining the cyclic damage, independent

of closure

• The critical driving forces ∆K*th and K*max are

dependent on the alloy microstructure, slip mode

and environment

• The two driving forces are independent of testing

methods

• The long crack growth behavior represents a

material property for a given environment

Fig. 5 - Plots of Kmax and ∆Kth for a variety

of alloys

Short Fatigue Cracks

Research shows that long crack growth behavior does

not accurately describe all fatigue damage, including

phenomena such as short crack growth, accelerations

in crack growth due to underloads, and retardations in

crack growth due to overloads. These deviations from

long crack growth behavior are caused by internal

stresses present at the crack tip. In the following

sections we discuss the characteristics and the role of

internal stresses that affect the short crack behavior.

The ∆K and Kmax analysis discussed previously can be

extended to analyze the growth of short cracks.
Volume 50, Number 4 61

There is no clear distinction between the nucleation of

a crack and its early growth, primarily due to the lack

of a conceptual definition of a nucleation event and the

inability to measure such an event. Therefore, the

crack nucleation stage can encompass all the growth

events of a vanishingly small crack below the limits of

experimental detection. For the sake of discussion, one

can define a short crack to be one that has a length

below the limit detectable by means of non-destructive

inspection techniques.

are made. It has been noted [46,47] that all these types

of short cracks are indistinguishable in the experimental data scatter. Furthermore, they all show faster

growth rates at stress intensities that are much lower

than the long crack growth thresholds commonly

represented in terms of ∆K.

Table 1 - Research that has noted that the driving force for fatigue crack

growth depends on two parameters (specifically, on ∆K and Kmax)

Year Authors Alloy Remarks

Erdogan Model

Semi-empirical crack growth model that related the rate of crack propagation to the 

size of two plastic zones: the alternating plastic zone that depended on K, and the 

pulsating zone based on Kmax.

1968

Schmidt

and Paris

Aluminum

alloy

Explained the experimentally determined Kth-stress ratio using closure concept, but 

plotted Kth and Kmax thresholds vs. stress ratio. Indicated that Kmax was constant where 

Kmin was less than Kcl but when Kmin was greater than Kcl, Kth was constant. Did not 

plot K vs. Kmax.

1973

Priddle Mild steels Similar explanation to Schmidt and Paris. Plotted Kthvs. Kmax curves but interpreted 

the results to be consistent with the concept of oxide-induced closure. 1981

Doker

and Marci

Aluminum and

titanium alloys

First to plot data in the form of Kthvs. Kmax, but explained their experimental results 

using closure concepts. Implied that there are two critical thresholds.

1981,

1983

Lal and 

Namboodhiri

Aluminum and

steel

Analyzed fatigue thresholds without invoking crack closure on the basis that for 

stress ratios greater than 0.6 the crack growth is K-controlled and for stress ratios

less than 0.6 it is Kmax-controlled. Did not plot K vs. Kmax.

1990

Vasudevan and

Sadananda

All

materials

Explicit statements that two independent driving forces K and Kmax are required for

crack growth, independent of crack closure. Applicable to the entire crack growth 

region and to variety of materials. (Details given in the text.)

1993

Wilkinson

and Roberts Copper

First dislocation model using K and Kmax for threshold fatigue that predicts the need 

for the two parameters and gives a fundamental reason for their occurrence. Predicted 

da/dN vs. K and K vs. Kmaxwithout invoking closure.

1996

There are several definitions of short cracks [44,45],

including mechanically short (where the crack is

smaller than the plastic zone size), physically short

(where the crack is shorter than a long crack), and

microstructurally short (where the crack is shorter than

the grain size of the material). These classifications

have been given to understand and quantify the

behavior of these cracks, which are considered to be

representative of the intrinsic characteristic of the

material. In comparison, long crack behavior is

considered intrinsic only after the closure corrections

Experimental observations have pointed out that short

crack behavior is anomalous; the source of this

anomaly has been attributed to the lack of plasticityinduced closure in the wake of a crack that has negligible length. Deviations in the short crack growth

behavior also have been explained in terms of: (1)

intrinsic material behavior, (2) small crack sizes that

result in larger contributions from non-singular,

second-order crack tip stress fields, which decrease

rapidly with increase in crack length contributing to

the crack tip driving force (3) plastic zone size that is

larger than the crack length resulting in violation of

small-scale yield condition (4) crack closure, and (5)

microstructural obstacles such as grain boundaries

giving rise to data scatter with discrete jumps in
62 Naval Research Reviews

growth rates. Regardless of cause, however, the

breakdown of similitude for short cracks (as compared

to long cracks) clearly shows that equal crack tip

forces do not result in equal crack growth rates. For

mechanically short cracks, this breakdown of the Ksimilitude has been interpreted in terms of large-scale

yielding and closure. For microstructurally short

cracks, it is due to crack tip plasticity influenced by the

microstructure.

After some careful analysis of the short crack literature, one can consider the short crack behavior from a

different view [46,47]. This can be envisioned by first

defining the behavior of long cracks to represent the

basic material characteristics and then recognizing

some of the salient differences between the long and

short crack growth characteristics:

(1) we recall that plasticity originated from a crack tip

does not cause crack closure [27-30], thus eliminating

this phenomenon as an issue between the short and

long cracks,

(2) there is a large experimental scatter in the short

crack data compared to the long cracks [45],

(3) short crack growth data for multiple samples taken

from the same alloy or from the same alloy with

different microstructures (due to heat treatments) tend

to fall within the same scatter band when compared to

the respective long cracks [44,48] data,

(4) the crack growth rate decreases and then increases

or arrests with an increase in ∆K for the short cracks,

while there is only a monotonic increase in crack

growth with long cracks [44], and

(5) there is a reproducibly unique threshold value for a

given R-ratio for the long cracks with minimum scatter

in the data [45], while there appears to be no unique

threshold for short cracks [44,45].

• Growth rates for short cracks can either accelerate or decelerate; in some cases, the short

cracks can cease growing altogether

• Growth rates for short cracks eventually

converge with the crack growth rates obtained

for long cracks during constant amplitude

fatigue tests

Based on the above differences between long and short

cracks, we have postulated [46,47] that:

Fig. 6 - Perturbations in long crack

behavior due to tensile or compressive

internal stresses

Figure 6 illustrates some examples of crack growth

behavior that deviate from the normal growth of a long

crack, in terms of a crack growth per cycle (da/dN) vs.

Kmax. These examples include short cracks growing

from a notch or other region of stress concentration, as

well as long cracks that have been subjected to a single

underload or overload. These plots clearly show two

phenomena:

• Short cracks should also satisfy the same long

crack thresholds for their growth

• Pre-existing internal stresses, or those that are

generated in-situ during cycling, contribute an

additional driving force that helps short cracks

satisfy the same long crack thresholds required

for crack growth.

• The two long crack growth thresholds ∆K*

th

and K*

max represent unique fatigue crack

thresholds that are applicable for a combination

of material, environment, and microstructure

• For a given microstructure and environment,

the long crack growth behavior under constant

amplitude loading represents the material crack

growth curve
Volume 50, Number 4 63

Hence, the assumption about the breakdown of similitude between short cracks and long cracks can be

rejected. The apparent anomaly in the short crack

growth behavior mainly arises from an insufficient

description of the total crack tip driving forces (due to

the lack of recognition of the internal stresses) and

from not considering the second threshold requirement

[K*

max] in addition to the normally considered threshold of [∆K*

th].

Tanaka et al. [49,50] have systematically studied the

crack growth behavior of short cracks nucleated from

notches, free surfaces, and specimen corners. Figure

7a shows crack growth data for small and long cracks

in a variety of specimen geometries in low carbon

steel. Interestingly, one can note that the short crack

growth data from free-surface specimens (designated

by open figures) fall within the same scatter band of

data for the sharp notch specimens (darkened figures).

While the internal stress gradients from notches can be

readily identifiable, the existence of such high stress

gradients is not obvious for free-surface specimens.

However, internal stress gradients must be present to

induce or sustain short crack growth. Therefore, we

postulate that for cases where there are no pre-existing

internal stresses, these gradients are generated in situ

by irreversible plasticity during cycling and are

therefore unavoidable. For example, internal stresses

can manifest themselves as intrusions and extrusions,

slip band intersections, etc., which lead to crack

nucleation and then short crack growth.

Figure 7a also shows crack growth data for long cracks

in the same material. The amount of internal stress can

be computed as the difference between the driving

forces for the long and the short cracks; this difference

is shown schematically in Fig. 6a as ∆Kint. When the

values for ∆Kint are normalized by the values for ∆KIc

for long cracks taken at the same growth rate, and then

plotted against a parameter (∆Ksc)

2, which is proportional to the length of the short crack, as in Fig. 6b,

then we can see that the functional form of the internal

stresses for all specimen geometries are similar.

This suggests that the internal stress gradients also

occur in smooth specimens (for example, via the

formation of intrusions and extrusions) and that all of

the different types of internal stresses decay rapidly

with distance from the source and show similar

functional forms. Therefore, short cracks grow in the

presence of internal stress fields that can accentuate

applied stresses to produce the driving force necessary

for crack growth to occur. From these analyses, one can

reach an important conclusion that there is no violation

of the similitude between the long and short cracks. This

analysis also can be extended to other types of deviations from the long crack growth, such as those arising

from overloads and underloads. We can then restate the

similitude concept with a modification:

• Equal crack tip forces cause equal crack growth

rates if all factors that contribute to the crack tip

driving forces are considered and if the crack

growth mechanism remains the same.

Type and Nature of Internal Stresses

Internal stress fields [47] can be either short-range or

long-range. Short-range stress fields occur in a material

near point defects such as vacancies or interstitials but

do not significantly affect the growth of short cracks. It

is important to consider long-range stresses, however,

because they do contribute to the growth of short cracks.

Fig. 7 - [a] Crack growth data for long and

short cracks in low carbon steel, and [b]

the same data normalized by long crack

data and then plotted against (∆Ksc)

2
64 Naval Research Reviews

must include all of the stresses applicable to the situation,

both applied and internal.

Stacey and Webster [51] studied the effect of compressive

residual stresses on long crack growth in an autofrettaged

thick-walled tube made of a low alloy steel. In addition,

measurements of compliance were used to determine the

stiffness change in the material due to the autofrettage

treatment. The fatigue

samples for this research

— some with the residual

stresses, and some without

— were sectioned from the

tube; each sample had a

crack starting at the inner

wall of the tube. Figure 9a shows the long crack growth

data for the two conditions. The difference between the

two curves gives the magnitude of the internal stresses

(∆Kint) that are induced by the autofrettage treatment.

Figure 9b shows the contribution of internal stresses to the

overall stress intensity as a function of crack length.

Figure 9c shows the values of stress intensity due to the

residual stresses (calculated from compliance measurements, using a weight function method).

AUTOFRETTAGE. A technique for

fabricating thick-walled tubes that

uses high pressure inside the tube to

induce compressive stresses on the

tube’s inner wall.

Fig. 8 - Some examples of internal stresses that affect fatigue life

Long-range stresses can arise from a number of

sources, including:

• dislocation pile ups at low- and high-angle

grain boundaries, with stresses varying with

distance as ƒ[1/r],

• microvoids and inclusions, with stresses

varying with distance as ƒ[1/r3],

• cracks-tip stresses, which vary as ƒ[1/√r],

• persistent slip bands,

• notch tip stress fields,

• transformation-induced stresses,

• residual stresses from cold working, shot

peening, ion implantation, machining, or other

mechanical processes, and

• thermal stresses from quenching, welding, etc.

The generic term “internal stress” encompasses all of

these types of stresses, some of which are depicted in

Fig. 8. A correct assessment of the crack tip forces
Volume 50, Number 4 65

A comparison of Figs. 9b and 9c shows that the internal

stresses estimated from the crack growth data are in

agreement with the results from calculated internal

stresses using the weight function method. This confirms that the compressive stresses created by the

autofrettage process generate the internal stress gradients necessary to reduce the crack tip driving forces.

Note that the length scales of the internal stress gradients are of the order of a few millimeters. High-precision methods like low angle X-ray and neutron diffraction may be used to measure the internal stress gradients ahead of a crack tip. To date, however, such results

are not available.

Linking Crack Nucleation to Short Cracks

and Long Cracks

FATIGUE LIMIT ∆σe.The level of

cyclic stress needed to nucleate a

crack. Once nucleated, a crack will

grow rapidly to failure unless ∆σe

is reduced continuously.

Fig. 9 - [a] Long crack growth data for low alloy

steel tubing (with and without autofrettage

treatment), [b] the contribution of internal stresses

to overall stress intensity as a function of crack

length, and [c] the value of stress intensity due to

residual stresses

Figure 10 uses this technique [55] to plot Usami’s data on

13Cr-steel alloy at different stress ratios [56]. The linear

portion of the change in stress (∆σ) vs. crack length (a)

Kitagawa and Takahashi found that for cracks that

were shorter than the critical crack size a*, the threshold condition for crack growth could be represented by

a critical stress ∆σth that approaches the smooth tensile

fatigue limit ∆σe for vanishingly small crack sizes. For

such short crack lengths, the fatigue limit is essentially

independent of the crack length. On the other hand, for

cracks that were longer than a*, the specimen’s crack

growth and fatigue life are dictated by the threshold

stress at a given stress ratio, ∆Kth ƒ(R). The intersection between the long and the extremely short crack

region defines the critical crack length a* for a given

stress ratio R. The critical crack length a* marks the

point where the growth behavior of short cracks

transitions to that of the long cracks. One can infer

from their analysis that a*ƒ(R) = (1/π) [∆Kth/∆σe].

Other researchers have attempted to link critical crack

size with characteristic microstructural scales in a

variety of materials [53,54].

Kitagawa and Takahashi [52] proposed a graphical

method to link the data for short and long cracks with

the fatigue limit ∆σe

, which was obtained from level of

stress vs. cycles to failure data for smooth fatigue

specimens. From a

Kitagawa-Takahashi

diagram (such as the

one shown in Fig. 10),

one can extract a

parameter called

critical crack size (a*).
66 Naval Research Reviews

plot at long crack lengths is given by linear elastic

fracture mechanics as ∆Kth = ∆σth √πa. At smaller

crack lengths, the curve approaches the fatigue limit

∆σe. As can be seen in Fig. 10, the parameters of

fatigue limit (∆σe), threshold change in stress intensity

(∆Kth), and critical crack size (a*) are all functions of

stress ratio R. As stress ratio R increases, the values of

a*, ∆Kth, and ∆σe decrease.

Each curve in Fig. 10 represents data for a single stress

ratio; data for different stress ratios can be collapsed

into one curve by normalizing each of the ordinates

with the respective ∆σe and a*. This unique curve is

characteristic of the material-microstructure-environment linking the region of crack nucleation and short

crack growth to the realm where long crack growth

occurs. As Usami has shown [56-58], a change in the

environment would affect the thresholds and therefore

shift the curves in the Kitagawa diagram.

As previously shown in Fig. 4, ∆Kth approaches a

limiting threshold value ∆K*

th at high stress ratios, and

at low values of stress ratio Kmax approaches K*max.

Similarly, one should expect asymptotic values for

∆σe. The two parametric requirements under the

nucleation (level of stress vs. cycles to failure) curve

are complicated by contributions from compressive

plastic flow, which have been addressed by Goodman

[59]. The question as to how the two parametric

requirements for crack growth in terms of ∆K and Kmax

translate into two parametric requirements in level of

stress vs. cycles to failure curves needs further investigation. Needless to say, at least for positive values of

stress ratio R, the Kitagawa diagram directly links the

behavior of the fracture mechanics specimen [crack

growth per cycle (da/dN) vs. change in stress intensity

(∆K)] to fatigue limits in smooth samples (obtained

from level of stress vs. cycles to failure curves) as a

function of stress ratio R.

There are additional physical connections between

short and long cracks that can be explained in terms of

the contribution from internal stresses. When the long

crack portion of the curve is extrapolated to the short

crack region (i.e., below the critical crack size a*); we

observe that an extremely short crack should be

growing at an enormously high stress level. The

Kitagawa diagram, however, indicates that applied

stresses approach a plateau, where short cracks nucleate and grow at the stress level of a fatigue limit ∆σe.

We have shown that the same thresholds are valid for

both short and long cracks. Hence internal stresses

must be generated in situ (if they are not pre-existing)

to allow the nucleation and growth of short cracks to

take place at the fatigue limit. The triangular region

defined by arrows in the upper left portion of Fig. 10

defines the minimum internal stress and the gradient

required for a short crack to grow at the fatigue limit

∆σe. For this case, the internal stresses follow a √r

dependence to satisfy the thresholds required to

produce crack growth. Thus the requirement for the

internal stresses follows directly from the Kitagawa

diagram; this is also consistent with the arguments

presented previously with reference to Fig. 5. Therefore, the two parametric analyses provide a unified

framework [55] for fatigue damage evaluation from the

stages of early crack nucleation, through the growth of

short cracks, to long crack growth.

Fig. 10 - Kitagawa-Takahashi diagram (i.e., a plot of crack

length vs. the threshold change in stress needed to

generate crack growth) for 13Cr-steel alloy

When the length of a crack is greater than the critical

crack length a*, applied stresses are sufficient to

sustain the crack growth with the aid of internal

stresses. The crack growth behavior then merges with

the long crack growth. Physically, a*

 defines the

critical crack size to which a short crack has to grow

before it can be treated as a long crack. It is in this

regime that the internal stresses either have to be

provided externally via stress concentrations or to be

generated internally via slip process. Because (1) both

nucleation and growth of short cracks occur in the

presence of augmented internal stresses, (2) a* is well

below the NDI limit for crack detection, as noted in
Volume 50, Number 4 67

Fig. 10, and (3) the demarcation between nucleation

and short crack growth stages is rather arbitrary; we

can conveniently define the crack nucleation stage to

occur until the crack length a reaches a*.

Summary

A unified approach to fatigue damage analysis has

been developed involving two driving forces for crack

growth: the change in stress intensity (∆K) and the

maximum stress intensity (Kmax). The analysis shows

that ∆K vs. Kmax curves represent the fundamental

materials behavior that is applicable to both long and

short cracks. Short cracks grow at different rates than

long cracks due to the presence of internal stress

gradients that primarily affect Kmax (an intrinsic

parameter in the unified framework for fatigue) and

not ∆K. Also, despite a wide diversity in the types of

internal stresses, their gradients are similar; this

indicates that it should be possible to predict the

magnitude of internal stresses using analytical methods. Thus, for a more reliable life prediction methodology one requires three main parameters: ∆K, Kmax and

Kinternal. Efforts must be focused to use these parameters to develop such models.

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15. A. Navarro and E.R. de los Rios, Phil. Mag., 10, 15-36 (1988).

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18. R.G. Forman, V.E. Kearney, and R.M. Engle, J. Basic Engg.,

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19. J. C. Newman, Jr, “FASTRAN-II: A Fatigue Crack Growth

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22. L. Lazzeri, A. Pieracci, and A. Salvetti; (Paper delivered at

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23. J. Harter, personal communication (1998).

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26. A.K. Vasudevan, K. Sadananda, and N. Louat, Scripta Metall.

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27. N. Louat, K. Sadananda, A.K. Vasudevan, and M. Duesbery,

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The Authors

Dr. K. Sadananda is a researcher in the Materials

Science and Technology Division at the Naval Research

Laboratory in Washington, DC. His broad areas of

expertise include the mechanics of dislocations behavior,

deformation in high-temperature order alloys and

intermetallics, alloy microstructure-load-environmental

interactions in gas turbine engine materials, creep-fatigueenvironmental interactions, and nanocrystalline coatings

for wear and corrosion applications. Dr. Sadananda has

published more than 150 articles covering many areas of

mechanics and metallurgy, and he has made more than

100 invited presentations in Europe, Asia, and the United

States. He has been awarded four NRL research

publication awards for papers published in International

Journal of Fracture, Metallurgical Transactions, and Metal

Science Reviews. Other awards include the Sigma Xi

Applied Science award in 1980 and the Burgess Memorial

Award from the Washington, DC, Chapter of ASM

International in 1998.

Dr. A.K. Vasudevan is a Scientific Officer at the Office

of Naval Research (ONR) Materials Science Division in

Arlington, Virginia. He has initiated and managed basic

and applied technical programs for high temperature

engine materials, casting, diamond films, Ospray

processing, condition-based maintenance, and

nanocrystalline materials for coating applications. Prior

to joining ONR in 1988, he was employed at ALCOA

Research Labs in Pittsburgh as a scientist developing high

strength aluminum alloys for aerospace and automobile

applications. These aluminum alloys are currently being

used in the Airbus A-333 and MD-11, as well as various

military aircraft, helicopters, and racecars. Dr. Vasudevan

has been actively involved with committees related to

various NATO, U.S. Air Force, and NASA technical

programs. He has published extensively and has given

talks in Europe and the United States on topics relating

to mechanical and physical properties associated with

microstructural features.
Profiles in Science

Dr. K. Sadananda

Materials Science and Technology Division

Naval Research Laboratory

Washington, DC

D

r. K. Sadananda has been one of the Naval Research Laboratory’s foremost researchers for the

past two decades. He has made extensive contributions in several areas of research related to

aircraft alloys, including high-temperature creep damage; dislocation theories that describe the

deformation of alloys; and fatigue damage in metals, ceramics, and composites. In addition, he

has developed specifications for the rejuvenation of turbine blades damaged by creep during service.

Dr. Sadananda’s broad areas of expertise include the mechanics of dislocations behavior, deformation in

high-temperature order alloys and intermetallics, alloy microstructure-load-environmental interactions in gas

turbine engine materials, creep-fatigue-environmental interactions, and nanocrystalline coatings for wear and

corrosion applications. He is currently involved in developing a unified theory for fatigue damage that encompasses damage from the crack nucleation stage to final fracture.

Dr. Sadananda has published more than 150 articles covering many areas of mechanics and metallurgy, and

he has made more than 100 invited presentations in Europe, Asia, and the United States. He has been awarded

four NRL research publication awards for papers published in International Journal of Fracture,

Metallurgical Transactions, and Metal Science Reviews. He received the Sigma Xi Applied Science award in

1980 for “his pioneering research on the understanding and development of new concepts on high temperature

crack growth in metals.” In 1998, he was presented the Burgess Memorial Award from the Washington, DC

Chapter of ASM International for his “outstanding research contributions and management of research in the last

five years.”
Naval Research Reviews, a theme-issue quarterly journal,

publishes articles about Department of the Navy Science

and Technology programs conducted by academia,

government laboratories, for-profit and nonprofit

organizations, and industry.

The journal is available on the World Wide Web at

http://www.onr.navy.mil/onr/pubs.htm

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