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7. Fatigue Fracture

Fracture surface of a bicycle spoke made of 7075-T6 aluminum alloy

25 × magnification

100 × magnification

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Introduction:

When metals are subjected to fluctuating load, the failure occurs at a stress level much lower than the fracture stress corresponding to a monotonic tension load.

With the development of the railway (<1900) much attention was given to the understanding of the fatigue failure phenomenon.

August Wöhler (1819-1914), director of Imperial Railways in Germany from 1847 to 1889 - First investigator to address fatigue tests on railway axles and small-scale specimens.

- Provided plots of the failure stress as a function of the number of cycles to failure

Useful for the total life prediction of a part subjected to constant amplitude cyclic loading: Wöhler S-N diagrams

The Wöhler approach was next extended to other areas : bridges, ships, machinery equipment …

It is still used to assess fatigue failure of modern structures (e.g. aircraft components) subjected to repeated loading.

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High-cycle fatigue when material failure occurs under a large number of cycles ( > 100000 ) :

Strains and stresses are within the elastic range.

Low-cycle fatigue failure (100-100000 cycles) :

Magnitude of fluctuating stresses does not remain in the elastic range.

Cannot be characterized by S-N curves.

To prevent fatigue failure of a structural part, in general ones needs :

(1) All loading events that a component will experience + number of times that each one occurs.

Laboratory tests for constructing the S-N diagram in given environmental conditions (inertial, thermal, pressure stresses …).

Calculation of the remaining life cycles

Significant plastic straining may occur throughout the structure, especially at stress concentrators.

(2) Empirical equation relating the fatigue crack growth da/dN with the crack tip SIF K .

(3) Material fracture toughness.

(4) Some estimate of the initial flaw size.

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1) Constant amplitude cyclic loading :

Five stress parameters that can defined the loading characteristics:

Cyclic stress amplitude :

Maximum stress :

Minimum stress :

max

min

min

max

min

max minS Alternatively, 2a max minS

SSa

Mean stress: 2mean max min

Stress Ratio: min maxR

Amplitude and mean stress stay constant.

Any two of the above quantities are sufficient to completely define cyclic loading.

Cyclic or Fluctuating load :

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Typical constant amplitude loading cyclesSt

ress

Time

1R

Stre

ss

Time

0R

Stre

ss

Time

0 1R

Stre

ss

Time

R St

ress

Time

1R

min maxR

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Stre

ss

Time

1R

For the same range, R=-1 is considered to be a less damaging cyclic load when evaluating the fatigue life of a structure.

Rotating-bending test using 4 point-bending to apply a constant moment to a rotating cylindrical specimen.

Example:

The maximum and minimum cyclic stresses are equal and 0mean

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Stre

ss

Time

0R

The minimum stress equals zero.

Pressurization and depressurization cycle of a pressurized tank

Example:

Stre

ss

Time

0 1R

Both maximum and minimum stresses are positive.

Example:

Preloaded bolt subjected to cyclic tensile stresses such that the minimum and maximum fatigue stresses are positive

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Stre

ss

Time

1R

R

Both maximum and minimum stresses are negative.

For the same stress range, R>0 is considered to be the most damaging cyclic load when evaluating the fatigue life of a structure.

Example:

Plate with a hole that undergone a sleeve cold expansion

= mandrelizing process that creates a massive zone of compressive residual stress field

fatigue tensile load applied

R= 1 : static loading

Mandrelized hole under fluctuating load with max =0.

The minimum stress is negative.

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Typical cyclic loads in real structures almost random in nature and vary in magnitude .

Stre

ss

Time

2) Random loading :

Complex and may contain any combination of the above cyclic cases.

Loading environment of a aircraft or space structure during is lifetime.

Example:

3) Fatigue spectrum :

All loading events and the number of times that each event occurs are reported .

Each event may be a function of several variables.

The irregular load sequence Sum of cycles Ni with associated Si and imean

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Different methods to establish a fatigue spectrum from time history data :

Load environments for a space component:

• Prelaunch cycles (acceptance , proof testing, …)

• Transportation cycles prior to flight

• Flight cycles

• On-orbit cycles due to on-orbit activities

• Thermal cycles

Rain flow method Fatigue and Fracture Mechanics of high risk parts, Bahram Farahmand (1997)

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Constant amplitude axial fatigue tests :

For the determination of the fatigue life of a metallic part:

- high number of cycling

- predominantly elastic stresses

Laboratory fatigue tests with typical types of specimens:

Recommended by the ASTM E-466

Designed such failure occurs in the middle region.

Diameter : D : 0.25- 2.5 cm

L : 5.3 cm, R : 10.16 cm

Area Wt : 0.07- 2.5 cm2

2<W/t <6

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The S-N diagram

Principle of similitude:

The life of a structural part is the same as the life of a test specimen

if both undergo the same nominal stress

Service loading of a bridge exposed to a fluctuating load:

Laboratory fatigue specimen subjected to the same nominal stress

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S-N curve definition:

Plot of stress amplitude S versus the number of cycles to failure Nf

Nf : dependant variable

maximun stress max

Endurance limit or fatigue limit

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• Other representations:

Semi-logarithmic plotting :

Log N

Log NLog max or Log S

andmax or S

Logarithmic plotting : and

most widely used

Typical S-N curve for ferrous alloys

Log

S

Log Nf

Endurance limit

- is not a constant and generally varies with R

- depends on the types of load: lower in uniaxial loading than in reverse bending

• Endurance limit:

- Affected by the degree of surface finish, heat treatment, stress concentration, corrosive environment

A well-defined fatigue limit is not always existing.

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S-N diagram for 4340 alloy steel:

Endurance limit of 43.3 ksi (1 ksi= 6.895 MPa) with R = -1

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S-N diagram for 2024-T4 aluminum alloy

Endurance limit not well defined.

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Linear cumulative damage

S-N diagram useful to determine the number of cycles of failure with a given constant amplitude applied cyclic stress .

Fatigue damage at a given stress level i

f i

n

N

= number of cycles applied at that stress level

Nfi = total number of cycles to fail the part at the same stress level

Total failure: 31 2

1 2 3... 1

f f f

nn nN N N

Total damage = result of several fluctuating stresses at different levels when

The contributing damage caused by each load environment should be evaluated.

1i i in N N

Palmgren-Miner rule (1945)

Based on the linear summation rule for damage (see eq. 2.12)

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• Application of the Palmgren-Miner rule:

A component of space structure made of allow steel :

- is subjected to fluctuating loads with different stress magnitudes:

- has the following S-N curve:

See launch and on-orbit fatigue spectrum given table 1

0 419 65 2 85 1 6 13

.maxlog N . . log R . R = 0, -0.1, -0.3 , -0.5 , -1

Including a safe-life factor of 4, will the part survive the load environment (R = -1) ?

1 ksi = 6.895 MPa

Decreasing R

(ksi

)

Decreasing R(ksi

)

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The number of cycles to failure, Nfi associated to the loading step i is given by:

0 419 65 2 85 1 6 13

max

.ifilog N . . log R .

First step:

1 20 5max . ksi 1

1

34 6

7.5 5f

nE

N E

1 7 5 5fN . E

Idem for the other loading steps (launch + on-orbit + thermal)

i

f i

n

N

Sum all

Multiply the expression by the safe-life factor of 4

1 3n

Check if the summation <1

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Stress Intensity factor range and crack growth rate

Time-dependent mode I loading

Fatigue crack propagation investigated using the fracture mechanics concepts:

The SIF factor is now time dependant :

IK t t a f a b

We define a maximum / minimum SIF

Imax

maxK a f a b

Imin

minK a f a b

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Time to grow the crack up to a certain length or until fracture of the part ?

Problem :

2IC thI Y

I I

K ΔKΔa ΔK σ, , ,ν,R,....

ΔN E ΔK ΔK E

F

The crack propagation law can be expressed by

Δa

ΔN: Average crack speed

Small scaling yielding

if

SIF characterizes the crack tip field.

Similitude principle: same behavior for cracks with the same FIC.

ICK Fracture thoughness

then

max minIΔK I IK K max min a f a b

thΔK : threshold value of SIF No crack growth below

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Number of cycles to grow the crack from a0 to a:

0

0 2IC thI Y

I I

N-NK ΔKΔK σ

, , ,ν,R,....E ΔK ΔK E

a

a

da

F

Difficult to derive theoretically !F

empirically obtained in order to fit experimental data

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Typical crack growth data and the curve fit for 2024-T861 aluminum alloy

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Typical fatigue crack growth behavior

Region II can be described by such a power law:

naA K

N

Earliest relation of Paris and Erdogan (1960), widely used.

Observed for a large class of materials

Region 1: slow crack advance

Region 2: intermediate, higher speeds

Region 3: fast crack growth, very short in time

thΔK : threshold value of SIF Similar to the endurance limit

A and n empirical parameters (see table 7.1)

85-90% 5-8% 1-2%

approximate % of life spent

is lineara

log( )N

versus log( K )

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Integration of Paris & Erdogan law:

0

0N-Na

na

da

A K

00 0IK a f a b

max min we have,Noting

IK a f a b 0

0II

aK K

a 0f a b f a bassuming that

Therefore by integration,

0

0 20 0N-N ln 2

I

a aif n

aA K

2 1

00

0 0

2N-N 1 2

2I

n

n

a aif n

n aA K

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Failure occurs at the crack length ac when

Imax

max c c ICK a f a b K

Solving for ac and reporting in the previous equations:

One obtains Nc : number of cycles to fracture

Others models than the model of Paris & Erdogan are reported:

More general, valid in all regions, takes into account the ratio R, the fracture toughness KIC , the limit thΔK

Forman-Newman- de Koning (FNK) law (1992):

1 1

1 11

pnn th

qn

IC

KC( f ) K

Ka

N KR

R K

f is a function that models the crack closure effect.

C, n, p, q are empirically constants

widely used in aerospace structures for life estimation of high risk fracture critical parts.