Fatigue of Materials

Dr. Richard Chung

Department of Chemical and Materials Engineering

San Jose State University

Learning Objectives• Explain why the fatigue problems are more profound in

polymers and metals than ceramics and composites• Describe three stages of a fatigue process in a material (crack

nucleation, crack growth and crack propagation)• Discuss how the fatigue crack propagation is determined by

the relationship between dC/dN (crack advance rate) and ΔK (cyclic stress intensity factor)

• Design and use a material having ΔK is less than ΔKth (fail-safe failure mode)

• Examine and discuss the physical meaning of striation formed on the fractured surface and determine the crack advance between cycles

• Determine the conditions of the slow crack growth region and fast crack growth region of the fractured surface

• Find the relationship between cyclic stress (or strain) amplitude and number of cycles to help design fatigue resistant material or applications

What is fatigue?

An engineering structure is often subjected to the repeated application of a stress below its yield strength of the material.

This cyclical stress may occur in the form of rotation, bending, or vibration.

Fatigue Testing

A common test to measure a material’s fatigue properties is to use a rotating cantilever beam.

A cylindrical beam is mounted in a motor-driven chuck with a load applied from the opposite end.

A fatigue mode (a sinusoidal cycle) of C-0-T-0 is repeatedly applied to the beam.

The maximum stress acting on the beam is governed by the following equation:

where l is the length of the beam, P is the load, and d is the diameter of the beam.

3

18.10

d

P

Example 1: A solid tool-steel shaft must be 96 inch long and must survive continuous operation for one year with an applied force of 12,500 pounds. The shaft is rotating one revolution per minute during operation. Design a shaft that will meet these requirements.

Solution:

No. of cycles= (1 cycle/min)(60 x 24 x 365min) = 5.256 x 105 cycles/yr.

From figure 6-19, the applied stress is around 72,000psi

Example 1: A solid tool-steel shaft must be 96 inch long and must survive continuous operation for one year with an applied force of 12,500 pounds. The shaft is rotating one revolution per minute during operation. Design a shaft that will meet these requirements.

Solution:No. of cycles= (1 cycle/min)(60 x 24 x 365min) = 5.256

x 105 cycles/yr.From figure 6-19, the applied stress is around 72,000psi

d = 5.54 inches Add a safety factor to the system:

d = 5.54 x1.05 = 5.82 inches

3

)12500.)(96)(18.10(000,72

d

lbsinpsi

TerminologyEndurance limit: is the stress below which that

failure by fatigue will never occur, this is a preferred design criterion.

Fatigue life: indicates how long (no. of cycles) a component survives a particular stress.

Fatigue strength: is applicable to a component has No endurance limit. It is the maximum stress for which fatigue will not occur at a particular number of cycles, in general, 500 million cycles.

Endurance ratio: the endurance limit is approximately ½ the tensile strength.

Definitions

max

minRatio Stress

R

R

RA

m

a

1

1Ratio Amplitude

A

AR

1

1

R

RA

1

1

22Stress gAlternatin minmax

Ra

2StressMean minmax

m

The difference between Point Stresses () and Nominal Stresses (S)

For simple axial loading, = S For bending, S

S=Mc/I; where M =bending moment, c = the distance from neutral axis to edge, and I = the area moment of inertia about the axis.

For notched specimen (No yielding), = KtS

where Kt = elastic stress concentration factor.

Stress vs. Life (S-N) Curves

An equation can be derived to represent an S-N curve

fa NDC logDfa NC log

DfC

Df

C

Ne

e

Ne

a

a

Bfa AN

bffa N )2('

A = 2b ’f B = b

Bfa AN

Plotting in Linear vs. Logarithmic scales

Three factors are necessary to cause fatigue failure:

A maximum tensile stress of sufficiently high value

A large enough variation or fluctuation in the applied stress

A sufficiently large number of cycles of the applied stress

Variables Affecting Fatigue In A Material

Stress concentration Corrosion (Environment) Temperature Overload Metallurgical structure (Microstructure) Residual stress (shot peening, presetting) Combined stress Surface condition

Stress Amplitude versus Mean Stress

Mean stress effects can be plotted in a diagram using stress amplitude versus mean stress.

Estimates of mean stress effects for un-notched specimens can be determined by Morrow equation or SWT equation (Smith, Watson, and Topper)

where ar= equivalent completely reversed stress f

m

aar

'1

aar max

The Palmgren-Miner Rule

13

3

2

2

1

1 jf

j

fff N

N

N

N

N

N

N

N

The fatigue failure of a material under a variable (multiple) amplitude loading is expected when such life fractions sum to unity.

In the case of creep-fatigue, a fracture criterion will be defined as:

1if

i

if

i

t

t

N

N

Initiation of Fatigue CracksF

ract

ure

Tou

ghn

ess

(KIC

)

Yield Strength (y)

Design based on strength

Design based on toughness

Crack Initiation and Propagation

Max

imu

m C

ycle

Str

ess

Number of Cycles

Ni Np

Ni Nt = Ni +Np

Ni = # of cycles for initiationNp= # of cycles for propagation

Crack Rate As A Function of The Stress-Intensity Range (K)

• When the length of the crack is small, the growth rate of the crack (a/N)is also small

• As the the length of the crack increases, the growth rate of the crack (a/N) also increases

• Under identical cyclic loading, larger initial cracks propagate to failure in short cycles

•

slope theis consatnt, theis where

)(

mA

KAdN

dc m

Example: A metal strip (4 inch wide and 0.2 inch thick) is loaded in a cyclic loading ranging from 6,000 to 43,000 lbs. A crack is found located in the center of the strip that extends through the thickness. For c = 0.1 and 0.4 inch , calculate K.

Answer:

Assume the geometry factor F = 1 cFSK i

inKsiKKK

inKsicSK

inKsicSK

c

inKsiKKK

inKsicSK

inKsicSK

c

psix

S

psix

S

9.514.84.60

4.8)4.0(500,7

3.60)4.0(750,53

0.4inchfor nscalculatio Repeat the

9.252.41.30

2.4)1.0(500,7

1.30)1.0(750,53

0.1inchfor rangeintensity stress Compute

500,742.0

000,6

750,5342.0

000,43

min2max22

minmin2

maxmax2

min1max11

minmin1

maxmax1

min

max

The Walker Equation

1)1(1

)1(1

11

1

1max

max

min

mm and )1(

)1(

)1(

)(

constant material a is γ

)1()1(

1

1

1

1

1

m

m

m

m

m

R

AA

KR

A

dN

dc

R

KA

dN

dc

KAdN

dc

R

KRKK

S

SR

The Forman Equation

• The R ratio has strong effects on the behavior of slow growth rate

•

))(1(

)(

)1(

)(

max

2222

KKR

KA

KKR

KA

dN

dc

c

m

c

m

thR

KK th

th

1)1(

Fatigue Life

dcdc

dNN

f

i

c

c

if )(

2)(m )2/1()(

2/12/1

mSFA

ccN

m

mi

mf

if

Summary• Fatigue failures are often focused on metals and

polymers.• Endurance limit and fatigue life can be used to help

prevent fatigue in materials• When applied stress magnitude increases, the

number of cycles decreases• Materials can fail by fatigue even when they contain

no cracks• A fatigue failure is based on the accumulation of

fatigue cycles used at low and high cyclic stresses• A final fractured surface resembles a ductile failure

pattern which includes three distinct stages: crack nucleation, crack growth and crack propagation.

• The spacing between the beach marks corresponds to the crack advance per cycle in materials.

Summary (cont’d)• As long as ΔK is less than ΔKth , the crack growth rate is not

going to increase• Using a material at ΔK value (less than ΔKth), a fail safe

fatigue design can be achieved.• The slow-crack growth surface in metals is generally smooth,

unless oxidation or abrasion have already developed.• The fast-crack region on a fractured surface is observed as

dull and fibrous resembling a tensile ductile failure.• Polymers can develop εDCG (shear band cracking

accompanied by crazing) prior to DCG (discontinuous crack growth)

• Temperature will have great effects on polymer and metal fatigue. High frequency of loading could yield hysteric heating and thermal softening which could significantly reduce the fatigue life/endurance limit in a material. Creep-fatigue could be a complicated process for material failure prediction.