FORMULA SHEET: Fatigue, Fracture Mechanics

Structural Mechanics

σ i=M x y i

I xx−

M y x i

I yy+ FA

ε x=σ x

E− νE (σ y+σz )+α ΔT

Fatigue: Machined Components

SmN=CS1mN1=S2

mN2m=

log( N2

N1)

log( S1S2 )N R={( S1Sa )

m

N 1 0.9 f ut≥SR≥Se

∞ SR<Se

Endurance limit estimates:

Se={0.25BHN ksi for BHN ≤400100 ksi for BHN>400

Steel

Se={ 0.5Sut for Sut≤200 ksi(1400MPa)100 ksi(700MPa) for Sut>200 ksi(1400MPa)

Cast Iron + Cast Steels:

Se={ 0.45Sut for Sut≤600MPa275MPa for Sut>600MPa

Stress concentrations

σ max=K t σK t=σ sc

σK f=

Se(un−notched )

Se(nothced )

K f=1+K t−1

(1+ ar )a=[ 300f u [ksi ] ]

1.8

×10−3∈.

K f' −1

K f−1=f ( f ut)

S1000' =

S1000K f

' Se' =

Se

K f

Mean stress:

σa

Se+σm

Sut=1

Modifying factors

Se=Se' C¿C loadC surf CTC rel…¿S103=S103

' CloadCTC rel

Size:

C¿¿{ 1.0 ,if d≤ 8mm1.189d−0.097 , if 8mm<d≤ 250mm

¿

Load:

Se ,axial=0.70 Se , bendingC load=0.7 if S−N curve is¿ bendingtests

Temperature:

CT=1.0 for T ≤450℃

1−5.8−3 (T−450 ) , for 450<T ≤550℃Reliability:

Fatigue: Large scale manufactured componentsN R=¿NC=2×10

6 cyclesN D=5×106 cycles

N L=100×106 cycles

Partial factor for fatigue:

Assessment Consequence of failureLow High

Damage tolerant 1.0 1.15

Safe life 1.15 1.35

Temperature:

ΔC , HT=ΔσC

EHT

E20° C

Grinding & TIG dressing:

Steel:

Δ σC=min {1.3×ΔσC

112Aluminium:

Δ σC=min {1.3×ΔσC

45Peening:

0.8

0.6

0.4

Δ σc=¿

Fracture MechanicsUniversal equation: K=βσ √πa

Plastic collapse: F pc=f y

Anett

Fracture:

K=K IcK Ic=βσ √π a fa f=1π (K Ic

βσ )2

σ f=K Ic

β √π a fr

acri=min {apc for plastic collapseaf for fracture

Stress concentration factors:

Centre cracked plate:

β=1+0.256 ( aW )−1.152( aW )

2

+12.2( aW )

3

Single edge crack:

β=1.12−0.23( aW )+10.56( aW )

2

−21.74( aW )

3

+30.42( aW )4

Double edge crack:

β=1.12+0.43( aW )−4.79( aW )

2

+15.46( aW )

3

LEFM

B ,W−a ,a≥2.5( K Ic

f yt)2

W ≥5.0( K Ic

f yt)2

Fracture toughness estimation: K IC=11.4 √C v

Lower limit: K IC=21.6 (C v )0.17

Note, C v in Joule, K IC in MPa .√mCrack growth

dadN

=C p¿¿

∫0

N

dN=∫ai

ae 1Cp (β Δσ √πa )mp

da

Pressure equipmentPdes=P+ ρgh