Stress Concentration - eFatigue Concentration © 2002-2014 Darrell Socie, All Rights Reserved 2 of 63 Stress Concentration Factor

Professor Darrell F. Socie

© 2004-2014 Darrell Socie, All Rights Reserved

Stress Concentration

Stress Concentration © 2002-2014 Darrell Socie, All Rights Reserved 1 of 63

Outline

1. Stress Concentration 2. Notch Rules 3. Fatigue Notch Factor 4. Stress Intensity Factors for Notches 5. Frost Data and Kf

6. Small Crack Growth 7. Small Notches


Stress Concentration Factor

Applied stress Local stress

ρ+σ=σ

a21appliedlocal

Inglis Solution 1910

2a

ρ


Circular Notch

λσ

λσ

σ σ θ

a

r σr

τrθ

σθ

σr

σθ

τrθ


Stresses

θ

−

+

λ−+

−

λ+=

σσ 2cos431

211

21 242

ar

ar

arr

θ

+

λ−−

+

λ+=

σσθ 2cos31

211

21 42

ar

ar

θ

+

−

λ−−=

στ θ 2sin231

21 24

ar

arr

Independent of size, dependant only on r/a


Stress Distribution For tension λ = 0 and θ = 90

0

1

2

3

0 1 2 3 4

stre

ss

σσr

σσθ

λσ

λσ

σ σ θ a

r

ra


Stress Ratio Effects

-4

-3

-2

-1

0

1

2

3

4

30 60 90 120 150 180 Angle

σ θ σ

λ = 1

λ = 0

λ = -1

λσ

λσ

σ σ

stresses around the circumference of a hole


Elliptical Notches

abaKT

221 =ρ

ρ+=

2a

2b

Sharp Notch: high KT high gradient

Blunt Notch: low KT low gradient

stre

ss

KT

KT


Stress Concentration in a Bar

1

2

3

4

5

r

Axial, σz

Tangential, σθ

Radial, σr ρ

σ σo


Bridgeman Analysis (1943)

r

σz

ρ a

σr

r

σz

ρ a

σr

Elastic stress distribution Plastic stress distribution

σys σys

ttancons2

rz =σ−σ

=τ


Stresses

ρ

−ρ++σ=σ

a2ra2aln1

22

oz

∫ σπ=a

0zz drr2P

ρ

+

ρ

+σπ=2a1ln

a21aP flow

2max

Pmax = Anet σflow CF

CF constraint factor


Constraint Factors

0 11 1.212 1.384 1.648 1.7320 2.63

2.96∞

CF a /ρ

Pmax = Anet σflow CF


Effect of Constraint

Stre

ss, σ

z

Strain, εz

Increasing constraint

Higher strength and lower ductility


Outline




Notch Rules Neuber

Glinka

Seeger

n1

22

t KEeSK

σ

σ+σ

=εσ=

n1

22

t Kn11

EdeSK

σ

σ+

+σ

=εσ= ∫

SKK*SKSK

KEeSK

p

t

y

tLimitp

n1

2**2

p

=σ

=

σ

σ+σ

=εσ=


Define Kσ and Kε after Yielding

S , e

σ , ε

Define: nominal stress, S nominal strain, e

notch stress, σ notch strain, ε

Stress concentration

Strain concentration S

K σ=σ

eK ε

=ε


Kσ and Kε

Stre

ss (M

Pa)

Strain

KtS

Kte

σ ε

SK σ

=σ

eK ε

=ε


Stress and Strain Concentration

Kt

Nominal Stress

Stre

ss/S

train

Con

cent

ratio

n

1K →σ

2tKK →ε

First yielding


Neuber’s Rule St

ress

(MPa

)

Strain

KtS

Kte

σ

ε εσ=eKSK tt

Stress calculated with elastic assumptions

Actual stress


Neuber’s Rule for Fatigue

2222eKSK tt ε∆σ∆

=∆∆

Stress and strain amplitudes

Elastic nominal stress

E2S

2e ∆

=∆

Substitute and rearrange

22E

2SKt

ε∆σ∆=

∆

The product of stress times strain controls fatigue life


SN Materials Data

10 1 10 102 103 106 107 105 104

Fatigue Life, Reversals

100

1000

10000

Stre

ss A

mpl

itude

, MP

a

93 steels

17 aluminums


εN Materials Data

93 steels

17 aluminums

10-4

10-2

10-3

0.1

10

1

Stra

in A

mpl

iture

1 10 102 103 106 107 105 104



1 106 107 105 10 102 103 104


1

10

102

105

104

103

2 2

E K

t ε ∆

σ ∆ or

2 S ∆

2 2 E

ε ∆ σ ∆


Glinka’s Rule St

ress

(MPa

)

Strain

KtS

Kte

σ

ε

Add in the ”missing” strain energy

εσ= ∫ d2eKSK tt


Outline




A Dilemma

Stress analysis and stress concentration factors are independent of size and are related only to the ratio of the geometric dimensions to the loads

Fatigue is a size dependant phenomena

How do you put the two together ?


Similitude


Fatigue of Notches

From Dowling, Mechanical Behavior of Materials, 1999


Notch Size

Kt Kf Kt

Kf

Large Notch Small Notch


Microstructure Size

Kt Kf

Kt Kf

Low Strength High Strength


Stress Gradient

Kt Kf

Kt

Kf

Low Kt High Kt


Kt vs Kf

0

2

4

6

8

10

0 2 4 6 8 10 Kt

Kf

Kf = Kt

Experiments


Peterson’s Equation

ρα

+

−+=

1

1K1K tf

mmMPa2070025.08.1

u

σ

=α

0

1

2

3

4

10-4 10-3 10-2 0.1 1 10 102 103

Kf

ρα

No effect when ρ << α

Full effect when ρ >> α


Pererson’s Constant

1000 500 2000 1500

0.1

0.03

0.7

0.3

Ultimate Strength, MPa

α, m

m


Static Strength

hole Kt = 2.5

slot Kt = 5

diamond Kt = 20

edge Kt = 20


1018 Steel Test Data

0

20

40

60

80

100

0 2 4 6 8 10

load

, kN

displacement, mm

edge diamond slot hole


Notched SN Curve

100

1000

10000

100

Cycles

Stre

ss A

mpl

itude

, MPa

101 102 103 104 105 106 107

Smooth specimen data

Notched specimen Kf

Stress concentrations are not very important at short lives


Outline




Smith - Miller

aK

D13.0l

πσ=

ρ>

lD69.71K

D13.0l

πσ

ρ+=

ρ<

Long crack

Short crack

Kfatigue


Cracks at Notches

D

a a D + a

KtS S S

a << D a >> D aSKK t π= ( )aDSK +π=


Stress Intensity Factors

0 0.1 0.2 0.3 0.4 0.5 0.6

1.0

0

3.0

2.0

Da

DSK

aSKK t π=

( )aDSK +π=


Cracks at Holes

10 1 1 12

Once a crack reaches 10% of the hole radius, it behaves as if the hole was part of the crack


Outline




Crack Growth Data

10-12

10-11

10-10

10-9

10-8

10-7

10-6

1 10 100

Cra

ck G

row

th R

ate,

m/c

ycle

mMPa,K∆

mKCdNda

∆=

∆KTH

Kc

a212.1KTH ππ

σ∆>∆

Nonpropagating cracks


Frost Data nucleation fracture

Rotating bending Notched bar

Notched plate

1 3 5 7 9 11 13 15

1.0

0.8

0.6

0.4

0.2

0

Kt

S nom

inal

S fat

igue

lim

it

tK1 nonpropagating cracks

Frost, “A Relation Between the Critical Alternating Propagation Stress and Crack Length for Mild Steel” Proceedings of the Institute for Mechanical Engineers, Vol. 173, No. 35, 1959, 811-836


Significance

DKS th

flπ

∆=

For Kt > 4, the notch acts like a crack with a depth D

Kt does not play a role for sharp notches !


Specimens with Similar Geometry

Kt = 2.4 Kt = 10.7

Ultimate Strength 780 MPa Yield Strength 660 MPa

25 25

2.5 2.5


Test Results

1

100

1000

Nom

inal

Stre

ss A

mpl

itude

1 10 100 103 104 105 106 107

Total Fatigue Life, Cycles

Strength Limited

Crack Growth Dominated Fatigue Strength Dominated

Threshold Stress Intensity Dominated

Kt = 2.4 Kt = 10.7


Outline




Small Crack Growth


Threshold


Normalized Thresholds


Growth from Notches


Cracks at Notches

D a crack tip plastic zone

notch plastic zone

notch stress field


Crack Growth

∆K or a

Cra

ck g

row

th ra

te

Long cracks Crack tip plasticity controls

Short cracks Notch plasticity controls


Closure Observations

a b

d c c 1st subcycle

d 500th subcycle

σ

ε

a

b

1026 steel

∆ε1/2 = 0.005

∆ε2/2 = 0.001


Closure Correlation


Outline




area


Small Notches


Threshold Stress Intensity


Hardness Corelation


Flaw Sensitivity


Fatigue Limit

Stress Concentration

Documents

Stress Concentration - eFatigue Concentration © 2002-2014 Darrell Socie, All Rights Reserved 2 of 63 Stress Concentration Factor