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Professor Darrell F. Socie

Stress Concentration

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

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

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

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

Notch Rules Neuber

Seeger

n1

22

t KEeSK

σ

σ+σ

=εσ=

n1

22

t Kn11

EdeSK

σ

σ+

=εσ= ∫

SKK*SKSK

KEeSK

p

t

y

tLimitp

n1

2**2

p

=

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

Kσ and Kε

Stre

ss (M

Pa)

Strain

KtS

Kte

σ ε

SK σ

eK 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

Fatigue Life, Reversals

1 106 107 105 10 102 103 104

Fatigue Life, Reversals

1

10

102

105

104

103

2 2

E K

t ε ∆

σ ∆ or

2 S ∆

2 2 E

ε ∆ σ ∆

ress

(MPa

)

Strain

KtS

Kte

σ

ε

Add in the ”missing” strain energy

εσ= ∫ d2eKSK tt

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

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

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

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

, 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

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

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

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

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

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

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

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

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

area

Small Notches

Threshold Stress Intensity