static strength and fracture stress concentration …static strength and fracture stress...

Static Strength and Fracture Stress Concentration Factors

Professor Darrell F. Socie Mechanical Science and Engineering

University of Illinois

© 2004-2013 Darrell Socie, All Rights Reserved

Fatigue and Fracture

Static Strength and Fracture © 2004-2013 Darrell Socie, All Rights Reserved 1 of 106

Static Strength and Fracture

Stress Concentration Factors Fracture Mechanics Approximate Stress Intensity Factors Ductile vs. Brittle Fracture


Stress Concentration

“Load flow” lines


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


Plane Stress – Plane Strain

thick plate thin plate

plane stress σz = 0 εz = 0 plane strain σz = 0 εz = 0


Thick or Thin ?

Plane strain

Plane stress


Transverse Strains

Longitudinal Tensile Strain

Tran

sver

se C

ompr

essi

on S

train

0 0.002 0.004 0.006 0.008 0.010 0.012

0.001

0.002

0.003

0.004

0.005

40 mm

20 mm

10 mm

5 mm

Thickness

100

x

z

y

ν = 0.3 ν = 0.5

25 D


Notch Stresses

t εx εz σx σz

7 0.01 -0.005 63.5 0 15 0.01 -0.003 70.6 14.1 30 0.01 -0.002 73.0 21.8 50 0.01 -0.001 75.1 29.3

x

z

y


Fracture Surfaces


Stress or Strain Control?

elastic material

plastic zone

Elastic material surrounding the plastic zone forces the displacements to be compatible, I.e. no gaps form in the structure.

Boundary conditions acting on the plastic zone boundary are displacements. Strains are the first derivative of displacement


Define Kσ and Kε

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


Notched Plate Experiments

1 2

1

1 2 1

10 1 4

1 2

1 2

Materials: 1018 Hot Rolled Steel 7075-T6 Aluminum

1/4 thick


1018 Stress-Strain Curve

0

100

200

300

400

500

0 0.1 0.2 0.3 0.4 0.5Strain

true fracture strain, εf = 0.86

true fracture stress, σf = 770 MPa


7075-T6 Stress-Strain Curve

0

100

200

300

400

500

600

700

0 0.05 0.1 0.15 0.2

Strain

true fracture strain, εf = 0.23

true fracture stress, σf = 720 MPa


1018 Steel Test Data

0

20

40

60

80

100

0 2 4 6 8 10

load

, kN

displacement, mm

edge diamond slot hole


0

20

40

60

80

100

0 2 4 6 8 10

7075-T6 Test Data

displacement, mm

load

, kN

edge diamond slot hole


Failure of a Notched Plate


Failures from Stress Concentrations

Net section stresses must be below the flow stress

Notch strains must be below the fracture strain


2D vs 3D

Plates and shells 2D stress state

Solids 3D stress state


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


1 4

1 4

1 2

1 4

1 8 R

1 4

1 16 R 1 R V notch

6

Notched Bars


0

5

10

15

20

25

30

0 1 2 3 4 5

1018 Steel Test Data

displacement, mm

V 1.6 3.2 25.4


7075-T6 Test Data

0

5

10

15

20

25

30

0 1 2 3 4 5displacement, mm

V 1.6 3.2 25.4


Conclusion

Net section area, state of stress and material strength control the failure load in a structure only in ductile materials. In brittle materials, cracks will form before the maximum load capacity of the structure is reached.


Fractures

1943 1972


Stress Concentration

2 a

ρ

ρ+=

a21KT

a ~ 10-3

for a crack

ρ ~ 10-9 KT ~ 2000


Fracture Mechanics Parameters

G strain energy release rate K stress intensity factor J J-integral R crack growth resistance


Strain Energy Release Rate

σ

σ

2a

U = 0

σ

σ

∆a

∆U = ∆Γ

strain energy ⇒ surface energy


Plastic Energy Term

σ

σ

∆a

∆U = ∆Γ + ∆P

strain energy ⇒ surface energy + plastic energy


Strain Energy Release Rate, G

aUG

∂∂

=

F x internal strain energy = external work

xF21U=

F

x

a


Strain Energy Release Rate, G

aU

t1G

∆∆

−=

F x F

x

a + ∆a

a + ∆a

a ∆U

t

G is the energy per unit crack area needed to extend a crack


Stress Intensity Factor, K

r θ

θθ

−θ

π=σ

23sin

2sin1

2cos

r2K

x

θθ

+θ

π=σ

23sin

2sin1

2cos

r2K

y

23cos

2sin

2cos

r2K

xyθθθ

π=τ

θ+−

ν+ν−θ

π=

2sin21

13

2cos

2r

G2Ku 2

θ+−

ν+ν−θ

π=

2cos21

13

2sin

2r

G2Kv 2

xu

x ∂∂

=ε

xv

y ∂∂

=ε

xv

yu

xy ∂∂

+∂∂

=γ


Design Philosophy

Stress < Strength

Stress Intensity < Fracture Toughness

σ < σy

K < KIc

Two cracks with the same K will have the same behavior


Fracture Toughness

πσ=

wafaKIC

fracture toughness

operating stresses

flaw size

flaw shape


K and G

EKG

2

=

( )E

K1G22ν−

=

plane stress

plane strain


Elastic-Plastic Stress Field

σy small scale yielding

plastic zone

2rp

θθ

+θ

π=σ

23sin

2sin1

2cos

*r2K

y

r r*


Critical Crack Sizes

2

f

ICcritical

K1a

σπ

=

What would the critical crack size be in a standard tensile test ?


Fracture Toughness

From M F Ashby, Materials Selection in Mechanical Design, 1999, pg 431


Measuring Fracture Toughness

displacement

forc

e

www.enduratec.com


Thickness Effects

From Wilhem “Fracture Mechanics Guidelines for Aircraft Structural Applications” AFFDL-TR-69-111


Plastic Zone Size

plane stress

plane strain

2

y sp

K21r

σπ=

2

y sp

K61r

σπ=


3D Plastic Zone

plane stress

plane strain


Fracture Surfaces


Thickness Requirements

0

20

40

60

80

100

0 5 10 15 20 25

thickness, mm

fract

ure

toug

hnes

s, M

Pa √

m

KIc 2

ys

cK5.2t

σ>


Size Requirements

p

2

ys

r50K5.2a,aW,t ≈

σ≥−

σys

KIc t, mm 2024- T3 345 44 40.7 7075 -T6 495 25 6.4 Ti-6Al-4V 910 105 33.3 Ti-6Al-4V 1035 55 7.1

4340 860 99 33.1 4340 1510 60 3.9

17-7 PH 1435 77 7.2 52100 2070 14 0.1


Strength, Toughness, Flaw Size

0

10

20

30

40

50

60

70

80

1400 1500 1600 1700 1800 1900 2000 2100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Yield Strength, MPa

Frac

ture

Tou

ghne

ss, M

Pa √

m

Flaw

Siz

e, m

m

0.5C-Ni-Cr-Mo steel

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2 y sσ=σ

2ysσ

=σ


Silver Bridge

Collapse, Wearne, P. TV Books, NY 1999


Source


Eyebar

12” wide 2” thickness

12

6


Material Properties

σu = 100 ksi σy = 75 ksi Working stress 50 ksi

CVN = 2.6 ft-lb at 32° F

CVN = 8.6 ft-lb at 165° F


Fracture Toughness

)lbft,inpsi()CVN(2E

K 232

IC −−=Barsom-Rolfe

)lbft(CVN5.15KIC −=Corten-Sailors

)lbft(CVN35.9K 65.1IC −=Roberts-Newton

Barsom-Rolfe inksi9.15KIC =

Corten-Sailors inksi0.25KIC =

Roberts-Newton inksi2.45KIC =

Average 28.7


Critical Crack Size

Assume a corner crack

( ) a212.1K 2IC π

πσ=

Let σ = σy a ~ 0.073 inches

Let σ = 50 a ~ 0.163 inches


Modern Aircraft Materials

Bucci et. al., “Need for New Materials in Aging Aircraft Structures” Journa of Aircraft, Vol. 37, 2000, 122-129


Summary

Both stress and flaw size govern fracture

πσ>

WafaKIc


Stress Intensity Factors

Analytical Theory of elasticity

Numerical Finite element

Experimental Compliance

Handbook Approximate



aMMK ts πσΦ

=

Ms free surface effects

Mt back surface effects

Φ crack shape effects


Ms free surface effects

Ms = 1.12

a

b

2a

c


0

1

2

3

4

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Mt back surface effects

a

b b2atan

ab2Mt

ππ

=

Mt

a / b


Φ crack shape effects

0

1

2

3

0 0.1 0.2 0.3 0.4 0.5

Q

c2a

2

2

π

Q=Φ

22

2/

0

22

ca1k

dsink1Q

−=

ββ−= ∫π2c

a

caca464.11Q

65.1

≤

+≅


Edge Cracked Plate in Tension

π

π−++ππ

=

b2acos

)b2asin1(37.0

ba02.2752.0

b2atan

ab2

baF

3


Edge Cracked Plate in Bending

π

π−+ππ

=

b2acos

)b2asin1(199.0923.0

b2atan

ab2

baF

4


Tension and Bending

0

2

4

6

8

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

baF

a / b

bending

tension


Boundary Conditions

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1

uniform stress

uniform displacement

baF

a / b


Handbook


Handbook

Stress Intensity Factors Handbook Y. Murakami Editor, Pergamon Press


Useful approximations

a2)12.1(K 2 ππ

σ=

Two free edges Semicircular shape

Corner crack

a a


Crack Shape

a

a71.0a212.1K πσ=ππ

σ=

a12.1K πσ=

Through crack

Semielliptical crack


Superposition

tension + bending

( ) ( )

( ) ( )

bendingtensiontotal

ijtotal

ijbendingtension

ij

ijbending

ijtension

ij

KKK

fr2

Kfr2

KK

fr2

Kf

r2K

+=

θπ

=θπ

+=σ

θπ

+θπ

=σ

Crack tip stresses:


Stress Gradients N

omin

al S

tress

x

a32

3112.1K tipcrackaverage π

σ+σ≅


Pressure Vessel

P

trP

=σ

trP

=σ

a1trPK π

+=

aPatrPK π+π=


Cracks at Notches

D

a a D + a

KtS S S

a << D a >> D



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

20 1 1 22

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


Summary

πσ>

WafaKIc

σ ~ σys / 2

a ~ 0.1 mm - 100 mm

21~Waf −


Fracture Behavior

LEFM

EPFM

Collapse


Failure Analysis Diagram

IcKK

f lowσσ

1

1

LEFM

EPFM

Collapse


Plastic Collapse

2W

2a

B

P

P

f lowB)aW(2P

σ<−

BW2PS= Nominal stress

−σ=

Wa1S flow


Fracture

IcKWafaS =

π

2W

2a

B

S

BW2PS=

π

=

WafW

Wa

KS Ic


Fracture vs. Collapse

π

=

−σ

WafW

wa

KWa1 Ic

flow

π

−=

σ WafW

wa

Wa1K

flow

Ic

Fracture and collapse equally likely


Material Properties

σys KIc

1020 250 200 0.8002024-T3 345 44 0.1287075-T6 495 25 0.051

Ti-6Al-4V 910 105 0.115Ti-6Al-4V 1035 55 0.053

4340 860 99 0.1154340 1510 60 0.040

17-7 PH 1435 77 0.05452100 2070 14 0.007

σys KIc


Fracture Diagram

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

W = 1m m1K

f low

Ic =σ

m1.0K

f low

Ic =σ

m5.0K

f low

Ic =σ

a / W

f low

Sσ


Fracture Diagram

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

W = 1m

m5.0K

f low

Ic =σ

W = 0.1m

a / W

f low

Sσ


Fracture vs. Collapse

W2atan

aW2

Wa

Wa1

WK

flow

Ic ππ

π

−=

σ


Fracture Diagram

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Plastic Collapse

Fracture WK

f low

c

σ

a / W small flaws high stresses

large flaws low stresses


Typical Stress Concentration

2TK

TK

Stress distribution Strain distribution

elastic

failure


Stress and Strain Concentration

KT

Nominal Stress

Stre

ss/S

train

Con

cent

ratio

n

1K →σ

2TKK →ε

First yielding


Failure Diagram – Ductile Material

0 1 a/w

ys2Tf K ε=ε

2T

fnetmax K

EAP ε=

fully plastic εf is large

netflowmax AP σ=strength limited

Strength limit is reached before cracking at the notch


Failure Diagram – Brittle Material

0 1 a/w

ductility limited

ysTf K ε=ε

T

fnetmax K

EAP ε=

elastic

εf is small

netflowmax AP σ=

Cracks form at the notch before the limit load is reached


Cracks at Notches

D

a a D + a

KtS S S

a << D a >> D


Cracks at Holes

20 1 1 22

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


Notch Fracture

r1.0K12.1K Tysc πσ=

For KT = 3 and r = 10 mm

18.0Kys

c >σ

to avoid fracture from the notch

What ratio of strength to toughness is needed to avoid fracture?


Material Properties

σys KIc

1020 250 200 0.8002024-T3 345 44 0.1287075-T6 495 25 0.051

Ti-6Al-4V 910 105 0.115Ti-6Al-4V 1035 55 0.053

4340 860 99 0.1154340 1510 60 0.040

17-7 PH 1435 77 0.05452100 2070 14 0.007

σys KIc


Summary

Fracture is a likely failure mode for all higher strength materials

Fracture is even more likely at stress concentrators

Fatigue and Fracture

static strength and fracture stress concentration …static strength and fracture stress...

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