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

Fatigue and Fracture

Static Strength and Fracture

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

Stress Concentration

“Load flow” lines

Circular Notch

σ σ θ

Stresses

λ−+

σσ 2cos431

21 242

λ−−

σσθ 2cos31

λ−−=

στ θ 2sin231

Independent of size, dependant only on r/a

Stress Distribution For tension λ = 0 and θ = 90

0 1 2 3 4

σσθ

σ σ θ a

Stress Ratio Effects

30 60 90 120 150 180 Angle

σ θ σ

λ = 1

λ = 0

λ = -1

stresses around the circumference of a hole

Elliptical Notches

221 =ρ

Sharp Notch: high KT high gradient

Blunt Notch: low KT low gradient

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

0 0.002 0.004 0.006 0.008 0.010 0.012

Thickness

ν = 0.3 ν = 0.5

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

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

σ , ε

Define: nominal stress, S nominal strain, e

notch stress, σ notch strain, ε

Stress concentration

Strain concentration S

K σ=σ

Kσ and Kε

Strain

Stress and Strain Concentration

Nominal Stress

1K →σ

2TKK →ε

First yielding

Notched Plate Experiments

10 1 4

Materials: 1018 Hot Rolled Steel 7075-T6 Aluminum

1/4 thick

1018 Stress-Strain Curve

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 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 2 4 6 8 10

displacement, mm

edge diamond slot hole

0 2 4 6 8 10

7075-T6 Test Data

displacement, mm

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

Axial, σz

Tangential, σθ

Radial, σr ρ

σ σo

Bridgeman Analysis (1943)

Elastic stress distribution Plastic stress distribution

σys σys

ttancons2

rz =σ−σ

Stresses

−ρ++σ=σ

a2ra2aln1

∫ σπ=a

0zz drr2P

+σπ=2a1ln

a21aP flow

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

ss, σ

Strain, εz

Increasing constraint

Higher strength and lower ductility

1 16 R 1 R V notch

Notched Bars

0 1 2 3 4 5

1018 Steel Test Data

displacement, mm

V 1.6 3.2 25.4

7075-T6 Test Data

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

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

∆U = ∆Γ

strain energy ⇒ surface energy

Plastic Energy Term

∆U = ∆Γ + ∆P

strain energy ⇒ surface energy + plastic energy

Strain Energy Release Rate, G

∂∂

F x internal strain energy = external work

xF21U=

Strain Energy Release Rate, G

∆∆

a + ∆a

a ∆U

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

Stress Intensity Factor, K

xyθθθ

θ+−

ν+ν−θ

2sin21

G2Ku 2

θ+−

ν+ν−θ

2cos21

G2Kv 2

x ∂∂

y ∂∂

xy ∂∂

+∂∂

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

wafaKIC

fracture toughness

operating stresses

flaw size

flaw shape

K and G

K1G22ν−

plane stress

plane strain

Elastic-Plastic Stress Field

σy small scale yielding

plastic zone

Critical Crack Sizes

ICcritical

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

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

3D Plastic Zone

plane stress

plane strain

Fracture Surfaces

Thickness Requirements

0 5 10 15 20 25

thickness, mm

Pa √

cK5.2t

Size Requirements

r50K5.2a,aW,t ≈

σ≥−

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

1400 1500 1600 1700 1800 1900 2000 2100

Yield Strength, MPa

Pa √

0.5C-Ni-Cr-Mo steel

3.2 y sσ=σ

Silver Bridge

Collapse, Wearne, P. TV Books, NY 1999

Source

Eyebar

12” wide 2” thickness

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

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

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

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

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Mt back surface effects

b b2atan

Φ crack shape effects

0 0.1 0.2 0.3 0.4 0.5

dsink1Q

ββ−= ∫π2c

caca464.11Q

Edge Cracked Plate in Tension

π−++ππ

b2acos

)b2asin1(37.0

ba02.2752.0

b2atan

Edge Cracked Plate in Bending

π−+ππ

b2acos

)b2asin1(199.0923.0

b2atan

Tension and Bending

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

bending

tension

Boundary Conditions

0 0.2 0.4 0.6 0.8 1

uniform stress

uniform displacement

Handbook

Stress Intensity Factors Handbook Y. Murakami Editor, Pergamon Press

Useful approximations

a2)12.1(K 2 ππ

Two free edges Semicircular shape

Corner crack

Crack Shape

a71.0a212.1K πσ=ππ

a12.1K πσ=

Through crack

Semielliptical crack

Superposition

tension + bending

( ) ( )

bendingtensiontotal

ijtotal

ijbendingtension

ijbending

ijtension

Crack tip stresses:

Stress Gradients N

3112.1K tipcrackaverage π

σ+σ≅

Pressure Vessel

a1trPK π

aPatrPK π+π=

Cracks at Notches

a a D + a

KtS S S

a << D a >> D

0 0.1 0.2 0.3 0.4 0.5 0.6

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

WafaKIc

σ ~ σys / 2

a ~ 0.1 mm - 100 mm

21~Waf −

Fracture Behavior

Collapse

Failure Analysis Diagram

f lowσσ

Collapse

Plastic Collapse

f lowB)aW(2P

σ<−

BW2PS= Nominal stress

−σ=

Wa1S flow

Fracture

IcKWafaS =

BW2PS=

Fracture vs. Collapse

KWa1 Ic

σ WafW

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 0.2 0.4 0.6 0.8 1

W = 1m m1K

Ic =σ

Fracture Diagram

0 0.2 0.4 0.6 0.8 1

W = 1m

Ic =σ

W = 0.1m

Fracture vs. Collapse

W2atan

Ic ππ

Fracture Diagram

0 0.2 0.4 0.6 0.8 1

Plastic Collapse

Fracture WK

a / W small flaws high stresses

large flaws low stresses

Typical Stress Concentration

Stress distribution Strain distribution

elastic

failure

Stress and Strain Concentration

Nominal Stress

1K →σ

2TKK →ε

First yielding

Failure Diagram – Ductile Material

0 1 a/w

ys2Tf K ε=ε

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

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

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

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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