1 lecture #19 failure & fracture. 2 strength theories failure theories fracture mechanics

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1 Lecture #19 Failure & Fracture

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Page 1: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

1

Lecture #19 Failure & Fracture

Page 2: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

2

Strength Theories

• Failure Theories

• Fracture Mechanics

Page 3: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

Failure

• = no longer able to perform design function– FRACTURE in brittle materials– YIELDING / excessive deformation in ductile

materials

Page 4: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

4

Stages of Cracking Failure

Page 5: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

5

Static Fatigue

Page 6: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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Bond and Microcracking

Page 7: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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Stress Conditions• Mechanical testing under

simple stress conditions• Design requires prediction of

failure for complex stress conditions– principal stresses (>>)

– biaxial stress state (=0)

Page 8: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

8

StrengthEnvelopeFor Concrete

Page 9: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

9

Simple Failure Theories

• Rankine 1=ft

• St. Venant 1= ft

• neither agree w/ experimental data

• either are rarely used

Page 10: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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Complex Failure Theories• Max Shear Stress

(Tresca)– ductile materials max= y

y

y

y

y/2 = max shear stress

at yield

1- 2 = -y

If 1< 0 and 2 > 0

1- 2 = y

If 1> 0 and 2 < 0

2 = y If 2 > 1 > 0

1 = y If 1 > 2 > 0

2 = -y If 1 < 2 < 0

1 = -y If 2 < 1 < 0

Page 11: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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Complex Failure Theories• Max Distortional Strain

Energy (octahedral shear stress, von Mises)– best agreement with

experimental data

– hydrostatic + distortional principal stresses

2231

232

221 2 ft

Page 12: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

12

Failure Theories• Mohr’s Strength

– both yielding & fracture

ft fc OR

ft = fc

Page 13: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

13

Failure Theories• Mohr’s Strength

Page 14: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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Failure Envelope• Mohr’s Strength

– failure envelope

Page 15: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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Effect of Confinement

Page 16: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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Comparison of Failure Theories–equivalent to Max Shear

Stress

if ft=fc

–ductile and modified

if ft fc (brittle)

Page 17: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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Fracture Mechanics• max stress criterion

not sufficient

• relationships between applied stress, crack size, and fracture toughness

• probability of failure, critical crack size

(size effect, variability of material properties)

• focus on linear fracture mechanics, tensile loading, brittle materials

• all materials contain flaws, defects, cracks

• concentrated stress at crack tip (see Fig. 6.7)

Page 18: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

Crack Growth

(a ) (b )

C ra c k p a tha ro u n da g g re g a te s

C ra c k p a thth ro u g ha g g re g a te s

Page 19: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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Fracture Mechanics• Theoretical cohesive strength

– fracture work resisted by energy to create two new crack surfaces

• Griffith Theory– flaw / crack size

sensitivity

0rE s

ft length crack 1/2

2

CC

E sft

Page 20: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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Fracture Mechanics

• stress concentration at crack tip (see Fig 6.9)

• for C>>

ltheoreticameasured ftft

C

Kfield

t 2max

21

max 21

Ct

Page 21: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

21

a2/12 r

K Iyy

Crack Tip

x

y

Stress Distribution

Stress Intensity Factor

Page 22: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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Fracture Mechanics• Three modes of

crack opening

• Focus on Mode I for brittle materials

Page 23: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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Fracture Mechanics

Page 24: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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Fracture Mechanics

23

22

23

21

2

23

21

2

21

coscossin

sinsincos

sinsincos

rK

z

y

x

0

yzxz

yxz

Page 25: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

25

Fracture Mechanics

• KI = stress intensity factor = F(C)1/2

– F is a geometry factor for specimens of finite size

• KI = KIC OR GI=GIC unstable fracture

• KIC= Critical Stress Intensity Factor

= Fracture Toughness

• GI=strain energy release rate (GIC=critical)

strainplane

EK

G

stressplaneE

KG

ICIC

ICIC

1

22

2

Page 26: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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F

Alpha

2 d

2 a

KI cc

Alpha = a/d

Page 27: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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Page 28: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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Flexure (Bending)

• Fracture– brittle materials

– nonlinear distribution • initiates as tensile failure

• flexural strength > tensile strength

• Yielding– similar as in tension– ductile materials

– first @ extreme fiber

– progresses inward

– gradual change masks proportional limit

Page 29: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

Failure Criterion

21

A lo n g th e s o l id c u rv e :g eo m e tr ic a lly s im ila r s p e c im e n s

A lo n g th e v e r t ic a l li n e :s p ec im e n s o f t h e s a m e s iz ew ith v a r ia b le n o tc h e s

log(

) N

log d( )

S treng th theory

LE FM

Size-effect o f concrete struc tu res

Page 30: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

30

Linear Fracture Mechanics

gC

K If 1

Non-Linear Fracture Mechanics

dgcg

Kc

f

Ifn

)()('

Page 31: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

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Crack

d

a cf

KI

Process Zone

Alpha = a/d

Page 32: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

Fracture specimens

R

P=2pt

2t

2a

R

P=2pt

2t

R

P=2pt

2t

2a r

Page 33: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

Specimen Apparatus

Page 34: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

Specimen Preparation

Page 35: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

Test Specimens

Page 36: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

Determination of Fracture Parameters

N = cn KIf / [g’(0)cf + g(0)d]1/2

N = cn P/(sr) - split tensile (eq. 5.12)N = cn P/(bd) - beam (eq. 5.13)• Linear Regression

– Y = AX + B– Y = cn

2 / [g’(0) N2]

– X = g(0) d / g’(0)– KIf = 1 / A1/2

– cf = B / A

Page 37: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

37

Spec#

b(in)

d(in)

2a0

(in)P

(lb)

1 3 6 0 13000 02 3 6 1 10000 0.1673 3 6 4 3500 0.667

F() g() g'() X (in)(g/g') d

Y (psi.in1/2)1/(g'2)

0.964 0.000 2.92 0.0000 1.620E-060.999 0.523 3.60 0.8711 2.219E-061.645 5.699 10.02 3.4125 6.512E-06

Page 38: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

Application of Fracture Method Strength Determination

• g( ) = c2nF2(

• Basic Geometry - split tensile– cn = 2/ ; = (1) 0.0, (2) 0.1667, or (3) 0.6667

– (1) F() = 0.964; g( ) = 0.0; g’( ) = 2.9195– (2) F() = 0.964 - 0.026+ 1.4722 - 0.2563

F() = 0.9994, g( ) = 0.5230; g’( ) = 3.6023– (3) F() = 2.849 - 10.451+ 22.9382 - 14.9403

F() = 1.6497, g( ) = 5.6997; g’( ) = 10.0214

• Basic Geometry - beam– cn = 1.5 s/ds/d ; = a/d

– F() = 1.122 - 1.40+ 7.332 - 13.083 + 14.04

Page 39: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

Failure Criterion

21

A lo n g th e s o l id c u rv e :g eo m e tr ic a lly s im ila r s p e c im e n s

A lo n g th e v e r t ic a l li n e :s p ec im e n s o f t h e s a m e s iz ew ith v a r ia b le n o tc h e s

log(

) N

log d( )

S treng th theory

LE FM

Size-effect o f concrete struc tu res

Page 40: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

Applications of Fracture Parameters Strength Determination

0.00

1.00

2.00

3.00

4.00

0.00 0.20 0.40 0.60 0.80 1.00

= a/d

N (

MP

a)

N = cn KIf / [g’(0)cf + g(0)d]1/2

Page 41: 1 Lecture #19 Failure & Fracture. 2 Strength Theories Failure Theories Fracture Mechanics

Applications of Fracture Parameters Strength Determination

Size effect on strength( 0 = 0.2; Bfu = 3.9 MPa = 566 psi; da = 25.4 mm = 1 in)

log (d/da) Specimen or structure size log (N / Bfu) N

d (mm or inch) (MPa or psi)

0.70 127 or 5 - 0.18 2.57 or 373

1.00 305 or 12 - 0.26 2.15 or 312

1.30 507 or 20 - 0.35 1.75 or 254