1 lecture #19 failure & fracture. 2 strength theories failure theories fracture mechanics
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Lecture #19 Failure & Fracture
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Strength Theories
• Failure Theories
• Fracture Mechanics
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Failure
• = no longer able to perform design function– FRACTURE in brittle materials– YIELDING / excessive deformation in ductile
materials
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Stages of Cracking Failure
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Static Fatigue
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Bond and Microcracking
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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)
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StrengthEnvelopeFor Concrete
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Simple Failure Theories
• Rankine 1=ft
• St. Venant 1= ft
• neither agree w/ experimental data
• either are rarely used
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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
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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
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Failure Theories• Mohr’s Strength
– both yielding & fracture
ft fc OR
ft = fc
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Failure Theories• Mohr’s Strength
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Failure Envelope• Mohr’s Strength
– failure envelope
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Effect of Confinement
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Comparison of Failure Theories–equivalent to Max Shear
Stress
if ft=fc
–ductile and modified
if ft fc (brittle)
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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)
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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
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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
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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
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a2/12 r
K Iyy
Crack Tip
x
y
Stress Distribution
Stress Intensity Factor
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Fracture Mechanics• Three modes of
crack opening
• Focus on Mode I for brittle materials
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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
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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
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F
Alpha
2 d
2 a
KI cc
Alpha = a/d
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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
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Failure Criterion
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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
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Linear Fracture Mechanics
gC
K If 1
Non-Linear Fracture Mechanics
dgcg
Kc
f
Ifn
)()('
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Crack
d
a cf
KI
Process Zone
Alpha = a/d
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Fracture specimens
R
P=2pt
2t
2a
R
P=2pt
2t
R
P=2pt
2t
2a r
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Specimen Apparatus
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Specimen Preparation
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Test Specimens
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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
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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
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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
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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
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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
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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