### linear fracture mechanics

```Lecture #19
Failure & Fracture
1
Strength Theories
• Failure Theories
• Fracture Mechanics
2
Failure
• = no longer able to perform design function
– FRACTURE in brittle materials
– YIELDING / excessive deformation in ductile
materials
Stages of Cracking Failure
4
Static Fatigue
5
Bond and Microcracking
6
Stress Conditions
• Mechanical testing under
simple stress conditions
• Design requires prediction of
failure for complex stress
conditions
– principal stresses (s1 > s2 > s3)
– biaxial stress state (s3=0)
7
Strength
Envelope
For Concrete
8
Simple Failure Theories
• Rankine
 s1=sft
• St. Venant
• neither agree w/
experimental data
• either are rarely used
 e1= eft
9
Complex Failure Theories
• Max Shear Stress
(Tresca)
– ductile materials
 tmax= ty
 s1-s3= sy
 s2-s3= sy
 s1-s2= sy
sy/2 = max shear stress
at yield
2 = y If 2 > 1 > 0
1- 2 = -y
If 1< 0 and 2 > 0
2 = -y If 1 < 2 < 0
1 = y If 1 > 2 >
1- 2 = y
If 1> 0 and 2 < 0
1 = -y If 2 < 1
<0
10
Complex Failure Theories
• Max Distortional Strain
Energy (octahedral shear
stress, von Mises)
– best agreement with
experimental data
– hydrostatic + distortional
principal stresses
s 1 - s 2 2  s 2 - s 3 2  s 1 - s 3 2 = 2s ft 2
11
Failure Theories
• Mohr’s Strength
– both yielding &
fracture
 sft  sfc OR
sft = sfc
12
Failure Theories
• Mohr’s Strength
13
Failure Envelope
• Mohr’s Strength
– failure envelope
14
Effect of Confinement
15
Comparison of Failure Theories
–equivalent to Max Shear
Stress
if sft=sfc
–ductile and modified
if sft  sfc (brittle)
16
Fracture Mechanics
• max stress criterion
not sufficient
• all materials contain flaws,
defects, cracks
• concentrated stress at crack
tip (see Fig. 6.7)
• relationships between applied stress, crack size, and
fracture toughness
• probability of failure, critical crack size
(size effect, variability of material properties)
brittle materials
17
Crack Growth
C rack path
aro und
aggregates
(a)
C rack path
through
aggregates
(b )
Fracture Mechanics
• Theoretical cohesive strength
– fracture work resisted by energy
to create two new crack
surfaces
• Griffith Theory
– flaw / crack size
sensitivity
s
s ft =
E s
r0
ft =
2 E s
C
C = 1/2 crack length
19
Fracture Mechanics
s ft
 s ft
measured
theoretica
l
• stress concentration at
crack tip (see Fig 6.9)
s max


= s t 1


C
2 




1
2




• for C>>
Kt =
s
max
s
field
= 2
C

20
Stress Intensity Factor
y
s yy =
a
KI
2 r
1/ 2
x
Crack Tip
Stress Distribution
21
Fracture Mechanics
• Three modes of
crack opening
• Focus on Mode I for
brittle materials
22
Fracture Mechanics
23
Fracture Mechanics
s x 


s y  =
s 
 z

 
 
 3   
cos   1 - sin   sin 
 


 2 
2
 2 


K1 


3



 
 

 
 
 cos    1  sin   sin 
2 r 
 2 
2
 2  
 sin    cos    cos  3  
 
 



2
2
 2  
s z =  s x  s y 
t xz = t yz = 0
24
Fracture Mechanics
• KI = stress intensity factor = Fs(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)
G IC =
K IC
G IC =
K IC
2
plane stress
E
2
1 -  2 
E
plane strain
25
2d
c
KI
c
2a
Alpha = a/d
F
Alpha
K I = N (a) 1 /2 F ( )
26
27
Flexure (Bending)
• Yielding
– similar as in tension
–
–
–
–
ductile materials
first @ extreme fiber
progresses inward
proportional limit
• Fracture
– brittle materials
– nonlinear s distribution
• initiates as tensile failure
• flexural strength > tensile
strength
28
Failure Criterion
S trength the ory
A lo n g th e solid c urv e :
g eo m e tric a lly sim ila r spe c im e ns
log ( s
N
)
LE FM
S ize -effe ct of
conc rete struc tures
2
A lo n g th e ve rtic al line :
sp ec im e n s o f the sa m e siz e
w ith va ria b le n o tc he s
log ( d )
1
Linear Fracture Mechanics
s =
K If
1
C g
Non-Linear Fracture Mechanics
s =
cn K If
g ' ( )c f  g ( )d
30
a
cf
Crack
Process
Zone
KI
d
Alpha = a/d
31
Fracture specimens
P=2 pt
R
R
R
2t
2t
2a
P=2 pt
2a
P=2 pt
2t
r
Specimen Apparatus
Specimen Preparation
Test Specimens
Determination of Fracture
Parameters
 sN = cn KIf / [g’(0)cf + g(0)d]1/2
 sN = cn P/(sr) - split tensile (eq. 5.12)
 sN = cn P/(bd) - beam (eq. 5.13)
• Linear Regression
–
–
–
–
–
Y = AX + B
Y = cn2 / [g’(0) sN2]
X = g(0) d / g’(0)
KIf = 1 / A1/2
cf = B / A
Spec
#
1
2
3
b
(in)
3
3
3
d
(in)
6
6
6
2a 0
(in)
0
1
4
P
(lb)
13000
10000
3500

0
0.167
0.667
g() g'() X (in) Y (psi.in 1/2 )
2
(g/g') d
1/(g's )
0.964 0.000 2.92 0.0000 1.620E -06
0.999 0.523 3.60 0.8711 2.219E -06
1.645 5.699 10.02 3.4125 6.512E -06
F()
37
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 + 1.4722 - 0.2563
F() = 0.9994, g( ) = 0.5230; g’( ) = 3.6023
– (3) F() = 2.849 - 10.451 + 22.9382 - 14.9403
F() = 1.6497, g( ) = 5.6997; g’( ) = 10.0214
• Basic Geometry - beam
– cn = 1.5 s/d ;  = a/d
– F() = 1.122 - 1.40 + 7.332 - 13.083 + 14.04
Failure Criterion
S trength the ory
A lo n g th e solid c urv e :
g eo m e tric a lly sim ila r spe c im e ns
log ( s
N
)
LE FM
S ize -effe ct of
conc rete struc tures
2
A lo n g th e ve rtic al line :
sp ec im e n s o f the sa m e siz e
w ith va ria b le n o tc he s
log ( d )
1
Applications of Fracture Parameters
Strength Determination
sN = cn KIf / [g’(0)cf + g(0)d]1/2
4 .0 0
2 .0 0
s
N
(M P a )
3 .0 0
1 .0 0
0 .0 0
0 .0 0
0 .2 0
0 .4 0
0 .6 0
 = a /d
0 .8 0
1 .0 0
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
d (mm or inch)
log (sN / Bfu)
sN
(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
```