Fracture and fatigue
Key point:
Preexisting surface flaws and preexisting internal cracks
play a central role in the failure of materials.
ISSUES TO ADDRESS...
• How do flaws in a material initiate failure?
• How is fracture resistance quantified; how do different
material classes compare?
• How do we estimate the stress to fracture?
• How do loading rate, loading history, and temperature
affect the failure stress?
Fracture mechanisms
•
Ductile fracture
– Occurs with plastic deformation and with high energy
absorption before fracture
•
Brittle fracture
– Little or no plastic deformation with low energy
absorption
– Catastrophic
1
An oil tanker that fractured in a brittle manner by crack propagation around its girth
Ductile vs Brittle Failure
• Classification:
Fracture behavior
%AR or %EL
Very Ductile Moderately Ductile
Large
Moderate
Brittle
Small
• Ductile fracture is usually desirable!
•
•
Ductile: warning before fracture
Brittle: no warning
2
Example: Failure of a Pipe
• Ductile failure:
--one piece
--large deformation
• Brittle failure:
--many pieces
--small deformation
Figures from V.J. Colangelo and F.A.
Heiser, Analysis of Metallurgical Failures
(2nd ed.), Fig. 4.1(a) and (b), p. 66 John
Wiley and Sons, Inc., 1987. Used with
permission.
Moderately Ductile Failure
• Evolution to failure:
necking
s
• Resulting
fracture
surfaces
(steel)
void
nucleation
void growth
and linkage
shearing
at surface
fracture
50
50mm
mm
100 mm
particles
serve as void
nucleation
sites.
From V.J. Colangelo and F.A. Heiser,
Analysis of Metallurgical Failures (2nd
ed.), Fig. 11.28, p. 294, John Wiley and
Sons, Inc., 1987. (Orig. source: P.
Thornton, J. Mater. Sci., Vol. 6, 1971, pp.
347-56.)
Fracture surface of tire cord wire
loaded in tension. Courtesy of F.
Roehrig, CC Technologies, Dublin,
OH. Used with permission.
3
Ductile vs. Brittle Failure
cup-and-cone fracture
brittle fracture
Adapted from Fig. 8.3, Callister 7e.
Fractured tensile test bars
”Cup and cone”
Fractured aluminium alloy I,
with a dimpled texture
Fractured aluminium alloy II, with a
typical cleavage texture
4
Scanning electron micrographs (a) spherical dimples characteristic of ductile
fracture, (b) parabolic-shaped dimples characteristic of ductile fracture.
Brittle Failure
Arrows indicate origin of crack
5
Transgranular fracture
Intergranular fracture
Brittle Failure
Principles of fracture mechanics
Flaws are Stress Concentrators or stress raisers
Results from crack propagation
• Griffith Crack
⎛a
s m = 2s o ⎜⎜
⎝ rt
rt
1/ 2
⎞
⎟⎟
⎠
= K t so
where
rt = radius of curvature of the
crack tip
so = applied stress
sm = stress at crack tip
Kt : stress concentration factor
6
Concentration of Stress at Crack Tip
Adapted from Fig. 8.8(b), Callister 7e.
Engineering Fracture Design
⎛a
s m = 2so ⎜⎜
⎝ rt
1/ 2
⎞
⎟⎟
⎠
= K t so
• Avoid sharp corners!
s
so
max
Stress Conc. Factor, K t = s
w
smax
r,
fillet
radius
o
2.5
h
Adapted from Fig.
8.2W(c), Callister 6e.
(Fig. 8.2W(c) is from G.H.
Neugebauer, Prod. Eng.
(NY), Vol. 14, pp. 82-87
1943.)
2.0
increasing w/h
1.5
1.0
0
0.5
1.0
sharper fillet radius
r/h
7
Crack Propagation
Cracks propagate due to sharpness of crack tip
• A plastic material deforms at the tip, “blunting” the crack.
Deformed region
bittle
plastic
Energy balance on the crack
• Elastic strain energy
• energy stored in material as it is elastically deformed
• this energy is released when the crack propagates
• creation of new surfaces requires energy
When Does a Crack Propagate?
•
Critical stress for crack propagates in a brittle material
i.e., sm > sc
or Kt > Kc
1/ 2
⎛ 2Egs ⎞
sc = ⎜
⎟
⎝ pa ⎠
where
•
•
•
•
•
E = modulus of elasticity
gs = specific surface energy
a = one half length of internal crack
Kc = sc/s0
For ductile => replace gs by gs + gp
where gp is plastic deformation energy
Example problem 8.1, p217
8
Fracture toughness
Critical stress for crack propagation (sc) and
crack length (a)
KC = Ysc ( p a)1/2
1. Y: geometry factor, on the order of 1
2. sc the overall applied stress at failure
3. a: the length of a surface crack (one-half the
length of an internal crack)
4. Unit of KIC: MPa • m1/2
a
KC: fracture toughness
the critical (c) value of the stress intensity factor at a
crack tip necessary to produce failure
a
The three modes of crack surface displacement
(a) mode I, opening or tensile mode
plane strain fracture toughness
p a)
KIC = Ysc (
1/2
(b) mode II, sliding mode
(c) mode III, tearing mode
9
Room-temperature mechanical properties for selected engineering materials
Fracture Toughness
Metals/
Alloys
Graphite/
Ceramics/
Semicond
Polymers
100
K Ic (MPa · m 0.5 )
70
60
50
40
30
C-C(|| fibers) 1
Steels
Ti alloys
Al alloys
Mg alloys
20
Al/Al oxide(sf) 2
Y2 O 3 /ZrO 2 (p) 4
C/C( fibers) 1
Al oxid/SiC(w) 3
Si nitr/SiC(w) 5
Al oxid/ZrO 2 (p) 4
Glass/SiC(w) 6
10
7
6
5
4
Diamond
Si carbide
Al oxide
Si nitride
PET
PP
3
PVC
2
1
0.7
0.6
0.5
Composites/
fibers
PC
<100>
Si crystal
<111>
Glass -soda
Concrete
PS
Glass 6
Polyester
10
Design Against Crack Growth
• Crack growth condition:
K ≥ Kc = Ys pa
• Largest, most stressed cracks grow first!
--Result 2: Design stress
--Result 1: Max. flaw size
dictates max. flaw size.
dictates design stress.
Kc
<
Y pamax
sdesign
amax
1 ⎛ Kc
< ⎜
p ⎜⎝ Ys design
⎞
⎟
⎟
⎠
2
amax
s
fracture
fracture
no
fracture
no
fracture
amax
s
Design Example: Aircraft Wing
• Material has Kc = 26 MPa-m0.5
• Two designs to consider...
Design A
sc =
• Use...
Design B
--use same material
--largest flaw is 4 mm
--failure stress = ?
--largest flaw is 9 mm
--failure stress = 112 MPa
Kc
Y pamax
• Key point: Y and Kc are the same in both designs.
--Result:
112 MPa 9 mm
(s
c
amax
4 mm
)= (s
A
c
amax
)
B
Answer: ( sc )B = 168 MPa
• Reducing flaw size pays off!
11
Loading Rate
• Increased loading rate...
• Why? An increased rate
gives less time for
dislocations to move past
obstacles.
-- increases sy and TS
-- decreases %EL
s
TS
sy
e
larger
e
smaller
TS
sy
e
Impact fracture testing
Charpy test of impact energy
1. A notched specimen – stress
concentrating
2. Loading applied rapidly
3. impact energy - the energy
necessary to fracture the test
specimen
Charpy test
Izod test
The swinging pendulum
The intial height h
The final height h’
12
Cu: fcc, ductile
1040 carbon steel: Fe-0.4C-0.75 Mn
Mg alloy: hcp, relatively brittle
J: joule. 1J = 1N • m
Toughness obtained in a tensile test
Fracture
Toughness = ∫ sde , the area under the s - e curve
Unit of toughness: N/m2
Or (N/m2) • (mm / mm) = N •m / (mm)3
Toughness: the energy necessary to fracture per unit volume of
material
13
Temperature dependence of the Charpy V-notch inpact energy and shear
fracture for an A283 steel.
Brittle
More Ductile
Schematic curves for the three general types of impact energyversus-temperature behavior
14
Ductile-to-Brittle Transtion Temperature (DBTT)
1. in bcc alloys
2. an abrupt drop in ductility and toughness as the temperature is lowered
Plain-carbon steels with various
carbon levels
Fe-Mn-0.05C alloys with various
manganese levels
Design Strategy: Stay Above The DBTT!
• Pre-WWII: The Titanic
Reprinted w/ permission from R.W. Hertzberg,
"Deformation and Fracture Mechanics of Engineering
Materials", (4th ed.) Fig. 7.1(a), p. 262, John Wiley and
Sons, Inc., 1996. (Orig. source: Dr. Robert D. Ballard,
The Discovery of the Titanic.)
• WWII: Liberty ships
Reprinted w/ permission from R.W. Hertzberg,
"Deformation and Fracture Mechanics of Engineering
Materials", (4th ed.) Fig. 7.1(b), p. 262, John Wiley and
Sons, Inc., 1996. (Orig. source: Earl R. Parker,
"Behavior of Engineering Structures", Nat. Acad. Sci.,
Nat. Res. Council, John Wiley and Sons, Inc., NY,
1957.)
• Problem: Used a type of steel with a DBTT ~ Room temp.
15
Fatique
•
Fatique = failure under cyclic stress
•
failure after N cycles
•
load < T.S
Fatique-testing apparatusfor making rotating-bending tests
16
• Stress varies with time.
-- key parameters are S, sm, and frequency
s
smax
sm
S
smin
time
S = stress amplitude, sm= mean stress,
sm = (smax + smin)/2
S = (smax – smin)/2
Load ratio R = smin/smax
Random stress cycle.
• Key points: Fatigue...
--can cause part failure, even though smax < sc.
--causes ~ 90% of mechanical engineering failures.
Typical fatique curve (S-N curve)
or fatigue strength
17
Fatigue Design Parameters
• Fatigue limit, Sfat:
S = stress amplitude
--no fatigue if S < Sfat
case for
steel (typ.)
unsafe
Sfat
safe
10
3
5
7
9
10
10
10
N = Cycles to failure
• Sometimes, the
fatigue limit is zero! (a
material does not display a
fatique limit)
S = stress amplitude
case for
Al (typ.)
unsafe
safe
10
3
5
Adapted from Fig.
8.19(b), Callister 7e.
7
9
10
10
10
N = Cycles to failure
Fatique S-N probability of failure curves for a 7075-T6 Al alloy
18
Fatigue Mechanism
• Crack grows incrementally
da
m
= (D K )
dN
typ. 1 to 6
~ (Ds ) a
increase in crack length per loading cycle
• Failed rotating shaft
crack origin
--crack grew even though
Kmax < Kc
--crack grows faster as
• Ds increases
• crack gets longer
• loading freq. increases.
Texture of the fatigue fracture surface – clamshell or beachmark texture
Intrusions and extrusions. SEM.
19
Transmission electron fractograph showing fatique
striations in aluminum
Each striationis thougth to represent the advance distance
of a crack front during a single load cycle, striation width
deponds on and incearses with increasing stress range.
Improving Fatigue Life
1. Decrease the mean stress
level
2. Impose a compressive surface
stress (to suppress surface cracks
from growing)
--Method 1: shot peening
--Method 2: carburizing
shot
C-rich gas
put
surface
into
compression
3. Remove stress concentrators.
bad
better
bad
better
20
SUMMARY
• Engineering materials don't reach theoretical strength.
• Flaws produce stress concentrations that cause
premature failure.
• Sharp corners produce large stress concentrations
and premature failure.
• Failure type depends on T and stress:
- for noncyclic s and T < 0.4Tm, failure stress decreases with:
- increased maximum flaw size,
- decreased T,
- increased rate of loading.
- for cyclic s:
- cycles to fail decreases as Ds increases.
- for higher T (T > 0.4Tm):
- time to fail decreases as s or T increases.
Homework 8.6 and 8.12, p246
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