Trends with Materials
•
Heat treatment will cause
embrittlement
•
•
Cast Iron
4140 Steel Q&T
•
•
•
More tempering provides
•
•
•
UTS = 1550 MPa
UTS = 950 MPa
Lower strength
More ductile failure
Larger % carbon in steel
•
•
Higher ductile-to-brittle
transition
Lower energy absorption
•
More brittle
From: Dowling, Callister
Intro to Fracture Mechanics
•
•
•
Recall our discussion of Theoretical Strength …
• Cohesive strength was estimated to be ~E/4 to E/8
• Way too high for most materials
• We rationalized that overprediction was due to flaws
Recall our discussion of Material Properties ...
• Some materials failed in a brittle manner, while some were
ductile
• Plastic deformation was introduced
We will now:
• Attempt to understand “quantitatively” the effect of defects
• Tie in to our discussion of ductile vs. brittle behavior
• Understand why brittle materials are more effected by these
defects
• Understand how ductile materials absorb some energy
Silver Bridge
Statistics
•
•
•
•
•
•
•
•
Completed 19 May 1928
Connects
• Huntington, VA to Middleport, OH
• Charleston, VA to Dayton, OH
Major east-west connection for US Route 35
Two lanes of automobile traffic
1750 feet in length
Steel Eyebar Suspension Bridge
Aluminum Paint (“Silver Bridge”)
A major east-west connection for US Route 35
connecting Charleston, WV and major cities in Ohio
Ohio River
4:58 PM December 15, 1967
Disaster
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•
•
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Second most deadly U.S. bridge disaster
64 hit the water
18 rescued
46 dead or missing
31 vehicles on the bridge
Wreckage
Wreckage
Wreckage
What’s Different About Silver Bridge?
•
•
•
First “eyebar” suspension bridge in the U.S.
First bridge that used high-strength, heattreated carbon steel
High Risk: new structure on a new scale, using
new materials.
Silver Bridge Collapse
Collapse, Wearne, P. TV Books, NY 1999
Source
Cause of Failure
•
•
•
Bridge Design?
Eyebar Manufacturing Quality?
Material Choice?
Bridge Design at Fault?
•
•
•
•
Steel Eyebar Suspension
Suspended “Bicycle Chain”
Weakest Link, No Redundancy
Cable Suspension has hundreds of links
Partially!
Failed Eyebar
Failure Evidence
John Bennet, US Bureau of Standards
•
“The Ohio River there is very heavily traveled so
the U.S. Corps of Engineers had taken all the
debris and just piled it on the shore – it was a
terrific mess.”
•
“Fortunately, each piece had been
photographed as they took it out.”
Failure Evidence
Photograph of Failed Eyebar 330
John Bennet, US Bureau of Standards
•
“Looking at it, the fractures on the
two sides were completely
different.
•
“One side was very straight,
almost like a saw cut.
•
“The other side was extensively
deformed, the metal bent and the
paint chipped off.
Eyebar Loading
12” wide
2” thickness
6
12
Eyebar 330 Failure Sketch1/8” corroded
crack
Brittle Appearance
12” wide
2” thickness
6
12
Ductile Appearance
Crack on Eyebar 330
Conditions of Failure
•
•
•
•
Crack formed by original forging operation
Quenched and tempered steel
Stress corrosion cracking
o
32 F ambient temperature
Wireless Sensors for Bridges
Fracture Mechanics Timeline
• Important parameters
• G : strain energy release
rate
• K: stress intensity factor
• Different but similar to
Kt
• R: crack growth
resistance
• J : J-integral
• Origins of modern fracture mechanics
date to 1920
• A.A. Griffith energy balance approach
• All energy that is created must be used
• George Irwin defines G in 1950’s
• H.M. Westergaard published solutions
using a stress based approach in 1939
(Irwin, 1958)
• Use K, which characterizes the magnitude of
the stresses in the vicinity of a crack, then can
use a universal stress field equation for
different geometries
• Irwin proposes R-curve in 1960
• Jim Rice introduces J-integral in 1968
Crack Propagation
Cracks propagate due to sharpness of crack tip
• A plastic material deforms at the tip, “blunting” the
crack.
deformed region
brittle
Energy balance on the crack
• Elastic strain energy-
plastic
• energy stored in material as it is elastically deformed
• this energy is released when the crack propagates
• creation of new surfaces requires energy
Griffith Approach
applied

Uncracked plate elastic energy
U el  12  2 w* l * t   2 E 2 w* l * t 
2

Introduce a crack which

l
Reduces elastic energy by:
U el2a
2a

2

 2 E 2a 2 * t

Increases total surface energy by:
  2 s * 2at
2w
applied
thickness, t

For a crack to grow, the energy provided by
new surfaces must equal that lost by elastic
relief
DUel2a = D
Griffith Approach
applied

Minimum criterion for stable crack growth:
Strain energy goes into surface energy
U  

 a a
l
Da
2a
2a
  2
2 

 2 a t  
a  2 E


 s 4 a t 
a
 2 a  2 E  s
2w
applied
thickness, t
 
2Es
a
Plastic Energy Term
applied
l

Previous derivation is for purely elastic material

Most metals and polymers experience some
plastic deformation

Strain energy (U) goes into surface energy (G)
& plastic energy (P)
U    P


 a a a
2a
2a
Da

2w
Orowan introduced plastic deformation energy,
p
 
applied
thickness, t

2 E ( s  p )
a
 p   s
Materials which exhibit plastic deformation
absorb much more energy, removing it from the
crack tip
When Does a Crack Propagate?
Crack propagates if above critical stress
i.e., m > c
or
where
•
•
•
•
Kt > Kc
1/ 2
 2E s 
c  

 a 
E = modulus of elasticity
s = specific surface energy
a = one half length of internal crack
Kc = c/0
For ductile => replace s by s + p
where p is plastic deformation energy
Energy Release Rate, G
applied

Irwin chose to define a term, G, that
characterizes the energy per unit crack area
required to extend the crack:
1 U
G
t a
l
G
2a
 a 2
Da
 
2w

E
EG
a
Comparing to the previous expression, we see
that:
G  2 s   p 
applied
thickness, t
Change in
crack area

Works well when plastic zone is small (“fraction
of crack dimensions”)
Experimentally Measuring G
x
Load cracked sample in elastic range
t
a
Load
F
a + Da
a
a + Da
Allow crack to grow length Da
Unload sample
Compare energy under the curves
Displacement
G
F
DU
Fix grips at given displacement
1 DU
t Da
x
 2 a
G Ic 
at fracture
E
G is the energy per unit crack area
needed to extend a crack
Experimentally measure combinations
of stress and crack size at fracture to
determine GIc
Stress Based Crack Analysis
•
Westergaard, Irwin analyzed fracture of cracked components using
elastic-based stress theory
• Defined three basic modes for crack loading
Mode I
opening
Mode II
in-plane shear
From: Socie
Mode III
out-of-plane shear
Stress Intensity Factor, K
K   a Y
Stress intensity factor
specimen geometry
flaw size
operating stresses
• Critical parameter is now based on:
• Stress
• Flaw Size
• Specimen geometry included using “correction factor”
• crack shape
• specimen size and shape
• type of loading (i.e. tensile, bending, etc)
Stress Intensity Factor, K
Stress Fields (near crack tip)
x 
K


3
cos  1  sin sin 
2 r
2
2
2
y
K


3
cos  1  sin sin 
2 r
2
2
2
 xy 
K


3
cos sin cos
2 r
2
2
2
Note: as r  0, stress fields  
Plane stress - too thin to support stress through thickness
 z  0 z  0
Plane strain - so thick that constrains strain through thickness
 z    x   y   z  0
Design Example: Aircraft Wing
• Material has Kc = 26 MPa-m0.5
• Two designs to consider...
Design A
--largest flaw is 9 mm
--failure stress = 112 MPa
• Use...
Kc
c 
Y amax
Design B
--use same material
--largest flaw is 4 mm
--failure stress = ?
• Key point: Y and Kc are the same in both designs.
--Result:
112 MPa
c
9 mm
amax
A  c
• Reducing flaw size pays off!

4 mm
amax
B
Answer: (c )B  168 MPa
Design Against Crack Growth
• Crack growth condition:
K ≥ Kc = Y a
• Largest, most stressed cracks grow first!
--Result 1: Max. flaw size
dictates design stress.
design 
Kc
Y amax
--Result 2: Design stress
dictates max. flaw size.
amax
1  K c

  Y design




amax

fracture
no
fracture
fracture
amax
no
fracture

2
Stress Intensity Factor, K
Displacement Fields

Displacement in x-direction
u

Displacement in y-direction
v
u
x
x
K
r
  3 

cos 
 1  2 sin2 
2 G 2
2  1 
2
K
r
  3 

sin 
 1  2 cos 2 
2 G 2
2  1 
2
v
y 
x
 xy 
u v

y x
Fracture Toughness, KC
Critical value of K
a
KC   c  a Y  
w
fracture toughness
specimen geometry
flaw size
(sometimes f(a/W))
Fracture strength
From: Callister
•
•
Fracture occurs when stress exceeds critical value, c
Geometric correction includes ratio between crack length and specimen width
•
•
•
•
Empirical mathematical expressions
Y calibration curves
Units are MPam or psiin
Measure of material’s resistance to brittle fracture
•
But some materials undergo large plastic deformations prior to failure….
New Concepts & Terms
•
•
•
•
Fracture Process
– Ductile vs. brittle
– Crack formation
– Crack propagation
• Stable
• Unstable
Ductile Fracture
– Cup-and-cone
– Shear vs. fibrous regions
Brittle Fracture
– Transgranular (cleavage)
– Intergranular
– Chevron and fan-like patterns
Impact Testing
– Charpy and Izod
– Falling Weight
– Maximum load
– Energy absorption
– Ductile-brittle transition
•
•
•
•
Stress Concentration Factor, Kt
– Inglis approach
Strain Energy Release Rate, G
– Elastic strain energy, U
– Surface energy, s
– Plastic deformation energy p
– Griffith & Irwin approaches
– Experimental measures of G
Stress Intensity Factor, K
– Mode I, II, III
– Fracture toughness
– Plain strain fracture toughness
– Plastic zone size
– Plane strain fracture toughness
testing
– Minimum dimensions
– Relation to G
Designing with K
– Critical crack size
–
Leak before break