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3 – Fracture of Materials
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Part of the top of an Aloha Airlines jet peeled of during the flight
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Liberty Ship failures involved both brittle fractures and fatigue fractures.
Of 2700 ships built, 400 suffered hull and deck fractures. 90 were
considered serious and in 20 ships fracture was essentially total with 10
ships breaking in half without warning.
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Outline
Fracture of Materials
Types of Fracture
Brittle Fracture
Ductile Fracture
Brittle to Ductile Transition
Fracture Mechanics
Stress Concentration
LEFM
K&G
Kc
Appendix - Griffith Theory
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Fracture
Fracture: separation of a body into pieces due to
stress, at temperatures below the melting point.
Steps in fracture:
Crack formation
Crack propagation
Depending on the ability of material to undergo plastic
deformation before the fracture two fracture modes
can be defined - ductile or brittle
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Ductile fracture - most metals (not too cold):
Extensive plastic deformation ahead of crack
Crack is “stable”: resists further extension unless
applied stress is increased
Brittle fracture - ceramics, ice, cold metals:
Relatively little plastic deformation
Crack is “unstable”: propagates rapidly without
increase in applied stress
Ductile fracture is preferred in most applications
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Brittle vs. Ductile Fracture
Ductile materials - extensive plastic deformation and
energy absorption (“toughness”) before fracture
Brittle materials - little plastic deformation and low
energy absorption before fracture
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Brittle vs. Ductile Fracture
(a)Brittle fracture; (b) Shearing fracture in single crystal
(c) Completely ductile (d) Ductile fracture
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Ductile Fracture
(a)Necking, (b) Cavity Formation,
(c) Cavity coalescence to form a crack,
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Ductile Fracture
(d) Crack propagation, (e) Fracture
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Ductile Fracture
(Cap-and-cone fracture in Al)
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Ductile Fracture
Scanning Electron Microscopy: Fractographic studies at high resolution.
Spherical “dimples” correspond to micro-cavities that initiate crack formation.
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Brittle Fracture
No appreciable plastic deformation
Crack propagation is very fast
Crack propagates nearly perpendicular to
the direction of the applied stress
Crack often propagates by cleavage –
breaking of atomic bonds along specific
crystallographic planes (cleavage
planes)
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Brittle Fracture
Brittle fracture in a mild steel
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Brittle Fracture
A. Transgranular fracture: Fracture cracks pass
through grains. Fracture surface have faceted
texture because of different orientation of cleavage
planes in grains.
B. Intergranular fracture: Fracture crack
propagation is along grain boundaries (grain
boundaries are weakened or embrittled by
impurities segregation etc.)
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Brittle Fracture
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Ductile-to-Brittle Transition
Ductile-to-Brittle Transition: As temperature
decreases a ductile material can become brittle
Alloying usually increases the ductile-to-brittle
transition temperature.
FCC metals remain ductile down to very low
temperatures.
For ceramics, this type of transition occurs at
much higher temperatures than for metals.
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Ductile-to-Brittle Transition
DBTT: Ductile-Brittle Transition Temperature
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Fracture Mechanics & Fracture Toughness
• Stress Concentration
• Energy Release Rate
• Crack Tip Plasticity
• Fracture Toughness
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Stress Concentration
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Stress Concentration
For a long crack oriented perpendicular to the applied
stress the maximum stress near the crack is:
m  2 (a/)1/2
(1)
where σ is the applied external stress, a is the halflength of the crack, and ρ the radius of curvature of the
crack tip. (note that a is half-length of the internal flaw,
but the full length for a surface flaw).
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Stress Concentration Factor
The ratio of the maximum stress and the nominal
applied tensile stress is denoted as the stress
concentration factor, Kt, where Kt can be calculated
by Equation 1. The stress concentration factor is a
simple measure of the degree to which an external
stress is amplified at the tip of a small crack.
m
a 1/ 2
Kt 
 2( )


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Stress Concentration
The stress distribution at the crack tip in a
thin plate for an elastic solid is given by:
(2)
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Stress Concentration
K: Stress Intensity Factor
K  f a
(3)
Unit of K: psiin, MN/(m3/2), or MPam
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Stress Concentration
Substitute K into Equation (3):
(4)
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Stress Concentration
Opening
In-plane Shear
Out-of-Plane Shear
Fracture Modes
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Energy Release Rate
Energy Release Rate: In 1956, Irwin defined an
energy release rate, G, which is a measure of the
energy available for an increment of crack extension:
G = - dU/dA
Since G is obtained from the derivative of a potential, it
is also called the crack extension force or crack
driving force
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Energy Release Rate
Griffith Approach:
G = 2a/E
Where  is the nominal stress, a is half-length of
crack, E is Young’s Modulus
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Relationship between K and G
Mode I
Plain Stress:
G = KI2/E
Plain Strain:
G = KI2(1-2)/E
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Plain Stress vs. Plain Strain
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Crack Tip Plasticity
Irwin Approach
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Crack Tip Plasticity
Irwin Approach
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Crack Tip Plastic Zone Shape
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Fracture Toughness
Kc: If we assume a material fails locally at some
combination of stresses and strains, then crack
extension must occur at a critical K value. This Kc value,
which is a measure of fracture toughness, is a material
constant that is independent of the size and geometry of
the cracked body.
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Fracture Toughness Measurement
MEASURE STRENGTH WITH A CRACK OF KNOWN
LENGTH
SHARP CRACK
LINEAR ELASTIC?
SMALL PROCESS ZONE RULES
ASTM STANDARDS: E-399
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SUMMARY
Types of Fracture:
Brittle Fracture
Ductile Fracture
Ductile Fracture: Crack initiation, crack growth, fracture,
fractography
Brittle Fracture: Intergranular & Transgranular
(Cleavage)
Brittle to Ductile Transition: DBTT
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SUMMARY – Cont’d
Stress Concentration
Stress distribution in front of the crack tip
Stress concentration factor
Linear Elastic Fracture Mechanics
K: Stress Intensity Factor
G: Energy Release Rate
Relationship between K & G
Crack tip plasticity
Fracture Toughness
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Appendix
- Griffith Theory for Brittle Fracture
For defect-free material:
th ~ E/10 is orders of
2 E
 th 
a0
magnitude higher than
usually observed
Where E – Young’s modulus
 – specific surface energy
a0 – lattice parameter
th – theoretical strength of the material
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Energy criterion for brittle fracture
(Griffith Theory)
Energy criterion: when rate of
change of the release of
elastic energy with respect to
crack size  rate at which
energy is consumed to create
new surfaces, fracture occurs
Onset:
d  2 2
a 
da   E
 d
 
4 s a 
 da
Where  – Griffith’s stress (the critical stress)
s – surface energy/per unit area
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Griffith Equation
2 E s
G 
a
• Griffith equation is only for glasses & ceramics –
brittle fracture, no plastic deformation occurs prior to
fracture
• Metals and polymers  plastic deformation occurs
before fracture  modified Griffith equation (Orowan
equation)
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