introduction to fracture mechanics - university of rhode island · 2020-05-28 · introduction to...
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Introduction to Fracture Mechanics
(Text section 5-12)
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Fracture Mechanics
Stress concentrations associated with cracks can
be very large
Materials can tolerate small cracks and flaws
When cracks grow to a critical size, sudden brittle
fracture can occur
Fracture mechanics is the study of the formation
and growth of cracks
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History of Fracture Mechanics
Theoretical foundation
– established by Inglis (1913), Griffith (1921)
First Application – World War II Liberty Ships
*
http://www.ssjeremiahobrien.org/
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• Built during World War II to replace ships sunk by German U-boats
• Between 1941 and 1945, over 2700 Liberty ships were built
• Inexperienced workers
• Low construction cost
• Quick to build
• Used welding instead of riveting (relatively new method)
• Early ships used a grade of steel that became brittle at cold temperatures
• Experienced cracks in the hull and deck
Liberty Ships(ref: en.wikipedia.org/wiki/Liberty_ships)
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SS Schenectady January 1943, Portland, Oregon
"Without warning and with a report which was heard for at least a mile, the deck and
sides of the vessel fractured just aft of the bridge superstructure. The fracture
extended almost instantaneously to the turn of the bilge port and starboard. The deck
side shell, longitudinal bulkhead and bottom girders fractured. Only the bottom
plating held. The vessel jack- knifed and the center portion rose so that no water
entered. The bow and stern settled into the silt of the river bottom."
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BREAKS IN TWO
“At 2:41 in the morning there was a loud pop and then a
tremendous tearing, grinding noise. Everybody but the watch was
in bed, but it brought us all out in a hurry.”
“She just seemed to bend in the middle a couple of times as she
rode the seas and then she just parted, the after and the bow end
separating about a third of the way back from the stern. The bow
end then disappeared in the darkness.
SS John P. Gaines November 1943, off the coast of Alaska
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Effect of Temperature on Brittle Failure of Steel
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Inglis (1913)
as b/a 0, y
On a microscopic level, localized
plastic deformation acts to reduce the
stress concentration factor and blunt
the crack tip, allowing the material to
tolerate small cracks (modeled using
elastic-plastic fracture mechanics)
Stress Concentration at Crack Tip
b
aaxy 21
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LEFM – Linear Elastic Fracture Mechanics
Apply theory of elasticity to determine the stress field at the tip of a crack for three types of crack propagation:
We’ll just consider Mode I
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Mode I – Crack Tip Stress Fields
2
3cos
2cos
2sin
2
2
3sin
2sin1
2cos
2
2
3sin
2sin1
2cos
2
r
a
r
a
r
a
xy
y
x
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Stress Intensity Factor
For mode I cracks, define:
Which gives the stress field equations as
aK I
2
3cos
2cos
2sin
2
2
3sin
2sin1
2cos
2
2
3sin
2sin1
2cos
2
r
K
r
K
r
K
Ixy
Iy
Ix
Units: MPa m
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Stress Intensity Factor
The stress intensity factor is a measure of the intensity of the stress field near the crack tip
For various load and geometric configurations, the stress intensity factor is given by
where is the stress intensity modification factor that can be found in design charts (Figs. 5-25 – 5-30)
aK I
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Stress Intensity Modification Factors
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Stress Intensity Modification Factors (cont.)
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Stress Intensity Modification Factors (cont.)
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Fracture Toughness
When the stress intensity reaches a critical value (called the Fracture Toughness), the crack will propagate
KI – Stress intensity factor
KIc – Critical stress intensity factor
(or fracture toughness)
Note, KI is a measure of the intensity of the stress field and KIc is a material property
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Fracture Toughness – Various Materials
Note inverse relation between yield strength and fracture toughness:
typically, as strength increases, fracture toughness decreases
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Fracture Criteria
Crack will propagate when the stress intensity
factor, KI, reaches a critical level, KIc
Factor of safety:
I
Ic
K
Kn
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Example 5-6
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Example 5-7