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