1. Trends with Materials Heat treatment will cause embrittlement
Cast Iron
4140 Steel Q&T
UTS = 1550 MPa
UTS = 950 MPa
More tempering provides
Lower strength
More ductile failure
Larger % carbon in steel
Higher ductile-to-brittle transition
Lower energy absorption
More brittle
From: Dowling, Callister

2. 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

3. Silver Bridge

4. 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

5. Ohio River

6. 4:58 PM December 15, 1967

7. Disaster Second most deadly U.S. bridge disaster 64 hit the water
18 rescued
46 dead or missing
31 vehicles on the bridge

8. Wreckage

9. Wreckage

10. Wreckage

11. What’s Different About Silver Bridge?
First “eyebar” suspension bridge in the U.S.
First bridge that used high-strength, heat- treated carbon steel
High Risk: new structure on a new scale, using new materials.

12. Silver Bridge Collapse
Collapse, Wearne, P. TV Books, NY 1999

13. Source

14. Cause of Failure Bridge Design? Eyebar Manufacturing Quality?
Material Choice?

15. Bridge Design at Fault? Partially! Steel Eyebar Suspension
Suspended “Bicycle Chain”
Weakest Link, No Redundancy
Cable Suspension has hundreds of links
Partially!

16. Failed Eyebar

17. 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.”

18. Failure Evidence

19. 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.

20. Eyebar Loading
12” wide
2” thickness 6 12

21. Eyebar 330 Failure Sketch 12” wide 2” thickness 6 12
1/8” corroded crack
Brittle Appearance
12” wide
2” thickness 6 12
Ductile Appearance

22. Crack on Eyebar 330

23. Conditions of Failure Crack formed by original forging operation
Quenched and tempered steel
Stress corrosion cracking
32oF ambient temperature

24. Wireless Sensors for Bridges

25. Fracture Mechanics Timeline
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 (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
Important parameters
G : strain energy release rate
K: stress intensity factor
Different but similar to Kt
R: crack growth resistance
J : J-integral

26. 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-
energy stored in material as it is elastically deformed
this energy is released when the crack propagates
creation of new surfaces requires energy
plastic

27. Uncracked plate elastic energy
Griffith Approach
applied
Uncracked plate elastic energy 2a
Introduce a crack which
Reduces elastic energy by:
Increases total surface energy by: l
For a crack to grow, the energy provided by new surfaces must equal that lost by elastic relief 2w
applied
DUel2a = DG
thickness, t

28. Griffith Approach l a thickness, t applied
Minimum criterion for stable crack growth:
Strain energy goes into surface energy a 2a 2a l 2w
applied
thickness, t

29. Plastic Energy Term l a thickness, t applied
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)
Orowan introduced plastic deformation energy, p
Materials which exhibit plastic deformation absorb much more energy, removing it from the crack tip a 2a 2a l 2w
applied
thickness, t

30. When Does a Crack Propagate?
Crack propagates if above critical stress
where
E = modulus of elasticity
s = 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
i.e., sm > sc
or Kt > Kc

31. Energy Release Rate, G l a thickness, t applied
Irwin chose to define a term, G, that characterizes the energy per unit crack area required to extend the crack: a 2a
Change in
crack area 2a l 2w
Comparing to the previous expression, we see that:
Works well when plastic zone is small (“fraction of crack dimensions”)
applied
thickness, t

32. Experimentally Measuring GF x
Load cracked sample in elastic range a
Fix grips at given displacement
Load
a + Da DU t
Allow crack to grow length a
a + a a
Unload sample
Compare energy under the curves
Displacement
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 F x

33. Stress Based Crack Analysis
Westergaard, Irwin analyzed fracture of cracked components using elastic-based stress theory
Defined three basic modes for crack loading
From: Socie

34. Stress Intensity Factor, K
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)

35. Stress Intensity Factor, K Stress Fields (near crack tip)
Note: as r  0, stress fields  
Plane stress - too thin to support stress through thickness
Plane strain - so thick that constrains strain through thickness

36. 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
Design B
--use same material
--largest flaw is 4 mm
--failure stress = ?
• Use...
• Key point: Y and Kc are the same in both designs.
9 mm
112 MPa
4 mm
--Result:
Answer:
• Reducing flaw size pays off!

37. Design Against Crack Growth
• Crack growth condition:
K ≥ Kc =
• Largest, most stressed cracks grow first!
--Result 1: Max. flaw size
dictates design stress. s
amax no
fracture
--Result 2: Design stress
dictates max. flaw size.
amax s no
fracture

38. Stress Intensity Factor, K Displacement Fields
Displacement in x-direction
Displacement in y-direction

39. Fracture Toughness, KC Critical value of K
specimen geometry
(sometimes f(a/W))
flaw size
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….

40. New Concepts & Terms 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
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