exam 2 grade distribution. stress-strain behavior (room t): ideal vs real materials ts
TRANSCRIPT
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Exam 2 Grade Distribution
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0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Grade
Fre
qu
ency
61.187.021.561.073.0
Median:Mode:
Max:Min:
Average:
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• Stress-strain behavior (Room T):
Ideal vs Real Materials
TS << TSengineeringmaterials
perfectmaterials
E/10
E/100
0.1
perfect mat’l-no flaws
carefully produced glass fiber
typical ceramic typical strengthened metaltypical polymer
• DaVinci (500 yrs ago!) observed... -- the longer the wire, the smaller the load for failure.• Reasons: -- flaws cause premature failure. -- Larger samples contain more flaws!
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Concentration of Stress at a Flaw
• First solved mathematically by Inglis in 1913• Solved the elasticity problem of an elliptical hole in an otherwise uniformly loaded plate
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Mathematical Solution of Stress Concentration
Where: t = radius of curvature
o = applied stress
eff = stress at crack tip
ott
oeff Ka
2/1
2
t
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Engineering Fracture Design
r/h
sharper fillet radius
increasing w/h
0 0.5 1.01.0
1.5
2.0
2.5
Stress Conc. Factor, Kt
max
o=
Avoid sharp corners!
r , fillet
radius
w
h
o
max
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Fracture Toughness
Define Stress Intensity:
aYK oI Stress Intensity Factor:
[Units: ksi√in]
Geometrically determined factor
Typically defined as: f (a/w)
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Examples
Source: T. L. Anderson, Fracture Mechanics Fundamentals and Applications, 3rd Edition, Taylor & Francis, 2005.
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Plane Strain Fracture Toughness, KIC
If you measure stress intensity you find:
Source: T. L. Anderson, Fracture Mechanics Fundamentals and Applications, 3rd Edition, Taylor & Francis, 2005.
1. That there is a critical value for a given geometry and material
2. That there is an effect of specimen thickness
3. That the critical values approaches a limit, KIC
KIC is a material constant
In fracture mechanics theory, a crack will not propagate unless the applied stress intensity is greater than the critical stress intensity, KIC
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Crack PropagationCracks propagate due to sharpness of crack tip • A plastic material deforms at the tip, “blunting” the
crack.
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
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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 = c/0
For ductile => replace s by s + p
where p is plastic deformation energy
212
/
sc a
E
i.e., m > c
or Kt > Kc
Griffith Crack Criterion
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• Crack growth condition:
• Largest, most stressed cracks grow first!
Design Against Crack Growth
K ≥ KIC = aY c
--Result 1: Max. flaw size dictates design stress.
maxaY
K ICdesign
amax
no fracture
fracture
--Result 2: Design stress dictates max. flaw size.
2
max
1
design
IC
Y
Ka
amax
no fracture
fracture
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• 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 = ?
• Key point: Y and Kc are the same in both designs.
Answer: MPa 168)( B c• Reducing flaw size pays off!
• Material has KIc = 26 MPa-m0.5
Design Example: Aircraft Wing
• Use...maxaY
K ICc
c amax A
c amax B
9 mm112 MPa 4 mm --Result:
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Loading Rate
• Increased loading rate... -- increases y and TS -- decreases %EL
• Why? An increased rate gives less time for dislocations to move past obstacles.
y
y
TS
TS
larger
smaller
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Charpy (or Izod) Impact Testing
final height initial height
• Impact loading: -- severe testing case -- makes material more brittle -- decreases toughness
(Charpy)
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• Increasing temperature... --increases %EL and Kc
• Ductile-to-Brittle Transition Temperature (DBTT)...
Temperature
BCC metals (e.g., iron at T < 914°C)
Imp
act
Ene
rgy
Temperature
High strength materials (y > E/150)
polymers
More Ductile Brittle
Ductile-to-brittle transition temperature
FCC metals (e.g., Cu, Ni)
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Fatigue• Fatigue = failure under cyclic stresses
• Stress varies with time. -- key parameters are S, m, and
frequency
max
min
time
mS
• Key points: Fatigue... --can cause part failure, even though max < c. --causes ~ 90% of mechanical engineering failures.
tension on bottom
compression on top
countermotor
flex coupling
specimen
bearing bearing
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Bauschinger Effect
• In most metals there is a hysteresis in stress-strain behavior
• Leads to accumulative damage to the microstructure under cyclic loading
• Think of a paper clip…
Source: Reed-Hill, Abbaschian, Physical Metallurgy Principles, 3rd Edition, PWS Publishing Company, 1994.
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Micromechanisms of Fatigue
Source: T. L. Anderson, Fracture Mechanics Fundamentals and Applications, 3rd Edition, Taylor & Francis, 2005.
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Formation of Slip Bands During Cyclic Loading
Source: Reed-Hill, Abbaschian, Physical Metallurgy Principles, 3rd Edition, PWS Publishing Company, 1994.
Copper single crystals – persistent slip bands
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Fatigue FractographyMicroscopic Fatigue Striations (SEM)
Source: Reed-Hill, Abbaschian, Physical Metallurgy Principles, 3rd Edition, PWS Publishing Company, 1994.
Source: T. L. Anderson, Fracture Mechanics Fundamentals and Applications, 3rd Edition, Taylor & Francis, 2005.
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Fatigue Fractography
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Fatigue Striations
Hardened Steel Sample tested in 3-point bending