Exam 2 Grade Distribution

0

2

4

6

8

10

12

14

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:

• 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!

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

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

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

Fracture Toughness

Define Stress Intensity:

aYK oI Stress Intensity Factor:

[Units: ksi√in]

Geometrically determined factor

Typically defined as: f (a/w)

Examples

Source: T. L. Anderson, Fracture Mechanics Fundamentals and Applications, 3rd Edition, Taylor & Francis, 2005.

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

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

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

• 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

• 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:

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

Charpy (or Izod) Impact Testing

final height initial height

• Impact loading: -- severe testing case -- makes material more brittle -- decreases toughness

(Charpy)

• 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)

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

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.

Micromechanisms of Fatigue

Source: T. L. Anderson, Fracture Mechanics Fundamentals and Applications, 3rd Edition, Taylor & Francis, 2005.

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

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.

Fatigue Fractography

Fatigue Striations

Hardened Steel Sample tested in 3-point bending