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Chapter 9 -

In the Name of God

1

Materials Science

Dr Mohammad HeidariDr Mohammad HeidariDr Mohammad HeidariDr Mohammad Heidari----RaraniRaraniRaraniRarani

Chapter 9 - 2

ISSUES TO ADDRESS...• How do cracks that lead to failure form?• How is fracture resistance quantified? How do the fracture

resistances of the different material classes compare?• How do we estimate the stress to fracture?• How do loading rate, loading history, and temperature

affect the failure behavior of materials?

Ship-cyclic loadingfrom waves.

Computer chip-cyclicthermal loading.

Hip implant-cyclicloading from walking.

Chapter 9: Mechanical Failure

Chapter 9 - 3

Fracture mechanisms

• Ductile fracture– Accompanied by significant plastic

deformation

• Brittle fracture– Little or no plastic deformation– Catastrophic

Chapter 9 - 4

Ductile vs Brittle Failure

Very Ductile

ModeratelyDuctile BrittleFracture

behavior:

Large Moderate%AR or %EL Small• Ductile fracture is usually more desirable than brittle fracture!

• Classification:

Ductile:Warning before

fracture

Brittle:No

warning

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Chapter 9 - 5

• Ductile failure:-- one piece-- large deformation

Example: Pipe Failures

• Brittle failure:-- many pieces-- small deformations

Chapter 9 - 6

• Evolution to failure:

• Resultingfracturesurfaces(steel)

50 mm

particlesserve as voidnucleationsites.

50 mm

100 mm

Moderately Ductile Failure

necking

σvoid

nucleationvoid growth and linkage

shearing at surface fracture

Chapter 9 - 7

Ductile vs. Brittle Failure

Cup-and-cone fracture in Aluminum

Brittle fracture in ductile steel

Chapter 9 - 8

Brittle Failure

Arrows indicate point at which failure originated

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Chapter 9 - 9

• Intergranular(between grains)

304 S. Steel (metal)Reprinted w/permission from "Metals Handbook", 9th ed, Fig. 633, p. 650. Copyright 1985, ASM International, Materials Park, OH. (Micrograph by J.R. Keiser and A.R. Olsen, Oak Ridge National Lab.)

• Intragranular(within grains)

316 S. Steel (metal)

Reprinted w/ permission from "Metals Handbook", 9th ed, Fig. 650, p. 357. Copyright

1985, ASM International, Materials Park, OH.

(Micrograph by D.R. Diercks, Argonne National Lab.)

Brittle Fracture Surfaces

Chapter 9 - 10

Chapter 9 - 11

• Stress-strain behavior (Room T):

Ideal vs Real Materials

TS << TSengineeringmaterials

perfectmaterials

σ

ε

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!

Reprinted w/ permission from R.W. Hertzberg, "Deformation and Fracture Mechanics of Engineering Materials", (4th ed.) Fig. 7.4. John Wiley and Sons, Inc., 1996.

Chapter 9 - 12

Theoretical fracture stress• Based on inter-atomic forces:

ππλσ E

x

Ec ≈=

0

• Based on specific surface energy, γs (J/m2):

2/1

0

=

x

E sc

γσ

x0: atomic radius

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Chapter 9 - 13

Flaws are Stress Concentrators!

ρt

Chapter 9 - 14

Effect of stress Concentration at Crack Tip

where ρt = radius of curvatureσo = applied stressσm = stress at crack tip

ott

o Ka σρ

σσ =

=

2/1

max 2

2/1

4

=a

E sc

γσ

Chapter 9 - 15

where– E = modulus of elasticity– γs = specific surface energy– a = one half length of internal crack

Crack propagates if above critical stress:

For ductile => replace γs by γs + γp

where γp is plastic deformation energy

212

/

sc a

E

πγ=σ

σm > σc

• Griffith Crack (Energy based method)

Chapter 9 - 16

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, K t=

• Avoid sharp corners!σ

r , fillet

radius

w

h

o

σmax

σmaxσ0

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Chapter 9 - 17

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

Chapter 9 -

Fracture Toughness

18

aYKI πσ=

Chapter 9 - 19

Fracture Toughness

Based on data in Table B.5,Callister & Rethwisch 3e.Composite reinforcement geometry is: f = fibers; sf = short fibers; w = whiskers; p = particles. Addition data as noted (vol. fraction of reinforcement):1. (55vol%) ASM Handbook, Vol. 21, ASM Int., Materials Park, OH (2001) p. 606.2. (55 vol%) Courtesy J. Cornie, MMC, Inc., Waltham, MA.3. (30 vol%) P.F. Becher et al., Fracture Mechanics of Ceramics, Vol. 7, Plenum Press (1986). pp. 61-73.4. Courtesy CoorsTek, Golden, CO.5. (30 vol%) S.T. Buljan et al., "Development of Ceramic Matrix Composites for Application in Technology for Advanced Engines Program", ORNL/Sub/85-22011/2, ORNL, 1992.6. (20vol%) F.D. Gace et al., Ceram. Eng. Sci. Proc., Vol. 7 (1986) pp. 978-82.

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibersPolymers

5

KIc

(MP

a ·

m0.

5)

1

Mg alloysAl alloys

Ti alloys

Steels

Si crystalGlass -soda

Concrete

Si carbide

PC

Glass 6

0.5

0.7

2

4

3

10

20

30

<100>

<111>

Diamond

PVC

PP

Polyester

PS

PET

C-C(|| fibers) 1

0.6

67

40506070

100

Al oxideSi nitride

C/C( fibers) 1

Al/Al oxide(sf) 2

Al oxid/SiC(w) 3

Al oxid/ZrO 2(p)4Si nitr/SiC(w) 5

Glass/SiC(w) 6

Y2O3/ZrO 2(p)4

Chapter 9 - 20

• Crack growth condition:

• Largest, most stressed cracks grow first!

Design Against Crack Growth

K ≥ Kc = aY πσ

-- Result 1: Max. flaw sizedictates design stress.

max

cdesign

aY

K

π<σ

σ

amaxno fracture

fracture

-- Result 2: Design stressdictates max. flaw size.

2

1

σπ<

design

cmax Y

Ka

amax

σno fracture

fracture

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Chapter 9 - 21

• 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 =σc• Reducing flaw size pays off!

• Material has Kc = 26 MPa-m0.5

Design Example: Aircraft Wing

• Use...max

cc

aY

K

π=σ

( ) ( )B max Amax aa cc σ=σ

9 mm112 MPa 4 mm-- Result:

Chapter 9 - 22

Loading Rate

• Increased loading rate...-- increases σy and TS-- decreases %EL

• Why? An increased rategives less time for dislocations to move past obstacles.

σ

ε

σy

σy

TS

TS

largerε

smallerε

Chapter 9 - 23

Brittle Fracture of Ceramics

• Characteristic Fracture behavior in ceramics– Origin point – Initial region (mirror) is flat

and smooth – After reaches critical

velocity crack branches• mist• hackle

Chapter 9 - 24

Impact Testing

final height initial height

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

(Charpy)

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Chapter 9 - 25

• Increasing temperature...-- increases %EL and Kc

• Ductile-to-Brittle Transition Temperature (DBTT) ...

Temperature

BCC metals (e.g., iron at T < 914°C): Low-strength steels

Impa

ct E

nerg

y

Temperature

High strength materials (σσσσ y > E/150)

polymers

More DuctileBrittle

Ductile-to-brittle transition temperature

Low-strength (FCC and HCP) metals (e.g., Al, Cu, Ni)

Chapter 9 - 26

• Fracture surface of A36 steel under V-notch Charpy impact test

Chapter 9 - 27

• Pre-WWII: The Titanic • WWII: Liberty ships

• Problem: Used a type of steel with a DBTT ≈ Room temp.

Reprinted w/ permission from R.W. Hertzberg, "Deformation and Fracture Mechanics of Engineering Materials", (4th ed.) Fig. 7.1(a), p. 262, John Wiley and Sons, Inc., 1996. (Orig. source: Dr. Robert D. Ballard, The Discovery of the Titanic.)

Reprinted w/ permission from R.W. Hertzberg, "Deformation and Fracture Mechanics of Engineering Materials", (4th ed.) Fig. 7.1(b), p. 262, John Wiley and Sons, Inc., 1996. (Orig. source: Earl R. Parker, "Behavior of Engineering Structures", Nat. Acad. Sci., Nat. Res. Council, John Wiley and Sons, Inc., NY, 1957.)

Design Strategy:Stay Above The DBTT!

Chapter 9 - 28

Fatigue• Fatigue = failure under cyclic stress.

• Stress varies with time.-- key parameters are S, σm, and

frequency

σmax

σmin

σ

time

σmS

• Key points: Fatigue...-- can cause part failure, even though σmax < σc.-- causes ~ 90% of mechanical engineering failures.

tension on bottom

compression on top

countermotor

flex coupling

specimen

bearing bearing

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Chapter 9 - 29

• Fatigue limit, Sfat:-- no fatigue if S < Sfat

Fatigue Design Parameters

Sfat

case for steel (typ.)

N = Cycles to failure103 105 107 109

unsafe

safe

S = stress amplitude

• Sometimes, thefatigue limit is zero! case for

Al (typ.)

N = Cycles to failure103 105 107 109

unsafe

safe

S = stress amplitude

Chapter 9 - 30

Fatigue Behavior of Polymers

• Fatigue limit:- PMMA, PP, PE

• No fatigue limit:- PET, Nylon (dry)

Chapter 9 - 31

• Crack grows incrementallytyp. 1 to 6

( ) a~ σ∆

increase in crack length per loading cycle

• Failed rotating shaft-- crack grew even though

Kmax < Kc-- crack grows faster as

• ∆σ∆σ∆σ∆σ increases• crack gets longer• loading freq. increases.

crack origin

Fatigue Mechanism

( )mKdNda ∆=

Chapter 9 - 32

Improving Fatigue Life

1. Impose a compressivesurface stress(to suppress surfacecracks from growing)

N = Cycles to failure

moderate tensile σmLarger tensile σm

S = stress amplitude

near zero or compressive σmIncreasing

σσσσm

--Method 1: shot peening

put surface

into compression

shot--Method 2: carburizing

C-rich gas

2. Remove stressconcentrators.

bad

bad

better

better

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Chapter 9 - 33

CreepSample deformation at a constant stress ( σσσσ) vs. time

Adapted fromFig. 9.35, Callister & Rethwisch 3e.

Primary Creep: slope (creep rate) decreases with time.

Secondary Creep: steady-statei.e., constant slope.

Tertiary Creep: slope (creep rate) increases with time, i.e. acceleration of rate.

σσ,ε

0 t

Chapter 9 - 34

• Occurs at elevated temperature, T > 0.4 Tm

Adapted from Figs. 9.36, Callister & Rethwisch 3e.

Creep

elastic

primarysecondary

tertiary

Chapter 9 - 35

• Strain rate is constant at a given T, σ-- strain hardening is balanced by recovery

stress exponent (material parameter)

strain rateactivation energy for creep(material parameter)

applied stressmaterial const.

• Strain rateincreasesfor higher T, σ

10

20

40

100

200

10-2 10-1 1Steady state creep rate (%/1000hr)ε ε ε ε s

Stress (MPa)427°C

538°C

649°C

Adapted fromFig. 9.38, Callister & Rethwisch 3e. (Fig. 9.38 is from Metals Handbook: Properties and Selection: Stainless Steels, Tool Materials, and Special Purpose Metals, Vol. 3, 9th ed., D. Benjamin (Senior Ed.), American Society for Metals, 1980, p. 131.)

−σ=εRTQ

K cns exp2ɺ

Secondary Creep

Chapter 9 - 36

Creep Failure• Estimate rupture time

S-590 Iron, T = 800°C, σ = 20 ksi• Failure:

along grain boundaries.

time to failure (rupture)

function ofapplied stress

temperature

L)t(T r =+ log20

appliedstress

g.b. cavities

• Time to rupture, tr

From V.J. Colangelo and F.A. Heiser, Analysis of Metallurgical Failures (2nd ed.), Fig. 4.32, p. 87, John Wiley and Sons, Inc., 1987. (Orig. source: Pergamon Press, Inc.)

L)t(T r =+ log20

1073K

Ans: tr = 233 hr

24x103 K-log hr

Adapted fromFig. 9.39, Callister & Rethwisch 3e. (Fig. 9.39 is from F.R. Larson and J. Miller, Trans. ASME, 74, 765 (1952).)

L(103K-log hr)

Str

ess,

ksi

100

10

112 20 24 2816

data for S-590 Iron

20

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Chapter 9 - 37

• Engineering materials don't reach theoretical strength.

• Flaws produce stress concentrations that causepremature failure.

• Sharp corners produce large stress concentrationsand premature failure.

• Failure type depends on T and stress:- for noncyclic σ and T < 0.4Tm, failure stress decreases with:

- increased maximum flaw size,- decreased T,- increased rate of loading.

- for cyclic σ:- cycles to fail decreases as ∆σ increases.

- for higher T (T > 0.4Tm):- time to fail decreases as σ or T increases.

SUMMARY