Chapter 9

Failure

Chapter 9

Ductile & Brittle Fracture, Griffithy Theory

Issues to Address...

• How do flaws in a material initiate failure?• How is fracture resistance quantified; how do different

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

affect the failure stress?

Ship-cyclic loadingfrom waves.

Computer chip-cyclicthermal loading.

Hip implant-cyclicloading from walking.

Adapted from Fig. 18.11W(b), Callister 6e. (Fig. 18.11W(b) is courtesy of National Semiconductor Corporation.)

Adapted from Fig.17.19(b), Callister 6e.

Mechanical Failure

Failure

• Fracture step

• Simple fracture

Separation of materials under the stresses

• Fatigue

Failure caused by dynamic and fluctuating stresses

• Creep

Material deformation at elevated service temperature

under static stresses• Fracture mode

Ductile fractureBrittle fracture

Crack formationCrack propagation

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

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

Figures from V.J. Colangelo and F.A. Heiser, Analysis of Metallurgical Failures (2nd ed.), Fig. 4.1(a) and (b), p. 66 John Wiley and Sons, Inc., 1987. Used with permission.

ex : Failure of A Pipe

Fracture Mechanisms

• Ductile fracture– Occurs with plastic deformation

• Brittle fracture– Little or no plastic deformation– Catastrophic!!!

Ductile vs. Brittle Failure

• Classification:Very

DuctileModerately

Ductile BrittleFracturebehavior:

Large Moderate%AR or %EL Small

• Ductilefracture is usuallydesirable!

Ductile:warning before

fracture

Brittle: No

warning

cup-and-cone fracture brittle fracture

Ductile vs. Brittle Failure

(1) Substantial plastic deformation before fracture

(2) Crack propagation is relatively slow

(3) Generally tough materials (more strain energy required for fracture)

(4) Pin-point fracture or Cup and cone fracture

Ductile Fracture

• Evolution to failure:necking void

nucleationvoid growth and linkage

shearing at surface

fracture

• Resultingfracturesurfaces(steel)

50 m

particlesserve as voidnucleationsites.

50 m

100 m

Moderately Ductile Failure

(1) Fast crack propagation catastrophic fracture

(2) Less tough materials

(3) Relatively flat fracture

- Transgranular - Intergranular

Brittle Fracture

Origin of Brittle Fracture

Arrows indicate point at which failure originated

Transgranular Fracture

• Intragranular (within grains)

316 S. Steel (metal)

Al Oxide(ceramic)

3mm

160mm

Intergranular Fracture

• Intergranular(between grains)

1mm

4mm

304 S. Steel (metal)

Polypropylene(polymer)

x2

sinT=T max λπ

10E

aE

π2λ

=σ∴

xλπ2

σxλπ2

sinσ=ax

E=E=σ

]x:=λ:π2[xλπ2

sinσ=σ

0c

cc0

c

≈

≈

∴ φ

Fracture Mechanics

anesatomic plseparate andmic bondsbreak ato

to requiredσtress tensile sCritical

C

notches & cracks sharp assuch defects the toDue →

σn lower thamuch isstrength fracture materials reality,in But C

materialsbrittlecompletelytheforonlyApplied

γ4=E

a2πσ→0=

dadW

)σ=σ(conditioncriticalgetfor

πaEσ

aγ4=W

=WaEπσ

=2×a×πE2σ

=2×width×area×E2σ

islossenergystrainElastic

s

2cT

c

22

sT

e2

22

22

-∴

∴

Griffith Theory

πa2

2a

1

• Stress-strain behavior (Room T):

TS << TSengineeringmaterials

perfectmaterials

• DaVinci (500 yrs ago!) observed...--the longer the wire, thesmaller the load to fail it.

• Reasons:--flaws cause premature failure.--Larger samples are more flawed!

Ideal vs. Real Materials

TS << TSengineeringmaterials

perfectmaterials

E/10

E/100

0.1

typical strengthened metaltypical polymer

typical ceramic

perfect mat’l-no flaws

carefully produced glass fiber

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.

t

Flaws are Stress Concentrators!_1Results from crack propagation

• Griffith Crack

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

0t21

t0m σK=)ρa

(σ2=σ

• Stress conc. factor:

BAD! Kt>>3NOT SO BAD

Kt=3

K t max /o

• Large Kt promotes failure:

• Elliptical hole ina plate:

• Stress distrib. in front of a hole:

o

2a

Flaws are Stress Concentrators!_2• Crack Tip

=

• Avoid sharp corners!

r/h

sharper fillet radius

increasing w/h

0 0.5 1.01.0

1.5

2.0

2.5

Stress Conc. Factor, Ktmaxo

=

Engineering Fracture Design

r , fillet

radius

w

h

o

max

Crack Propagation

• Cracks propagate due to sharpness of crack tip• A plastic material deforms at the tip, “blunting” the crack.

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

brittle

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

For ductile => replace s by s + pwhere p is plastic deformation energy

i.e., m > c21

sc )

πaEγ2

(=σ

s + p) = Gc : Critical Strain energy release rate

• Condition for crack propagation:

• Values of K for some standard loads & geometries:

2a2a

aa

πaK=σ πaσ1.1K=

K ≥ Kc

Stress Intensity Factor:--Depends on load &geometry.

Fracture Toughness:--Depends on the material,

temperature, environment, &rate of loading.

inpsiormMPa

:Kofunits

Geometry, Load, & Material

Kcmetals

Kccomp

Kccer Kc

poly

incr

easi

ng

Fracture Toughness

• Crack growth condition:

πaYσ• Largest, most stressed cracks grow first!

--Result 1: Max flaw sizedictates design stress.

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

design

KcY amax

amax 1

KcYdesign

2

K ≥ Kc

amax

no fracture

fracture

amax

no fracture

fracture

Design Against Crack Growth

• 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...max

cc πaY

K=σ

• Key point: Y and Kc are the same in both designs.--Result:

( ) ( )BmaxcAmaxc a=a σσ

9 mm112 MPa 4 mm

Answer: MPa168=)σ( Bc• Reducing flaw size pays off!

Design ex : Aircraft Wing

mMP26=K aC• Materials has

Impact Testing

final height initial height

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

(Charpy)

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

• Why? An increased rategives less time for disl. tomove past obstacles.

y

y

TS

TSlarger

smaller

Loading Rate

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

• Ductile-to-brittle transition temperature (DBTT)...

BCC metals (e.g., iron at T < 914C)

Imp

ac

t E

ne

rgy

Temperature

FCC metals (e.g., Cu, Ni)

High strength materials (y>E/150)

polymers

More Ductile Brittle

Ductile-to-brittle transition temperature

Temperature

Effect of Carbon in steel

Temperature

Effect of Carbon in steel

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

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

Design Strategy : Stay Above The DBTT!

• Fatigue = failure under cyclic stress.

tension on bottom

compression on top

countermotor

flex coupling

bearing bearing

specimen

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

max

min

time

mS

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

Fatigue

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

• Sometimes, the fatigue limit is zero!

Sfat

case for steel (typ.)

N = Cycles to failure103 105 107 109

unsafe

safe

S = stress amplitude

case for Al (typ.)

N = Cycles to failure103 105 107 109

unsafe

safe

S = stress amplitude

Fatigue Design Parameters

·Ferrous alloys : Fe and Ti alloys

· Nonferrous alloys :Aluminum, Copper, Magnesium

• Crack grows incrementally

dadN

K mtyp. 1 to 6

~ aincrease in crack length per loading cycle

• Failed rotating shaft--crack grew even though

Kmax < Kc--crack grows faster if

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

crack origin

Fatigue Mechanism

C-rich gasput

surface into

compression

shot

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

--Method 1: shot pinning

2. Remove stressconcentrators. bad

bad

better

better

--Method 2: carburizing

N = Cycles to failure

moderate tensile mlarger tensile m

S = stress amplitude

near zero or compressive m

Improving Fatigue Life

• Case Hardening:--Diffuse carbon atoms

into the host iron atomsat the surface.

--Example of interstitialdiffusion is a casehardened gear.

• Result: The "Case" is--hard to deform: C atoms

"lock" planes from shearing.--hard to crack: C atoms put

the surface in compression.

Processing Using Diffusion

Creep

Sample deformation at a constant stress (s) vs. time

0 t

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.

timeelastic

primary secondary

tertiary

T < 0.4 Tm

INCREASING T

0

strain,

• Occurs at elevated temperature, T > 0.4 Tmelt• Deformation changes with time.

Creep: Temperature Effect

0 t

• Most of component life spent here.• Strain rate is constant at a given T,

--strain hardening is balanced by recovery

stress exponent (material parameter)

strain rateactivation energy for creep(material parameter)

applied stressmaterial const.

• Strain rateincreasesfor larger T,

10

20

40

100

200

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

s

Stress (MPa)427C

538C

649C

s K2

n exp QcRT

.

Secondary Creep

• Failure:along grain boundaries.

time to failure (rupture)

function ofapplied stress

temperature

T(20 log t r ) L

L(103K-log hr)

Str

ess

, ksi

100

10

112 20 24 2816

data for S-590 Iron

20appliedstress

g.b. cavities

• Time to rupture, tr

• Estimate rupture timeS 590 Iron, T = 800C, = 20 ksi

T(20 log t r ) L

1073K

24x103 K-log hr

Ans: tr = 233hr

Creep Failure

• Engineering materials don't reach theoretical strength.

• Flaws produce stress concentrations that cause premature failure.

• Sharp corners produce large stress concentrationsand premature failure.

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

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

-for cyclic :cycles to fail decreases as increases.

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

Summary