failure - kocwcontents.kocw.net/kocw/document/2015/hanyang/... · 2016-09-09 · failure •...
TRANSCRIPT
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