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Adventures in Structural Design Using Theoretical, Computational, and
Experimental Fracture Mechanics
Roberto Ballarini
Leonard Case Jr. Professor of EngineeringCase Western Reserve University
University of MinnesotaApril 10, 2006
ThemeWhat I enjoy and have been doing.
•Mechanism-based design guidelines.•Exploration and modeling of failure mechanisms.
•Whatever it takes experimentation.
2 μm2 μm
Design of anchor boltsDesign guidelines for spur gears
Fatigue of MEMS materials/structuresMechanical testing
of nanofibers
( ) ( )θπ
θπ
σ IIij
IIIij
Iij f
rKf
rK
22+=
( ) criticalIIIIII FKKKF =,,
Stress field near crack tip
Initiation criterion
Linear Elastic Fracture MechanicsKI = mode I
stress intensity factor
KII = mode II stress intensity
factor
θ
Direction of growth
0=localIIK
crIKabaFIK == πσ)/(
m
IKCdNda
⎟⎠⎞⎜
⎝⎛Δ=
nIDK
dtda=
CRACK GROWTH MECHANISMS
Fast fracture
High cycle fatigue
Stress corrosion
ACI Code:2hfP tu ≈
ANCHOR BOLTS
h
2/32/3 hEGhKP fIcu =≈
Ballarini et al., Proc. Roy. Soc. Lond.,1985
PcK
θ
Stress intensity factor analysis
Design Curve
Crack Paths
Results of RILEM Round-Robin Tests
Now an undergradcan do these calculations
• Thin-rim gears desired for reduced weight.• Stress fields and failure characteristics significantly
different for thin-rim gears compared to conventional gears.• Designed according to standards published by AGMA.• Catastrophic failures have occurred in thin-rim gears.
Thick rim Thick rim --““benignbenign””
tooth tooth
fracturefracture
Thin rim Thin rim --catastrophic rim catastrophic rim
fracturefracture
Spur GearsSpur Gears
ObjectivesObjectives
Develop fracture mechanicsDevelop fracture mechanics-- based design guidelines to based design guidelines to
prevent rim fracture failure modes prevent rim fracture failure modes in gear tooth bending fatigue.in gear tooth bending fatigue.
hhmmBB ==bb
Definition of Backup Ratio (mB )Definition of Backup Ratio (mB )
b
h
Bending stress index
Crack mouthCrack tip
UserUser--defined defined initial crackinitial crack
Quarter-
pointrosette
Final mesh of Final mesh of initial crackinitial crack
Simulatedcracktrajectory
Initialcrackregion
Predicted crack pathPredicted crack path
Crack Modeling Using Finite Element MethodCrack Modeling Using Finite Element Method
Definition of Initial Crack Location (θ0 )Definition of Initial Crack Location (θ0 )
θθ00
PitchPitchradiusradius
Applied Applied tooth loadtooth load
Stress Intensity FactorsStress Intensity Factors
x
Initial crack mouth
Initial crack tipNew crack tip
Crack increment size
Predicted angle = Predicted angle = f f ( ( KKII II / / KKII ))
y
Gear tooth fillet
KI = mode I stress intensity
factor
KII = mode II stress intensity
factor
mIKC
dNda
⎟⎠⎞⎜
⎝⎛Δ=
Growth rate
Crack Propagation Angle and Growth RateCrack Propagation Angle and Growth Rate
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−= 2 ⎟⎠⎞
⎜⎝⎛±
4
8+ KK
KK
II
I2
II
I
m1tanθ
Tooth load at Tooth load at HPSTCHPSTC
SlotsSlots
FixedFixedinnerinner--hubhubB.C.sB.C.s
Typical Finite Element Gear ModelTypical Finite Element Gear Model
Load Case Locations for FEMLoad Case Locations for FEM
Load case123456
78
9101112
1314
1516
1718
Tooth 1 Tooth 2 Tooth 3
0.26-mm crack size, 68 N-m driver gear torque.
Effect of Initial Crack Location on Crack PathEffect of Initial Crack Location on Crack Path
Gear Parameters:Gear Parameters:•• 28 teeth28 teeth•• 8 pitch8 pitch•• 1.75" pitch rad1.75" pitch rad•• 2020°°
pressure anglepressure angle
•• mmBB = 1.0= 1.0
Effect of Initial Crack Location on Crack PathEffect of Initial Crack Location on Crack Path
Initial crackInitial cracklocation:location:
θθ
00 = = 120120°°
Failure mode:Failure mode:ToothTooth
fracturefracture
Effect of Initial Crack Location on Crack PathEffect of Initial Crack Location on Crack Path
Initial crackInitial cracklocation:location:
θθ
00 = = 114114°°
Failure mode:Failure mode:ToothTooth
fracturefracture
Effect of Initial Crack Location on Crack PathEffect of Initial Crack Location on Crack Path
Initial crackInitial cracklocation:location:
θθ
00 = = 109109°°
Failure mode:Failure mode:ToothTooth
fracturefracture
Effect of Initial Crack Location on Crack PathEffect of Initial Crack Location on Crack Path
Initial crackInitial cracklocation:location:
θθ
00 = = 104104°°(max tensile)(max tensile)
Failure mode:Failure mode:ToothTooth
fracturefracture
Effect of Initial Crack Location on Crack PathEffect of Initial Crack Location on Crack Path
Initial crackInitial cracklocation:location:
θθ
00 = = 9999°°
Failure mode:Failure mode:ToothTooth
fracturefracture
Effect of Initial Crack Location on Crack PathEffect of Initial Crack Location on Crack Path
Initial crackInitial cracklocation:location:
θθ
00 = = 9494°°
Failure mode:Failure mode:ToothTooth
fracturefracture
Effect of Initial Crack Location on Crack PathEffect of Initial Crack Location on Crack Path
Initial crackInitial cracklocation:location:
θθ
00 = = 8888°°(root centerline)(root centerline)
Failure mode:Failure mode:ToothTooth
fracturefracture
Effect of Initial Crack Location on Crack PathEffect of Initial Crack Location on Crack Path
Initial crackInitial cracklocation:location:
θθ
00 = = 8383°°
Failure mode:Failure mode:ToothTooth
fracturefracture
Effect of Initial Crack Location on Crack PathEffect of Initial Crack Location on Crack Path
Initial crackInitial cracklocation:location:
θθ
00 = = 7878°°
Failure mode:Failure mode:RimRim
fracturefracture
Effect of Initial Crack Location on Crack PathEffect of Initial Crack Location on Crack Path
Initial crackInitial cracklocation:location:
θθ
00 = = 7373°°
Failure mode:Failure mode:RimRim
fracturefracture
Effect of Initial Crack Location on Crack PathEffect of Initial Crack Location on Crack Path
Initial crackInitial cracklocation:location:
θθ
00 = = 6868°°
Failure mode:Failure mode:RimRim
fracturefracture
Effect of Initial Crack Location on Crack PathEffect of Initial Crack Location on Crack Path
Mode Istress
intensityfactor,
KI (ksi√in)
0
20
40
60
80
Crack Length, in0.0 0.1 0.2 0.3 0.4 0.5
Mode IIstress
intensityfactor,
KII (ksi√in)
-2
0
2
4
Stress Intensity FactorsStress Intensity Factors
θ0 =120°99°
88°83°
θ0 =68°
θ0 =83°
Backup ratio:Backup ratio:mmBB = = 1.01.0
Tooth/rim fractureTooth/rim fracturetransition:transition:
θθ
00 = = 8181°°
Effect of Backup Ratio on Crack PathEffect of Backup Ratio on Crack Path
Backup ratio:Backup ratio:mmBB = = 1.11.1
Tooth/rim fractureTooth/rim fracturetransition:transition:
θθ
00 = = 7676°°
Effect of Backup Ratio on Crack PathEffect of Backup Ratio on Crack Path
Backup ratio:Backup ratio:mmBB = = 1.21.2
Tooth/rim fractureTooth/rim fracturetransition:transition:
θθ
00 = = 7171°°
Effect of Backup Ratio on Crack PathEffect of Backup Ratio on Crack Path
Backup ratio:Backup ratio:mmBB = = 1.31.3
Tooth/rim fractureTooth/rim fracturetransition:transition:
All tooth fracturesAll tooth fractures
Effect of Backup Ratio on Crack PathEffect of Backup Ratio on Crack Path
Backup ratio:Backup ratio:mmBB = = 1.01.0
Tooth/rim fractureTooth/rim fracturetransition:transition:
θθ
00 = = 8181°°
Effect of Backup Ratio on Crack PathEffect of Backup Ratio on Crack Path
Backup ratio:Backup ratio:mmBB = = 0.90.9
Tooth/rim fractureTooth/rim fracturetransition:transition:
θθ
00 = = 8686°°
Effect of Backup Ratio on Crack PathEffect of Backup Ratio on Crack Path
Backup ratio:Backup ratio:mmBB = = 0.80.8
Tooth/rim fractureTooth/rim fracturetransition:transition:
θθ
00 = = 9191°°
Effect of Backup Ratio on Crack PathEffect of Backup Ratio on Crack Path
Backup ratio:Backup ratio:mmBB = = 0.70.7
Tooth/rim fractureTooth/rim fracturetransition:transition:
θθ
00 = = 9797°°
Effect of Backup Ratio on Crack PathEffect of Backup Ratio on Crack Path
Backup ratio:Backup ratio:mmBB = = 0.60.6
Tooth/rim fractureTooth/rim fracturetransition:transition:
θθ
00 = = 102102°°
Effect of Backup Ratio on Crack PathEffect of Backup Ratio on Crack Path
Backup ratio:Backup ratio:mmBB = = 0.50.5
Tooth/rim fractureTooth/rim fracturetransition:transition:
θθ
00 = = 107107°°
Effect of Backup Ratio on Crack PathEffect of Backup Ratio on Crack Path
Design MapDesign Map
T = tooth fracturesT = tooth fracturesR = rim fracturesR = rim fracturesC = compressionC = compression
Initial crack location, θ0 (deg)60708090100110120
Backupratio,mB
0.50.60.70.80.91.01.11.21.3
CRRRRRRRRRTTT
RRTT
CRRRRRRRTTTTT
CRRRTT
CRRRTT
CCRRRTTTTTTTT
CCRRTTTTTTTTT
CCRTTTTTTTTTT
CCTTTTTTTTTTT
Mode Istress
intensityfactor,
KI (ksi√in)
0
1
2
3
4
5
6
7
8
9
Mode I Stress Intensity FactorsMode I Stress Intensity Factors
0.5
0.71.0 1.3Backup ratio, mB
68
78
88
99
109
120
Initialcrack
location,θ0 (deg)
Gear Parameters:Gear Parameters:
•• 28 teeth28 teeth•• 8 pitch8 pitch•• 1.75" pitch rad1.75" pitch rad•• 2020°°
press anglepress angle
•• 500 lb tooth load500 lb tooth load•• 0.030" crack size0.030" crack size
Mode Istress
intensityfactor,
KI (ksi√in)
0
1
2
3
4
5
6
7
8
9
Mode I Stress Intensity FactorsMode I Stress Intensity Factors
0.5
0.71.0 1.3Backup ratio, mB
68
78
88
99
109
120
Initialcrack
location,θ0 (deg)
Gear Parameters:Gear Parameters:
•• 28 teeth28 teeth•• 8 pitch8 pitch•• 1.75" pitch rad1.75" pitch rad•• 2020°°
press anglepress angle
•• 500 lb tooth load500 lb tooth load•• 0.030" crack size0.030" crack size
•• AISI 9310 steelAISI 9310 steel•• ΔΔKKthth = 5 ksi= 5 ksi√√inin
Design MapDesign Map
T = tooth fracturesT = tooth fracturesR = rim fracturesR = rim fracturesN = no fractureN = no fracture
Initial crack location, θ0 (deg)60708090100110120
Backupratio,mB
0.50.60.70.80.91.01.11.21.3
NNNNRRRRRRTTT
RRTT
NNNNNNRRTTTTT
NNNNTT
NNNNNT
NNNNNNNTTTTTT
NNNNNNNTTTTTT
NNNNNNNTTTTTT
NNNNNNNTTTTTT
Test GearsTest Gears
Backup ratio = 3.3
Notch inserted in tooth fillet
Backup ratio = 1.0
Backup ratio = 0.3
AISI 9310 SteelCase carburized and ground
Effective Case Depth 0.032 in.
Crack growth gage
Spur gear rig at NASA Glenn
Specimen
Spur gear rig at NASA Glenn
PE
PE
PE
Backup ratio = 3.3Backup ratio = 3.3
Backup ratio = 1.0Backup ratio = 1.0
Backup ratio = 0.5Backup ratio = 0.5
E = ExperimentP = Predicted
Validation of Finite Element ModelingValidation of Finite Element Modeling
Fracture and Fatigue of MEMS Polysiliconand Silicon Carbide
Analog Devices GyroscopeiMEMS
Gyro Die Showing the Rate Sensor and Integrated Electronicshttp://www.analog.com/technology/mems/gyroscopes/index.html
MEMS Device-Fuel AtomizerMotivation• Reduce cost through batch fabrication• Achieve desired tolerances using a
precise silicon micromachining technology
Operation• Fuel enters the spin chamber
through tangential slots• Fuel swirls in the spin chamber
and exits through the orifice in a hollow conical spray
• Swirling produces sprays with wider spray angles as compared to plain orifice atomizers
Ant Carrying a (1000 µm)2
Microchip
Or is it a Palm Pilot?
silicon substrate
polysiliconSiO2SiO2
SiO2SiO2
crack surface
a)
b)
c)
d)
INDENTATION CRACKING
ORIGINAL OBJECTIVES
Characterize strength, fracture toughness, high cycle fatigue
and environmentally assisted crack growth
in poly-Si, poly-SiC,and SiC
at scales relevant to MEMS devices.
•Develop (micron size) on-chip specimens.•Generate data.
•Study mechanisms.•Formulate predictive models.
CHALLENGES
•Experiments are difficult to design, execute and interpret.
crIKabaFIK == πσ)/(
??? m
IKCdNda
⎟⎟⎠
⎞⎜⎜⎝
⎛Δ=
??? nIDK
dtda =
CRACK GROWTH MECHANISMS
Fast fracture
High cycle fatigue
Stress corrosion
If applicable, how sensitive are the parametersto processing procedures?
Two types of on-chip specimenshave been developed:
•Loading through electrostatic actuation•Loading through fabrication-induced residual stress
acr2μm
Say acr
=1μm
Say tlife
=10yrs
Then vcr
<10-15
m/s !!!
Why subcritical crack initiation andgrowth should be studied in MEMS
CVD Polysilicon - Effects of Deposition Temperature550°C 580°C 615°C
1100°C 570/615°C
all filmsare ~2-6 µmthick, anddepositedon SiO2
Fracture Mechanics Specimen
Actuator
anchor pads
movable comb drive
fixedcomb drive
MEMS Fracture Mechanics Specimenintegrated with
MEMS Loading Device Actuator
(Proc. Royal Soc. A, 455, 3807-3823, 1999)
produces ~0.7 mN
Fracture Device (notched specimen)
specimenspecimen
actuator
500 μm
ADVANTAGES OF THIS “ON-CHIP” SPECIMEN
•No need for external loading device.•Resonance loading can be used to study very high cycle fatigue.
•Uncracked ligament size of the same order as dimensions of typical MEMS components.
CURRENT LIMITATIONS
•Low “yield”, but improving
Sharpe and Bagdahn, Proc. Fatigue 2002
Fatigue of Notched Polysilicon
DIFFICULTIES IN DETERMINING ENVIRONMENTAL EFFECTSUSING THESE TESTS
•Tests involve cyclic loading, not constant load.
•Tests involve tension and compression.
100 μmSpecimen T 100 μm100 μmSpecimen T 100 μmSpecimen C100 μm100 μmSpecimen C
VARIATIONS ON A THEME
R-ratioand mean
stresseffects
Dynamic Fatigue Resultslow-cycle fatigue
1
2
3
4
5
6
-4 -3 -2 -1 0 1
Load Ratio, RLow
-Cyc
le F
atig
ue S
tren
gth,
σm
ax(G
Pa)
PolySi
thickness
Test Ambient3.5 μm air (105
Pa)5.7 μm air (105
Pa)5.7 μm
vacuum (10 Pa)
polysilicon
indent
2h = 500 μm
w=60 μmaA
silicon substrate
polysilicon
SiO2 anchors
B pre-crack
Stress Intensity and Stress vs. Crack Length
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.0 5 9 13 17 21 25 29 33 37 41 45 49 53 57
Crack Length [um]
Str
ess
Inte
nsi
ty
[MP
a m
1/2 ]
0.05.010.015.020.025.030.035.040.045.050.0
Str
ess
[MP
a]
Stress Intensity Stress*
)(* απσ FaK =
σ* = σresidual /(1+4aV(α)/2h)
PASSIVE DEVICE ASSOCIATEDWITH CONSTANT TENSION
(Science, 2002)
5 um
indent
substrate
pre-crack
beam
residual tensile stressbeam anchor (to substrate)
crack tip
1 um
INDENTATION CRACK
CWRU
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 5 10 15 20 25 30 35 40 45 50 55 60
Crack Length [μm]
Stre
ss In
tens
ity [M
Pa m
1/2 ]
= no propagation = propagation to failure
45 MPa
56 MPa
69 MPa
FINE-GRAINED POLYSILICONFRACTURE TOUGHNESS
FRACTURE TOUGHNESS DATA(MPa-m1/2)
Multilayered silicon 0.79<KIc
<0.84Fine-grained silicon
0.76<KIc
<0.86SiC
2.80<KIc
<3.41
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 5 10 15 20 25 30 35 40 45 50 55 60
Crack Length [μm]
Stre
ss In
tens
ity [M
Pa m
1/2 ]
= no propagation = propagation to failure
69 MPa
FINE-GRAINED SILICONSTATIC FATIGUE STUDY
90% RH
K between 0.62-
0.86 MPa-m1/2
No growth in 30 daysV< 3.9 x10-14
m/sSame results for eight multipoly
specimens
Stre
ss
Schematic Bend Strength Tests
Monotonic Increasing Amplitude Fatigue
1 Comp/Mono Fixed Δσ
Fatigue/Mono High T Hold/Mono
Time
Stre
ss
0
σm Δσσcr
10 min.= 5x106cycles
TimeSt
ress
0
σcrσhold
10 min.
Δσ=(σmax -σmin )
Time
Stre
ss
0σmax = σcr
σmin
σm
~1 min.(5x105cycles)
Time
0
σcr
σmin
Time
Stre
ss
0
σcr
Increasing Amplitude Fatigue B-doped polysilicon (no sputtered Pd)
Mean Stress, σm
(GPa)
Fatig
ue S
tren
gth,
σcr
(GPa
)
1.5
2
2.5
3
3.5
-2 -1 0 1 2
Monotonic Bend Strength after cycling with a fixed (low) amplitude
2
2.5
3
3.5
4
4.5
-5 -4 -3 -2 -1 0 1 2 3
Mean Stress, σm
(GPa)
Mon
oton
ic S
tren
gth,
σcr
it(G
Pa)
σa = 1.0 GPa
Time
Stre
ss
0
σcrit
Time
Stre
ss
0
σm Δσσcrit
10 min.= 6x106cycles
σa = Δσ/2
1.0 0
- 1.0
Mean Stress, σ m(GPa)
1.6
1.4
1.2
1.0
0.8
0.6N
orm
aliz
ed S
tren
gth,
σcr
it(G
Pa)
0 0.2 0.4 0.6 0.8
Fatigue Amplitude, σa (GPa)
Effects on Monotonic Bend Strength of mean stress σm , and fatigue amplitude σa
Fatigue Amplitude, σa
(GPa)
Mea
n St
ress
, σm
(GPa
)
low high
high
l
ow
low
high
tens
ile
com
pres
sive
weakening
weakening
strengthening
strengthening
no effect
no effect
(not measured)
undoped polysilicon
B-doped polysilicon
Mechanical Testing of Collagen Fibers (Nanotechnology)
• Most abundant protein in the human body.• One of the basic components of bone, ligaments,
tendons, teeth, skin.• Collagen monomer:
– Triple helical structure made of three chains of amino acids.
– The monomers assemble into fibrils.
Collagen Fibrils
Rho et al., 1998
Hierarchical Structure of Bone
Nanofiber
Testing Device
Journal of the Royal Society Interface, 2006.
Force: 0.1-100 µNDisplacement: 1-5 µm
A B
anchor
Actuation Force
fibril
anchorte ther beam s
canals
hole etched in substrate
stiffening com b drive
anchor
Actuation Force
fibril
anchorte ther beam s
canals
hole etched in substrate
stiffening com b drive
Labeling fibrils using fluorescent antibodies
1. Imaging using SEM2. Labeling
fibril
Primary antibody
Secondary antibody
Alexa Fluor 568
fibril
Primary antibody
Secondary antibody
Alexa Fluor 568
Fluorescently Labeled Collagen Fibers (Negative Image)
Different dilutions of the fibrils were imaged using SEMto determine the appropriate dilution at which individual
fibrils were distinguishable. The fibrils were labeled with fluorescentantibodies to achieve contrast and brightness under optical microscope
for 5 minutes. Anti-fading agents beingtried to allow 30 minutes of manipulation time.
Manipulation usingmicropipette
2 μm2 μm
0
20
40
60
80
100
120
140
160
0 1 2 3 4 5
1st test2nd test3rd test4th test
with fibril
1st test2nd test
without fibril
Displacement (μm)
Appl
ied
Vol
tage
(V)
0
20
40
60
80
100
120
0 5 10 15 20 25 30 35
Strain (%)S
tress
(MP
a)
200 μm
5 μmA
B D
C
0
10
20
30
40
50
60
0 1 2 3 4 5
Displacement (μm)
Forc
e (μ
N)
extra force needed to strain fibril
with fibrilwithout fibril
0
10
20
30
40
50
60
0 1 2 3 4 5
Displacement (μm)
Forc
e (μ
N)
extra force needed to strain fibril
0
10
20
30
40
50
60
0 1 2 3 4 5
Displacement (μm)
Forc
e (μ
N)
extra force needed to strain fibril
with fibrilwithout fibril
F
E
fixed fingers
overlap
movablefingers
tether beams
anchor
anchoranchor anchor
0
20
40
60
80
100
120
140
160
0 1 2 3 4 5
1st test2nd test3rd test4th test
with fibril
1st test2nd test
without fibril
Displacement (μm)
Appl
ied
Vol
tage
(V)
0
20
40
60
80
100
120
0 5 10 15 20 25 30 35
Strain (%)S
tress
(MP
a)
200 μm
5 μmA
B D
C
0
10
20
30
40
50
60
0 1 2 3 4 5
Displacement (μm)
Forc
e (μ
N)
extra force needed to strain fibril
with fibrilwithout fibril
0
10
20
30
40
50
60
0 1 2 3 4 5
Displacement (μm)
Forc
e (μ
N)
extra force needed to strain fibril
0
10
20
30
40
50
60
0 1 2 3 4 5
Displacement (μm)
Forc
e (μ
N)
extra force needed to strain fibril
with fibrilwithout fibril
F
E
fixed fingers
overlap
movablefingers
fixed fingers
overlap
movablefingers
tether beams
anchor
anchoranchor anchor
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