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Frontiers of Fracture
Mechanics
Adhesion and Interfacial Fracture
Contact Damage
Biology, Medicine & Dentistry –
The Next Frontiers For Mechanics
• One of the current challenges in materials & mechanics is how to make adequate connections to biology and medicine
• This requires new knowledge and teaming approaches beyond the boundaries of our current efforts
– medicine, dentistry
– biology, chemistry, physics
– materials and mechanics
The Benefits of The Mechanics
Approach
• Theoretical/computational mechanics enable quantitative predictions of cause and effect
– However biological systems are complex
– Two approaches are often needed
• Experimental mechanics provides new insights and measurements
– Enables model validation
– Enables new clinical solutions
Background and Introduction
• This class presents selected examples at
the frontiers of fracture mechanics
• These include:
– Problems involving adhesion and interfacial
fracture
– Problems involving contact damage
• This focus is on applications of fracture
mechanics in biomedical applications
Path to understanding the effects of
multiple loadings on dental structure
P
Ceramic
foundation
Adhesive
layer
Tooth on tooth contact Hertzian contact on idealized
Multilayer Structure
Basic FEA on idealized 3-layer
structure
Ceramic
Foundation
Ceramic
foundation
Adhesive
layer
FAILURES CAUSED BY HERTZIAN INDENTATION
Cone crack
Subsurface crack
Sub-surface crack is the major clinical failure mode.
Loading rate effects on 3-layer
structure
Plot of critical load Pm for radial cracking as function of loading
rate P for coatings of soda-lime glass (filled symbols) and silicon (unfilled symbols) bonded to
polycarbonate substrates.
(Lee et al. 2002)
Contribution of Lawn’s approach
•Experimental study demonstrates the existence of
loading rate effect
•Silicon does not exhibit slow crack growth (SCG) but
silicon tri-layer shows loading rate effect
•SCG model can only explain part of loading rate effect
•Material properties of the foundation and join layers
may play important role
Approach of the Current Work
• Develop understanding of the
constitutive behavior of individual layers
• Integrate the constitutive behavior into
mechanics models
• Predict critical loads in multilayer
structure
• Develop understanding of loading rate
effects on deformation and cracking
Experiments• Studied joins, foundations & multilayers with real dental materials
• Measured constitutive behavior for individual layers (monotonic
compression tests)
• Studied cracking of multilayers (monotonic Hertzian contact tests)
IndenterCapacitance
Gage
Specimen
Indenter
Specimen
Capacitance
Gage
Hertzian contact TestsCompression Tests
Loading rate effect of critical load
measurements
Loading rate 100 N/s
IndenterCapacitance
Gage
Specimen
Dental cement RelyX
Z100 dental
restoration
Glass
Loading rate effect of critical load
on dental multilayer
IndenterCapacitance
Gage
Specimen
Glass
Z100 dental
restoration
Dental cement RelyX
1 10 100 1000200
400
600
800
1000
1200
1400160018002000
Cri
tical
load
(N
)
Loading Rate (N/s)
1 mm
4 mm
~100 μm
9 mm
Rate Dependent Young’s
Moduli of Cement &
Foundation Layers
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.140
50
100
150
200
250
300
350
0.000010.0010.01
Str
ess
(
MP
a)
Strain
0.1
Strain rate (s-1
)
Indenter
Specimen
Capacitance
Gage
Dental Cement RelyX ARC
(1 mm thick and 7 mm diameter)
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.140
50
100
150
200
250
300
350
0.00050.0001
0.0010.01
Str
ess
(
MP
a)
Strain
Strain rate (s-1
)
Dental Restoration Z100
(4 mm thick and 9 mm diameter)
Elasticity Analysis
• Coating/substrate bi-layer model
– B and d are dimension coefficients
2
log /t f
B dP
E E
ceramic
foundation
a
P
Ceramic
foundation Adhesive
layer
Bi-layer Slow Crack Growth
solution• Power law crack
growth model
– K is stress intensity factor = ya1/2
– N and v0 are determined through four point bending tests
• Bi-layer SCG analytical solution (Lee. et al)
1
11
N
mP A N P
0
N
IC
da Kv v
dt K
1 10 100 1000200
400
600
800
1000
12001400160018002000
Experimental data
Bi-layer SCG model analytical solution
Best linear fit of experimental data
Cri
tical
load
(N
)
Loading rate (N/s)
Rupture time
Dental cement RelyX
Dental cement RelyX
SCG model for top
glass layer
RDEASCG Model
(Rate Dependent Elastically-
Assisted SCG)
0
Rt
t dt D
E
t
Glass
Z100 dental
restorationGlass
Z100 dental
restoration
E
Finite element
analysis (FEA)
E
IndenterCapacitance
Gage
Specimen
Herztian contact
Test data
D
1 / 2 1 / 2
0
0/ 2 1
N N N
Ic f
N
K c cD
N v
Integration
Rt
Rc tPP
Critical load cP
Prediction of Loading rate effect of critical
load with RDEASCG model
1 10 100 1000200
400
600
800
1000
12001400160018002000
Experimental data
3-layer RDEASCG model FEA results
Bi-layer SCG model analytical solution
Best linear fit of experimental data
Crit
ica
l lo
ad
(N
)
Loading Rate (N/s)Clinically relevant loading rates
Summary – Loading Rate
Effects• Experimental
-- studied constitutive behavior of individual layers
-- studied rate-dependent critical loads in multilayer structure
-- in situ study gives good observation of deformation and
cracking in dental multilayer
• Modeling
-- implemented rate-dependent Young’s modulus into
multilayer structure with real dental material
-- developed analytical model for reasonable prediction of
deformation and cracking in dental multilayer under monotonic
loading
CYCLIC CONTACT EXPERIMENTS
• Model multilayers are subjected to cyclic Hertzian indentation for 106 cycles
• Peak load levels lower than observed crack initiation loads under monotonic loading
Model multilayer Experimental set-up
Glass
Epoxy
Ceramic-filled Polymer
8 mm
3 mm
P
IndenterCapacitance
Gage
Specimen
CYCLIC HERTZIAN INDENTATION
Surface deformation after crack nucleation
Hertzian cone crack Subsurface crack
(Soboyejo et al, 2001)
0.8 mm, P=60 N 3.18 mm, P=70 N 8 mm, P=90 N
0.8 mm, P=60 N 3.18 mm, P=70 N 8 mm, P=90 N
MAXIMUM PRINCIPAL STRESS DISTRIBUTION
1 mm
Ceramic
Polymer
Cement
high tensile stress area causes sub-surface crack
DEFORMATION DURING CYCLIC LOADING
200 mm Indenter
1(120) ~ -.0085 mm
200(120) ~ -.0089 mm
1100000 (120) ~ -.0105 mm
Two Sources — Time dependent response of epoxy layer
— Accumulation of plastic strain in epoxy layer
P ~ 20-120N
Deformation During Cyclic
Loading
200 mm Indenter
P
-16
-14
-12
-10
-8
-6
-4
-2
0
2
0.00E+00 2.00E+05 4.00E+05 6.00E+05 8.00E+05
Time (sec)
Sharp Change in
sample compliance
(
m)
20-130 N 20-135 N 20-140 N 20-145 N
Sample loaded for million cycles at each load level at 5 Hz loading rate
Subsurface Pop-In Conditions
0
10
20
30
40
50
60
70
80
90
-11 -8 -5 -2
After 2 X 106Cycles
After 3 X 106Cycles
After 106Cycles
(m)
P(N
)
200 mm Indenter
P
Pop-In Criteria(Lawn, 2001)
10
100
1000
0.1 1 10 100
Indenter Radius (mm)
Cri
tica
l L
oa
d (
N)
Cone Crack
Radial Crack
Cone Crack Critical Load [Lawn,2001]
Radial Crack Critical Load [Lawn, 2001]
POSSIBLE MECHANISMS OF
SUB-SURFACE CRACKING
Mechanics-driven crack growth – due to
flow of the cement into the crack
Stress corrosion cracking
Mechanical fatigue
HYDRAULIC FRACTURE TEST
glass
glass
p
oil
Glass cracks under certain pressure.
p
p2R
a
crack
hole
c
=c/a
NORMALIZED STRESS INTENSITY FACTOR
CEMENT FLOWS INTO CRACK
a0
2p0H0
W
W2H0
c
2
CRACK GROWTH RATE
HpapKapKapKa
pH
dt
da
ccc 0000
0
2 cos1
4/2/tan
1
/sin
1
3
2
HpK
pH
dt
da
ca 0
2
3
0
2 cos1
3
8
If H= 1μm, p0=2 MPa, Kc=0.5 MPa-s, μ=100 Pa-s, cos=0.005 N/m, then
da/dt = 0.27 μm/s
Summary – Cyclic Contact
• Pop-in conditions are strongly influenced by ball
size in Hertzian indentation experiments
• Ball size effects explained using simple contact
mechanics models
• Hydraulic fracture may be caused by water
within cracks
• Mechanics model developed for hydraulic
fracture modeling
• Mechanistically-based fatigue model needed
Modeling of Water Diffusion
Cracking is shown in top ceramic layer after the dental multilayers
immersed in water for some time.
Major observations:
1. cracking occurs after some time, and becomes more extensive as time
increases.
2. cracking first occurs near the edge of the sample and propagates
parallel to the edge.
3. cracking initiates from the bottom surface of the top layer.
after 6 days after 15 days after 70 days
Courtesy of Prof. V.P. Thompson, NYU Dental School
Real Questions
Is the stress induced by water diffusion high enough to cause
the cracking?
Can the crack pattern be understood by water diffusion
induced SCG?
Thermal diffusion analogy
Water diffusion Heat transfer
Water induced foundation expansion Thermal expansion
MAXIMUM PRINCIPAL STRESS DISTRIBUTION
DURING WATER DIFFUSION
dentin-like polymer
dental ceramic or glass
dental cement
Maximum Principal Stress (Pa)
Maximum tensile stress appears at the bottom surface of the top layer.
Peak tensile stress first appears near the edge and is parallel to the edge.
Symmetry line
Maximum Principal Stress Distributions in Dental Crown Multilayer
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1
Location, x/R
t/ = 0.01
0.02
0.05
0.1
0.2
0.5
1
center edge
3/*
scVE
days 289
/m 10
m 105~
ydiffusivit
scalelength 212-
232
s
MPa300~3/ 1%~ GPa100~ **
scsc VEVE
Slow Crack Growth
TheoryCrack growth rate: N
ICKKvdtda // 0
Stress intensity factor:2/1aK y
DdttRt
N
0
12/
012/
N
i
N
N
IC
avN
KD
y
Driving force to form subsurface crack:
independent of time and load
Criteria to form subsurface crack:
N
tNdtt
/1
0
SCG in Top Glass Layer Due to Water Diffusion
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1
t/ = 0.010.02
0.05
0.1
0.2
0.5
1
center edgeLocation, x/R
3/*/1
/1
0
sc
N
Nt
N
VE
dt
Water diffusion is very important in determining the lifetime of dental multilayers.
Summary – Modeling of Water
Diffusion Model predicts the cracking in the top ceramic layer after the dental
multilayers immersed in water for some time.
Model also predicts the major observations:
1. cracking occurs after some time, and becomes more extensive as time
increases (due to stresses associated with water distribution).
2. cracking first occurs near the edge of the sample and propagates
parallel to the edge (consistent with stresses and slow crack growth).
3. cracking initiates from the bottom surface of the top layer (ditto).
after 6 days after 15 days after 70 days
Courtesy of Prof. V.P. Thompson, NYU Dental School
BIO-INSPIRATION - TOOTH STRUCTURE
Enamel
Dentin
Dentin-Enamel-Junction (DEJ)
ELASTIC MODULUS DISTRIBUTION IN
DENTIN-ENAMEL JUNCTION (DEJ)
G.W. Marshall Jr., et al. J Biomed Mater Res 54, 87-95, 2001
MECHANICAL PROPERTIES OF
DENTAL MATERIALS/MULTILAYERS
Real tooth
Dental ceramic E: 50~200 GPa
Dentin-like polymer E=20 GPa
Dental cement E=5 GPa
Enamel E=65 GPa
Dentin E=20 GPa
Dentin-Enamel Junction (DEJ): Graded
Dental restoration
DENTAL CROWN RESTORATION FGM DESIGN
Dental ceramic E: 50-200 GPa
Dentin-like polymer E: 10-30 GPa
Dental cement E: 2-13 GPa
Functionally graded layer
MAXIMUM PRINCIPAL STRESS DISTRIBUTION
Maximum Principal Stress (Pa)
dentin-like polymer
ceramic
cement
dentin
DEJ
enamel
0.1 mm
cementFGM
dentin-like polymer
ceramic
Current crown
Natural tooth
FGM design
Maximum Principal Stress
ZicorniaGlass
Mark II
Dicor MGC
Dicor
Empress 2
Young’s modulus of ceramic (GPa)
Max
imum
pri
nci
pal
str
ess
(MP
a)
FGM
polymer
ceramic
polymer
ceramic
Natural Enamel
0
20
40
60
80
100
120
0 50 100 150 200 250
Effects of Different Distributions of Young’s Modulus
E=20 GPa
E=72 GPa
40
45
50
55
1 2 3
Ceramic
FGM
Max
imu
m p
rin
cip
al s
tres
s (M
Pa)
I II III
Different Young’s modulus distribution
40
50
60
70
0 20 40 60 80 100
Cement thickness (μm)
Yo
un
g’s
mo
du
lus
(GP
a)
I
II
III
Effects of FGM on Fracture Toughness
Ceramic
Cement
a
2112 ,,/ , EEfaKI
222 / cc Ka
crack
Stress intensity factor:
Critical defect length:
E1
E2
Comparison of Critical Defect Lengths
Empress 2
Zicornia
GlassDicor-MGC
Dicor
Mark II
Enamel
Cri
tica
l D
efec
t L
ength
(
m)
Maximum Principal Stress in the Ceramic (MPa)
Zicornia
Empress 2
Mark II
Dicor-MGC
Glass
Dicor
FGM
0
500
1000
1500
2000
2500
0 20 40 60 80 100 120
Brazil-nut Sandwich Sample Test in Determining
Interfacial Toughness between a Dental Cement
Composite and Glass Substrate
Schematics of (a) teeth contact during
chewing and (b) Brazil-nut sandwich
samples under contact loading
Schematic illustration of Brazil-nut
sandwich sample and setup for
fracture testing
Experimental Results of Brazil-nut Sandwich
Sample Test
0
1
2
3
4
5
6
7
0 5 10 15 20
COMPRESSION ANGLEIN
TE
RF
AC
IAL
TO
UG
HN
ES
S (
J/M
2)
0100200300400500600700800900
0 5 10 15 20
COMPRESSION ANGLE
Max
. F
orc
e (
N)
Compression
Angle Applied Force
Geometry of Brazil-nut Sandwich Sample in
Numerical Simulation
Interfacial crack
Glass Substrate
E (GPa) ν
70 0.2
Epoxy Interlayer
E (GPa) ν
10 0.35
Introduction to Cohesive Zone Models for
Interfacial Failure
• Cohesive Zone Ahead of an Interfacial Crack Tip
• Various Kinds of Cohesive Zone Laws
Cohesive Bond Rupture Leads to Physical Crack Growth
Cohesive Bond Structure
σ
δ
σ
δ
σ
δ(a) Exponential Type (b) Bilinear Type (c) Parabolic Type
Interfacial Potential and Tractions
2
2
1( ) exp {[1 ] [ ]exp
1 1
n n n tn n
n n n t
q r qr q
r r
Xu and Needleman (1994) proposed an interfacial potential of the
following form
to derive the normal and tangential traction as follows
n
n
T
t
t
T
and
which gives
2 2
2 2
1exp { exp [ ][1 exp ]}
1
n n n t n tn
n n n t n t
qT r
r
2
2
12 { }exp exp
1
n n t n n tt
n t t n n t
qT q
r
Interfacial Properties Employed in Simulation
COHESIVE
ENERGY (J/M/M)
NORMAL
COHESIVE
STRENGTH (MPA)
TANGENTIAL
COHESIVE
STREGNTH (MPA)
CRITICAL
INTERFACIAL
SEPARATION
(micron)
SET 1 16.3 6 7 1
SET 2 17.6 6.5 7.6 1
SET 3 19 7 8.2 1
SET 4 15 5.5 6.4 1
For simplicity, the cohesive energy and critical interfacial separation
are assumed to be equal in both normal and tangential direction
Comparison between Simulation and
Experimental Results
0
1
2
3
4
5
6
7
0 5 10 15 20
COMPRESSION ANGLE
INT
ER
FA
CIA
L
TO
UG
HN
ES
S (
J/M
^2)
Exp
SET 1
SET 2
SET 3
SET 4
0
200
400
600
800
1000
0 5 10 15 20
COMPRESSION ANGLE
MA
X.
FO
RC
E (
N)
EXP
SET 1
SET 2
SET 3
SET 4
Fracture Toughness of
Glass/Cement and Zirconia/Cement
Interface
0
5
10
15
20
25
30
35
40
-5 15 35 55
y
Inte
rfa
cia
l F
ra
ctu
re T
ou
gh
ness
Gc (
J/m
2)
Glass/Epoxy
Zirconia/Epoxy
substrate
substrate
21 Adhesive
Glass/Epoxy Interface Fracture
Resistance:
Models vs. Experiment
0
2
4
6
8
10
12
0 20 40 60 80
y
Inte
rfa
cia
l F
ra
ctu
re
To
ug
hn
ess
Gc (
J/m
2)
f = 45, b = 170
f = 45, b = 175
f = 45, b = 180
Zone Model
Experiments
0
5
10
15
20
25
30
35
40
-10 10 30 50 70
y
Inte
rfa
cia
l F
ra
ctu
re T
ou
gh
ness
Gc (J
/m2)
45,170
45,175
45,180
Zone Model
Experiments
)tan1( 2
0 yGG
Evans and Hutchinson
(1989) Acta Metall. Mater.,
37(3), 909-916.
Glass/Cement
Zirconia/Cement
yyy
2
0
2
0
2
0 tan1
)1/)(tan1([1tan
GG
G
G
InterfaceLl
CrackH
Crack
2dd
Crack w
F
)tan1(cos
)tancos(sin
)tan1(cos
)]tan)cos())(sin(tancos[(sin2
),,,(
22
22
2
0
y
y
y
yyy
h
h
G
G
Microstructure Effect : FIB Finite elements simulations carried out to
study the crack path selection, using the He &
Hutchinson criteria, were not successful to fully
capture the crack path selection. Knowledge of
microstructure is needed for further
improvement in model.
In the following model clusters of zirconia
clusters have been assumed to exist in the
epoxy which affect the crack path selection.
Zirconia
Clusters
Glass
Epoxy
Glass
He MY, Hutchinson JW
(1989), Int. J. Solids
Structures, 25(9), 1053-1067.
Focused Ion Beam Images of
Interfacial Cracks in Ceramic/Cement Interface
Interfacial Crack Glass
Epoxy
Concluding Remarks
• This class presents an introduction to contact damage in dental multilayers
• Complex loading and geometry idealized to provide insights into mechanisms
• Rate-dependent slow crack growth model used to describe underlying physics
• Bio-inspired design concept presented for the design of robust interfaces
• Interfacial fracture mechanisms explored using a combination of models and experiments
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