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