Buckling Delamination with Application to Films and Laminates

Thermal gradient withinterruption of heat transferacross crack.

Mixed mode interface crack

Buckling delamination

Compression in filmproducing buckling

No crack driving forcedue to film stress; Unless

Show Volinsky movie

Mechanics of thin films and multilayers Application areas electronics, coatings of all kinds.

Example: Buckle Delaminations

Good Delaminations on Patterned substrates

Thermal Barrier Coatings (TBCs) Application to jet and power generating turbines

Blades taken from an engineshowing areas of spalled-offTBC

Ceramic (Zirconia)

Ceramic (Alumina)

Metal bond coat

Straight-sided

Buckle Delaminations: Interface cracking driven by buckling Three Morphologies: Straight-sided, Varicose and Telephone Cord

Propagation of a buckle delamination along a pre-patterned tapered region of low adhesion betweenfilm and substrate. In the wider regions the telephonecord morphology is observed. It transitions to thestraight-sided morphology in the more narrow regionand finally arrests when the energy release rate dropsbelow the level needed to separate the interface.

Computer simulations

Experimental observations200nm DLC film on silicon

A Model Problem—Mode I Buckling Delamination of Symmetric Bi-layer

,P

2b

L

CP

P

C

0

1strain energy=

2 C C C CSE Pd P P

unbuckled state buckled state

2 2 2 2

2 22( ) ,

12 2 12C

C C

PEh hP hw L L

b hwE b

2h

Buckling stress of clamped beam length 2b.w is the width perpendicular to the plane.Overall buckling of the entire laminate (with thickness 2h) will not occur if b>L/4.

2

4C

C

P LSE P

whE

2 3

3

1

2 6 C

SE EhG

w b b

Energy release rate under prescribed Δ

two crack tips

G

C

This neglects the very slightincrease of P as the bucklingamplitude increases

unbuckled buckled

Fixed b

,P

bb b

Mode I Buckling Delamination of Symmetric Bi-layer: continued

The energy release rate G can be re-written in the following non-dimensional form:

3/ 2

3 25/ 2 2

124 3 1 , where

GL b

E h L

0

0.5

1

1.5

0 1 2 3 4 5

3/ 2

5/ 2

GL

E3/ 23/ 2

5/ 2

8 3The maximum of is 3 1.288

5 5

occuring for = 5/3

GL

E

unbuckled

stable under

prescribed

unstable under

prescribed

3/ 2

5/ 2ICG L

E

3/ 2

5/ 2If 1.288, the crack will advance

if is such that . If is to the left of the

peak the crack is unstable under prescribed

and it will jump to the value of associated with

to the

IC

IC

IC

G L

Eb G G b

b

G

right of the peak. If is then increased,

the crack grows stabily with associated with

to the right of the peak.IC

b

G

Abbreviated Analysis of the Straight-Sided Buckle Delamination A 1D analysis based on vonKarman plate theory (Derived in Class) Propagation direction

h

Film pre-stress

Buckle deflection:

1( ) 1 cos( / )

2w y y b

Average stress in buckled film:

22

112C

hE

b

In-plane compatibility condition

2

2 2

1

1 1

2 8

b

C bw dy

E b

At edge of buckle:3 2

12

( ) ,12 2C

E hN h M

b

Energy release rate and mode mix along sides from basic solution:

1

( )( 3 )sides C C

hG

E

Buckle amplitude:4

13 Ch

4 3( / ) tantan

4 tan 3( / )

h

h

Energy-release rate can also be obtained fromdirect energy change calculation

Mode mix depends on the amplitude ofThe buckle

Energy release rate alongpropagating front

2

1

1

2

b

front sides Cb

hG G dy

b E

sidesG

frontG

Plots are given on next overhead

BASIC ELASTICITY SOLUTION FOR INFINITE ELASTIC BILAYER WITH SEMI-INFINITE CRACK (Covered in earlier lecture)

Equilibrated loads. General solution for energyrelease rate and stress intensity factors availablein Suo and Hutchinson (1990)

1 1,E

2 2,E

hP

M

moment/length

force/length interfacedelamination crack

Infinitely thick substrate--Primary case of interest for thin filmsand coatings on thick substrates

Dundurs’ mismatch parameters for plane strain:

1 22

1 2

,(1 )D

E E EE

E E

1 2 2 1

1 2 2 1

(1 2 ) (1 2 )1,

2 (1 ) (1 ) 2(1 )D

E

For homogeneous case: 0D D If both materials incompressible: 0D is the more important of the two parameters for most bilayer crack problems D

Take 0D if you can. It makes life easier!

1 1,E

2 2,E

hP

M

interfacedelamination crack

Basic solution continued:

Energy release rate

2 2

31

112

2

P MG

E d d

Stress intensity factors:(see Hutchinson & Suo (1992) if secondDundurs’ parameter cannot be taken to be zero)

( 0)D

1/ 2 3/ 2

1/ 2 3/ 2

1cos 2 3 sin

21

sin 2 3 cos2

I

II

K Pd Md

K Pd Md

where ( )D is shown as a plot and is tabulated in Suo & Hutch.

Note: For any interface crack between two isotropic materials,

2

2 2

1 2

1 1 1

2D

I IIG K KE E

2/(1 )E E

45

50

55

60

65

70

-1 -0.5 0 0.5 1D

52.1o

0D

no mismatch

Interface toughness—the role of mode mix,I IIK K

Experimental finding: The energy release rate required to propagatea crack along an interface generally depends on the mode mix, often withlarger toughness the larger the mode II component.

C

C

Interface Toughness: ( )

Propagation condition: ( )G

2C ( ) ( )Jm

Liechti & Chai (1992) data for an epoxy/glassinterface.

A phenomenological interface toughness law

2( ) 1 tan ((1 ) )C IC

( )C

in degrees

.1, .2, .5, 1

/IIC IC

1 no mode dependence

<<1 significant mode dependence

0.1

0

0.5

1

1.5

2 4 6 8 10

C=(b/b

C)2

sides

curved front

-80

-60

-40

-20

0

2 4 6 8 10

C=(b/b

C)2

D=-1/2, 0, 1/2

mode mix along the delamination sides

0

2

4

6

8

10

2 4 6 8 10

C=(b/b

C)2

=0.2,0.4, 0.6, 0.8, 1

Energy release rate and mode mix on sides of Straight-sided buckle delamination

22

112C

hE

b

21

12Cb E

h

2

01

1

2

hG

E

Stress atonset of buckling

Half-width atonset of buckling

Energy/areaavailable forrelease inplanes strain

Half-width of straight-sided delamination

Impose:2( ), ( ) 1 tan ((1 ) )ICG f f

0 ( )

1 1 3C CIC

G f

dStability of crack front requires: 0.

db ( )

i.e. if tip "accidentally" advances, it is no longer critical.

G

f

See earlier slide for interface toughness function

Stable configurations(half-widths, b) of1D delaminations

0D Mode II

Caution! This plot is difficult to interpret becauseeach axis depends on

Inverse determination of interface toughness, stress (or modulus)by measuring buckling deflection and delamination width

Gside 12S4b

4

2Db

2 4b

2

41

2 4SSG Sb

N0 Db

2

S4b

2

Straight-sided delamination without ridge crack on flat substrate

front (SS)

side

S stretching stiffness

D bending stiffness

Applies to any multilayer film with arbitrarystress distribution

The basic results can be written as:

If bending and stretching stiffness of the film are known, then the energy release rates and theresultant pre-stress can be determined by measurement of the deflection and the delamination width.

If resultant pre-stress is known, then the equations can be used to determine film modulus andrelease rates in terms of deflection and delamination width– see Faulhaber, et al (2006) for an example.