Buckling Delamination with Application to Films and LaminatesThermal 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 multilayersApplication 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 bucklingThree 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 Δ

2b

L

CPP

ΔCΔ

( )0

1strain energy=2 C C C CSE Pd P P

Δ= Δ = Δ + Δ − Δ∫

unbuckled state buckled state

2 2 2 2

2 22( ) ,12 2 12

CC C

PEh hP hw L Lb hwE b

π π= Δ = =

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

CP LSE PwhE

= − + Δ

( )2 3

3

12 6 C

SE EhGw b b

π

Δ

∂= − = Δ − Δ

∂

Energy release rate under prescribed Δ

two crack tips

G

CΔ Δ

This neglects the very slightincrease of P as the bucklingamplitude increases

unbuckled buckled

Fixed b

,P Δ

bb b+ Δ

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 , whereGL bE h L

ξ ξ ξπ

− − Δ= − =

Δ

0

0.5

1

1.5

0 1 2 3 4 5ξ

3/ 2

5/ 2

GLEΔ3/ 23/ 2

5/ 2

8 3The maximum of is 3 1.2885 5

occuring for = 5/3

GLE

ξ

⎛ ⎞ =⎜ ⎟Δ ⎝ ⎠

unbuckled

stable underprescribed Δ

unstable underprescribed Δ

3/ 2

5/ 2ICG L

EΔ

3/ 2

5/ 2If 1.288, the crack will advance

if is such that . If is to the left of thepeak the crack is unstable under prescribed and it will jump to the value of associated with

to the

IC

IC

IC

G LE

b G G b

bG

<Δ

=

Δ

right of the peak. If is then increased,the crack grows stabily with associated with

to the right of the peak.IC

bG

Δ

Abbreviated Analysis of the Straight-Sided Buckle DelaminationA 1D analysis based on vonKarman plate theory (See next 2 pages) Propagation direction

h

Film pre-stress

Buckle deflection:

( )1( ) 1 cos( / )2

w y y bδ π= +

Average stress in buckled film:22

112ChEb

πσ ⎛ ⎞= ⎜ ⎟⎝ ⎠

In-plane compatibility condition

( )2

2 2

1

1 12 8

b

C bw dy

E bπσ σ δ

−′− = =∫

At edge of buckle:3 2

12( ) ,

12 2CE hN h M

bπ δσ σΔ = − =

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

1

( )( 3 )sides C ChGE

σ σ σ σ= − +

Buckle amplitude:4 13 Ch

δ σσ

⎛ ⎞= −⎜ ⎟

⎝ ⎠

4 3( / ) tantan4 tan 3( / )

hh

δ ωψω δ

+=

− +

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

12

b

front sides Cb

hG G dyb E

σ σ−

= = −∫

sidesG

frontG

Plots are given 3 slides ahead

Discussed in class

Digression—Von Karmen nonlinear plate theory applied to clamped wide plates

x

y

••

( )u x

( )w xPlate is infinite in z direction. Deformation is plane strain with 2/(1 )E E ν= −

Strain-displacement relations:1 22mid-surface strain: ; mid-surface curvature:u w wε κ′ ′ ′′= + =

Stress-strain relations: (for plate of thickness h)

/ 2 / 2 3

/ 2 / 2, , /12

h h

h hN dy Eh M ydy D D Ehσ ε σ κ

− −≡ = ≡ − = =∫ ∫

( )p x

Equilibrium equations: (obtained from principle of virtual work)

Moment equil.: ; Horizontal equil.: 0M Nw p N′′ ′′ ′− = =

Finite deflection solution for buckling of clamped-clamped beam (wide plate)

h δ

2b

Notation: average compressive stress in unbuckled beam: /average compressive stress in buckled beam: /deflection at center of buckle: (0)beam length 2 ( )

With the left end fixed,

C C

N hN h

wb b x b

σσ

δ

= −= −

== − ≤ ≤ impose a displacement on the right end and then hold that end fixed.

The compressive stress in the unbuckled beam is /(2 ).u

E bσ= −Δ

= Δ

By equilibrium, N is independent of x

Moment equil. 0; ( 0); clamped BC's 0,CDw h w p w w x bσ′′′′ ′′ ′⇒ + = = ⇒ = = = ±

This is an eigenvalue problem with Cσ as the eigenvalue. Note that this stress will be independent of the amplitude of w.

Continued on next slide

Von Karmen nonlinear plate theory applied to clamped wide plates--continued

2 3 1 4

22

For the lowest eigenvalue, the BCs 0, sin 0, .

Thus, the stress in the buckled beam and the deflection shape are: & ( ) 1 cos12 2

C

C

hc c b c cD

h xE w xb b

σ

π δ πσ

⎛ ⎞⇒ = = = =⎜ ⎟⎜ ⎟

⎝ ⎠

⎛ ⎞⎛ ⎞ ⎛ ⎞= = +⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

Relation between stress in buckled beam, stress in unbuckled beam and deflection

( )1 22

With and measured from the unstressed state and measured from the unbuckled stressed state,

Now integrate the above equation from -b to b using ( ) ( ) 0 :

4

C C

C

u w u

N Eh h h Eh u w

u b u b

Eb

ε σ σ

σ σ

′ ′= ⇒ − = − + +

− = =

′⇒ − =22

2 4 116 3

b

bC

w dx E or hb

π δ σδσ−

⎛ ⎞⎛ ⎞= = −⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

∫

1 2 3 4General solution sin cosC Ch hw c c x c x c xD D

σ σ⎛ ⎞ ⎛ ⎞⇒ = + + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

2

Finally, we will need the moment at :

( ) ( ) ( )24

This completes the finite deflection for the clamped-clamped wide plate.

x b

EM b Dw b M bbπ δ

=

⎛ ⎞′′= ⇒ = ⎜ ⎟⎝ ⎠

hδ

C

σσ1

BASIC ELASTICITY SOLUTION FOR INFINITE ELASTIC BILAYER WITH SEMI-INFINITE CRACK(Covered in earlier lectures and included here again for completeness)

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

1 1,E ν

2 2,E ν

hP

Mmoment/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 EEE E

αν

−= =

+ −

1 2 2 1

1 2 2 1

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

Eμ ν μ νβ μμ ν μ ν ν

− − −= =

− + − +

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

Take 0Dβ = if you can. It makes life easier!

1 1,E ν

2 2,E ν

hP

M

interfacedelamination crack

Basic solution continued:

Energy release rate

2 2

31

1 122

P MGE d d

⎛ ⎞= +⎜ ⎟

⎝ ⎠

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

( 0)Dβ =

1/ 2 3/ 2

1/ 2 3/ 2

1 cos 2 3 sin21 sin 2 3 cos2

I

II

K Pd Md

K Pd Md

ω ω

ω ω

− −

− −

⎡ ⎤= +⎣ ⎦

⎡ ⎤= −⎣ ⎦

where ( )Dω α 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 12

DI IIG K K

E Eβ ⎛ ⎞−

= + +⎜ ⎟⎝ ⎠

2/(1 )E E ν= −

45

50

55

60

65

70

-1 -0.5 0 0.5 1αD

52.1o

0Dβ =

•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

ψψ

Γ= Γ

2C ( ) ( )Jmψ −Γ

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

A phenomenological interface toughness law

( )2( ) 1 tan ((1 ) )C ICψ λ ψΓ = Γ + −

( )C ψΓ

in degreesψ

.1, .2, .5, 1λ =

λ

/IIC ICΓ Γ

1 no mode dependence<<1 significant mode dependence

λλ

= ⇒⇒

0.1λ ≅

0

0.5

1

1.5

2 4 6 8 10σ/σ

C=(b/b

C)2

sides

curved front

-80

-60

-40

-20

0

2 4 6 8 10σ/σ

C=(b/b

C)2

αD=-1/2, 0, 1/2

mode mix along the delamination sides

0

2

4

6

8

10

2 4 6 8 10σ/σ

C=(b/b

C)2

λ=0.2,0.4, 0.6, 0.8, 1

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

22

112ChEb

πσ ⎛ ⎞= ⎜ ⎟⎝ ⎠

21

12Cb Eh

πσ

=

2

01

12

hGE

σ=

Stress atonset of buckling

Half-width atonset ofbuckling

Energy/areaavailable forrelease inplanes strain

Half-width of straight-sided delamination

Impose: 2( ), ( ) 1 tan ((1 ) )ICG f fψ ψ λ ψ= Γ = + −

0 ( )

1 1 3C CIC

G f ψσ σσ σ

⇒ =Γ ⎛ ⎞⎛ ⎞− +⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

dStability of crack front requires: 0.db ( )

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

Gf ψ

⎛ ⎞<⎜ ⎟

⎝ ⎠

See earlier slide for interface toughness function

Stable configurations(half-widths, b) of1D delaminations

0Dα =Mode II

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

Pure mode II

Illustration of Spread of Delamination if no mixed mode dependence ( 1 & )ICGλ = = Γ

0Scenario: Given & initial delamination flaw with length 2 .Monotonically increase the pre-stress (the stress in the unbuckled film), .

IC bσ

Γ

ICΓ

0b

G

b

0Cσ σ<

0Cσ σ=

0

0

buckling stressassociated with

C

bσ =

This is the pre-stress, ,at which crack will advance.

σ

0

Note that once the interface crack advances,and it will spread dynamically without

limit. For mode-independent interface toughnessthe condition to ensure no "wholesale" delamination

is , or

IC

IC

G

G σ

> Γ

< Γ2

.2

Stresses well above this level can be tolerated if theinterface toughness has a significant mixed modedependence.

ICh

E< Γ

0

Since the delamination becomes mode II as itspreads, the above simple criterion against

can generalized whenthere is mode-dependence of the toughness by the requirement, .IICG < Γ

complete delamination

But such a criterionwould not exclude localized delaminations suchas telephone cord delaminations.

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

Gside =12

Sπ4

δb

⎛ ⎝

⎞ ⎠

4

+ 2Dπb

⎛ ⎝

⎞ ⎠

2 π4

δb

⎛ ⎝

⎞ ⎠

2

412 4SSG S

bπ δ⎛ ⎞= ⎜ ⎟

⎝ ⎠

N0 = Dπb

⎛ ⎝

⎞ ⎠

2

+ Sπ4

δb

⎛ ⎝

⎞ ⎠

2

Straight-sided delamination without ridge crack on flat substratefront (SS)

side

S stretching stiffnessD 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.

Energy Released as a Function of MorphologyThree morphologies:

2/ ( / )C Cb bσ σ =

Film under equi-biaxial stress

Energy/area:2

0 (1 )hU

Eσ

ν=

−

Energy/area in buckled film averagedover one full wavelength: U

Euler (straight-sides) mode is only possible modeFor / 6 :Cσ σ <

Telephone cord morphology has lowest energy and releasesthe most energy/area.

For / 7.5 :Cσ σ >

DLC on silicon—tapered low adhesion interface: propagates from right to left

Moon et al 2004

Metal or Ceramic Films on Compliant Substrates (Polymer or Elastomer)Cotterell & Chen, 2000; Yu & Hutch, 2002; Parry, et al.,2005

Analytical Fact: Edges of buckle delamination is effectively clamped if substrate modulusis larger than 1/3 of film modulus (i.e. clamped plate model is valid)

Highly compliant substrate has three effects:1) Stabilizes straight-sided buckle delamination and tends to eliminate telephone cord morphology.2) Significant film rotation occurs at edges of delamination and larger buckling deflections.3) Relaxation of stress along bonded edges of delamination (shear lag effect) amplifies energy released.

Ni films on polycarbonatesubstrates (Parry, et al.)

greater rotationalong edges

Shear lag relaxation of stressin bonded film

clamped model

Compliant substrate:simulations and exps.