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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.
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