Buckling Delamination with Application to Films and Laminates
No crack driving force
due to film stress; Unless
Thermal gradient with
interruption of heat transfer
across crack.
Mixed mode
interface crack
Buckling
delamination
Compression in film
producing buckling
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
Ceramic
(Zirconia)
Ceramic
(Alumina)
Metal
bond coat
Blades taken from an engine
showing areas of spalled-off
TBC
Buckle Delaminations: Interface cracking driven by buckling
Three Morphologies: Straight-sided, Varicose and Telephone Cord
Computer simulations
Straight-sided
Experimental observations
200nm DLC film on silicon
Propagation of a buckle delamination along a prepatterned tapered region of low adhesion between
film and substrate. In the wider regions the telephone
cord morphology is observed. It transitions to the
straight-sided morphology in the more narrow region
and finally arrests when the energy release rate drops
below the level needed to separate the interface.
A Model Problem—Mode I Buckling Delamination of Symmetric Bi-layer
L
P, Δ
P, Δ
2h
2b
buckled state
unbuckled state
P
PC
b
b + Δb
strain energy=SE = ∫ Pd Δ =
This neglects the very slight
increase of P as the buckling
amplitude increases
ΔC
Δ
Δ
0
1
PC Δ C + PC ( Δ − Δ C )
2
π 2 Eh 2
PC
π 2 h2
PC = 2(hw)
L=
L
, ΔC =
12 b 2
2hwE
12 b 2
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.
PC 2 L
SE = −
+ PC Δ
4 whE
G
Energy release rate under prescribed Δ
1 ∂SE
G=−
2 w ∂b
=
Δ
π 2 Eh3
6b
3
unbuckled
buckled
( Δ − ΔC )
Fixed b
two crack tips
ΔC
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:
GL3/ 2
b 12Δ
−3
−2
4
3
1
,
where
=
ξ
−
ξ
ξ
=
(
)
E Δ5 / 2
h π 2L
8 ⎛3⎞
GL3/ 2
The maximum of
is
3⎜ ⎟
E Δ 5/ 2
5 ⎝5⎠
3/ 2
= 1.288
1.5
GL3/ 2
EΔ 5/ 2
1
stable under
prescribed Δ
occuring for ξ = 5/3
GIC L3/ 2
EΔ 5/ 2
0.5
GIC L3/ 2
If
< 1.288, the crack will advance
E Δ 5/ 2
if b is such that G = GIC . If b is to the left of the
peak the crack is unstable under prescribed Δ
and it will jump to the value of b associated with
GIC to the right of the peak. If Δ is then increased,
the crack grows stabily with b associated with
GIC to the right of the peak.
unstable under
prescribed Δ
0
0
1
unbuckled
2
ξ
3
4
5
Abbreviated Analysis of the Straight-Sided Buckle Delamination
A 1D analysis based on vonKarman plate theory (See next 2 pages)
Propagation direction
G front
Film pre-stress
Gsides
h
Average stress in buckled film:
Buckle deflection:
1
w( y ) = δ (1 + cos(π y / b) )
2
Buckle amplitude:
⎞
δ
4⎛ σ
=
−
1
⎜
⎟
h
3 ⎝ σC
⎠
π2
⎛h⎞
σC =
E1 ⎜ ⎟
12 ⎝ b ⎠
2
h
(σ − σ C )(σ + 3σ C )
E1
1
1 b 2
π2 2
(σ − σ C ) = ∫−b w′ dy = δ
E1
2
8b
At edge of buckle:
E1h3 π 2δ
ΔN = (σ − σ C )h , M =
12 2b 2
Energy release rate and mode mix along sides from basic solution:
Gsides =
In-plane compatibility condition
4 + 3(δ / h) tan ω
tanψ =
−4 tan ω + 3(δ / h)
Energy release rate along
propagating front
G front =
1 b
h
2
=
σ
−
σ
G
dy
(
)
sides
C
2b ∫−b
E1
Discussed in class
Energy-release rate can also be obtained from
direct energy change calculation
Mode mix depends on the amplitude of
The buckle
Plots are given 3 slides ahead
Digression—Von Karmen nonlinear plate theory applied to clamped wide plates
Plate is infinite in z direction. Deformation is plane strain with
y
E = E /(1 −ν ) p ( x)
2
w( x)
•
Strain-displacement relations:
•
mid-surface strain: ε = u ′ + 12 w′2 ; mid-surface curvature: κ = w′′
u ( x)
Stress-strain relations: (for plate of thickness h)
N ≡∫
h/2
−h / 2
σ dy = Ehε , M ≡ − ∫
h/2
−h / 2
σ ydy = Dκ ,
D = Eh3 /12
δ
h
Equilibrium equations: (obtained from principle of virtual work)
Moment equil.: M ′′ − Nw′′ = p; Horizontal equil.: N ′ = 0
By equilibrium, N is independent of x
Finite deflection solution for buckling of clamped-clamped beam (wide plate)
2b
Notation: average compressive stress in unbuckled beam: σ = − N / h
average compressive stress in buckled beam: σ C = − N C / h
deflection at center of buckle: δ = w(0)
beam length = 2b (−b ≤ x ≤ b)
With the left end fixed, impose a displacement u = −Δ on the right end and then hold that end fixed.
The compressive stress in the unbuckled beam is σ = E Δ /(2b).
Moment equil. ⇒ Dw′′′′ + σ C h w′′ = 0; ( p = 0); clamped BC's ⇒ w = w′ = 0, x = ±b
This is an eigenvalue problem with
σC
as the eigenvalue. Note that this stress will be independent of the amplitude of w.
Continued on next slide
x
Von Karmen nonlinear plate theory applied to clamped wide plates--continued
⎛ σCh ⎞
⎛ σCh ⎞
x ⎟⎟ + c4 cos ⎜⎜
x ⎟⎟
General solution ⇒ w = c1 + c2 x + c3 sin ⎜⎜
D
D
⎝
⎠
⎝
⎠
⎛ σCh ⎞
For the lowest eigenvalue, the BCs ⇒ c2 = c3 = 0, sin ⎜⎜
⎝
D
b ⎟⎟ = 0, c1 = c4 .
⎠
π2
δ⎛
⎛h⎞
⎛ π x ⎞⎞
Thus, the stress in the buckled beam and the deflection shape are: σ C =
E ⎜ ⎟ & w( x) = ⎜1 + cos ⎜
⎟⎟
12 ⎝ b ⎠
2⎝
⎝ b ⎠⎠
2
Relation between stress in buckled beam, stress in unbuckled beam and deflection
With u and w measured from the unstressed state and u measured from the unbuckled stressed state,
(
N C = Ehε ⇒ − σ C h = −σ h + Eh u′ + 12 w′2
)
Now integrate the above equation from -b to b using u ( −b) = u (b) = 0 :
⇒ σ −σC =
π2 ⎛δ ⎞
E b 2
′
w
dx
E⎜ ⎟
=
4b ∫− b
16 ⎝ b ⎠
2
or
δ =h
⎞
4⎛ σ
− 1⎟
⎜
3 ⎝ σC
⎠
δ
h
Finally, we will need the moment at x = b:
E ⎛π ⎞
M (b) = Dw′′(b) ⇒ M (b) = ⎜ ⎟ δ
24 ⎝ b ⎠
This completes the finite deflection for the clamped-clamped wide plate.
2
1
σ
σC
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 energy
release rate and stress intensity factors available
in Suo and Hutchinson (1990)
Infinitely thick substrate-Primary case of interest for thin films
and coatings on thick substrates
moment/length
M
Dundurs’ mismatch parameters for plane strain:
E − E2
E
αD = 1
, E=
E1 + E2
(1 −ν 2 )
βD =
force/length
delamination crack
interface
E2 ,ν 2
E
1 μ1 (1 − 2ν 2 ) − μ2 (1 − 2ν 1 )
, μ=
2 μ1 (1 −ν 2 ) + μ2 (1 −ν 1 )
2(1 +ν )
For homogeneous case:
αD
E1 ,ν 1
P
αD = βD = 0
If both materials incompressible:
βD = 0
is the more important of the two parameters for most bilayer crack problems
Take
β D = 0 if you can. It makes life easier!
h
Basic solution continued:
Energy release rate
M
1 ⎛ P2
M2 ⎞
+ 12 3 ⎟
G=
⎜
d ⎠
2 E1 ⎝ d
E1 ,ν 1
P
delamination crack
E = E /(1 −ν )
2
1 ⎡ −1/ 2
−3/ 2
+
Pd
ω
Md
cos
2
3
sin ω ⎤⎦
⎣
2
1 ⎡ −1/ 2
K II =
Pd sin ω − 2 3Md −3/ 2 cos ω ⎤⎦
⎣
2
where
ω (α D ) is shown as a plot and is tabulated in Suo & Hutch.
Note: For any interface crack between two isotropic materials,
1 − βD2 ⎛ 1
1 ⎞
2
2
+
G=
⎜
⎟ ( K I + K II )
2 ⎝ E1 E2 ⎠
interface
E2 ,ν 2
Stress intensity factors: ( β D = 0)
(see Hutchinson & Suo (1992) if second
Dundurs’ parameter cannot be taken to be zero)
KI =
h
70
65
βD = 0
60
no mismatch
55
o
52.1
50
•
45
-1
-0.5
0
0.5
α
1
D
K I , K II
Interface toughness—the role of mode mix
Experimental finding: The energy release rate required to propagate
a crack along an interface generally depends on the mode mix, often with
−2
larger toughness the larger the mode II component.
Γ C (ψ ) ( Jm )
Interface Toughness:
Γ C (ψ )
Propagation condition: G = Γ C (ψ )
A phenomenological interface toughness law
λ ≅ 0.1
ΓC (ψ ) = Γ IC (1 + tan 2 ((1 − λ )ψ ) )
ΓC (ψ )
Γ IIC / Γ IC
ψ
λ = .1, .2, .5, 1
Liechti & Chai (1992) data for an epoxy/glass
interface.
ψ in degrees
λ
λ = 1 ⇒ no mode dependence
λ <<1⇒ significant mode dependence
Energy release rate and mode mix on sides of Straight-sided buckle delamination
π2
⎛h⎞
σC =
E1 ⎜ ⎟
12 ⎝ b ⎠
2
Stress at
onset of
buckling
1.5
0
mode mix along the delamination sides
-20
sides
bC
π E1
=
h
12 σ
2
1 σ 2h
G0 =
2 E1
Half-width at
onset of
buckling
Energy/area
available for
release in
planes strain
1
-40
α =-1/2, 0, 1/2
D
0.5
-60
curved front
Pure mode II
-80
0
2
4
6
8
σ/σ =(b/b )
C
2
G = Γ IC f (ψ ),
6
8
σ/σ =(b/b )
C
f (ψ ) = 1 + tan 2 ((1 − λ )ψ )
G0
f (ψ )
=
σC ⎞
Γ IC ⎛ σ C ⎞ ⎛
⎜1 −
⎟ ⎜1 + 3
σ ⎠⎝
σ ⎟⎠
⎝
4
10
2
C
10
8
See earlier slide for interface toughness function
⇒
2
C
Half-width of straight-sided delamination
Impose:
10
6
Stable configurations
(half-widths, b) of
1D delaminations
Mode II
4
αD = 0
λ =0.2,0.4, 0.6, 0.8, 1
2
d ⎛ G ⎞
⎜
⎟ < 0.
db ⎝ f (ψ ) ⎠
i.e. if tip "accidentally" advances, it is no longer critical.
Stability of crack front requires:
0
2
4
6
8
σ/σC=(b/b C)
Caution! This plot is difficult to interpret because
each axis depends on σ
2
10
Illustration of Spread of Delamination if no mixed mode dependence
(λ = 1 & G = Γ IC )
Scenario: Given Γ IC & initial delamination flaw with length 2b0 .
Monotonically increase the pre-stress (the stress in the unbuckled film), σ .
This is the pre-stress, σ ,
at which crack will advance.
G
Note that once the interface crack advances,
G > Γ IC and it will spread dynamically without
limit. For mode-independent interface toughness
the condition to ensure no "wholesale" delamination
is G0 < Γ IC , or
σ 2h
< Γ IC .
2E
Stresses well above this level can be tolerated if the
Γ IC
interface toughness has a significant mixed mode
dependence.
σ = σ C0
σ <σ
Since the delamination becomes mode II as it
0
C
spreads, the above simple criterion against
complete delamination can generalized when
there is mode-dependence of the toughness by
σ = buckling stress
0
C
associated with b0
b0
b
the requirement, G0 < Γ IIC . But such a criterion
would not exclude localized delaminations such
as telephone cord delaminations.
Inverse determination of interface toughness, stress (or modulus)
by measuring buckling deflection and delamination width
Straight-sided delamination without ridge crack on flat substrate
S ∼ stretching stiffness
D ∼ bending stiffness
front (SS)
The basic results can be written as:
side
1 ⎛π δ ⎞
GSS = S ⎜
⎟
2 ⎝4 b⎠
4
1 ⎛ π δ⎞
⎛ π ⎞ ⎛ π δ⎞
= S
+ 2D
⎝ b⎠ ⎝ 4 b⎠
2 ⎝ 4 b⎠
4
Gside
⎛π⎞
⎛π δ ⎞
N 0 = D⎝ ⎠ + S ⎝
b
4 b⎠
2
2
2
2
Applies to any multilayer film with arbitrary
stress distribution
If bending and stretching stiffness of the film are known, then the energy release rates and the
resultant 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 and
release rates in terms of deflection and delamination width– see Faulhaber, et al (2006) for an example.
Energy Released as a Function of Morphology
Three morphologies:
DLC on silicon—tapered low adhesion interface: propagates from right to left
Film under equi-biaxial stress
Energy/area:
σ 2h
U0 =
E (1 −ν )
Energy/area in buckled film averaged
over one full wavelength:
For σ / σ C < 6 :
U
Euler (straight-sides) mode is only possible mode
σ / σ C = (b / bC ) 2
For σ / σ C > 7.5 :
Telephone cord morphology has lowest energy and releases
the most energy/area.
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 modulus
is 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.
Shear lag relaxation of stress
in bonded film
Compliant substrate:
simulations and exps.
Ni films on polycarbonate
substrates (Parry, et al.)
clamped model
greater rotation
along edges