薄膜力学8-断裂2

GSS
l
2 0
2E f
Z ( , )
02 h f
Ef
l 2Z ( , )h f
For a thin substrate with the lower surface fixed:
ks ~
s
hs
l2
E f h f hs
s
Effect of substrate constraint
1 When = = 0, s = 0.5; 0.5 For < 0, s < 0.5 (less singular); For > 0, s > 0.5 (more singular); For 1, s 1.

-1 0 1
Penetration or deflection?
Elastic shear-lag approximation
0 0
x h f
u Ef x
ksu
(x)
0
S
0
ks 2u u 2 x E f hf
u(0) 0 ( S / 2) 0
Average cgth
From the shear-lag model,
S GSS tanh 2E f l
2 0 l
l2
E f hf ks
For thick substrates, compare the steady-state energy release rate with numerical solutions. For S :
h xx (r ,0) C r
s
The stress singularity exponent, s, depends on the elastic mismatch (Zak and Williams, 1963): 2 2 cos( s ) 2 (1 s) 0 2 s 1 1
Film Substrate

G p s
Gd i
When both conditions are satisfied, the crack is likely to penetrate into the substrate if
Gp s Gd 1 i
Both the energy release rates depend on the elastic mismatch between the film and the substrate.
A three-dimensional process: crack grows in both parallel and perpendicular directions, with a curved crack front.
Cutting and Channeling
Cutting: a crack growing perpendicular to the interface; a plane strain problem, assuming long in the parallel direction.
35 30 25 20 15 10 5 0 -1
Beuth [21] XFEM = /4
GSS Z ( , )
02 h
Ef
Z 1.976
-0.5 0 Elastic mismatch, 0.5 1
Beuth, IJSS 29, 1657-1675 (1992); Huang et al., Engineering Fracture Mech. 70, 2513-2526 (2003).
Dundurs parameters: f s 1 s f 1 E f Es f s 1 s f 1 E f Es

0.25
f s 1 s f 1 f s 1 s f 1
When = = 0 or a/h 0:
F
0
0
K I 1.12 a
• Stiff film on compliant substrate: SIF increases monotonically; • Compliant film on stiff substrate: SIF attains a maximum near the interface
Ratcheting underlayer: the constraint is gradually lost over thermal cycles, analogous to creep.
Crack growth modulated by creep
Tensile Film
Creeping layer
Steady-state channeling cracks
The channel front maintains its shape as it advances, and the cross section (opening) profile behind the front attains the equilibrium shape of a plane-strain crack. Thus, the steady-state energy release rate of a channeling crack can be obtained from the plane strain problem, and the fracture resistance is the film toughness.
S 2u ( S / 2) tanh Ef l
Energy release rate:
0l
l2
E f hf ks
02l 1 S GSS 0 tanh 2 2E f l
Xia and Hutchinson, JMPS 48, 1107-1131 (2000).
Rigid substrate
For a stationary long crack: When t 0, Steady-state velocity:

~a Free-standing film
a
h

G1 ~
2
E
a

a h
Film on elastic substrate
G2 ~
2
E
h
~h
The energy release rate increases if the crack penetrates into the substrate or if the film debond.
1 h GSS 0 ( z )dz 0 2h
From dimensional analysis:
or
1 h GSS G(a)dz h 0
GSS Z ( , )
02 h
Ef
Numerical results
40
Dimensionless energy release rate, (G E *)/(2h) 0 SS 1
Film
h
0
Substrate
Steady-state energy release rate
z Film
0
Film
(z)
x
0
h
Substrate Substrate (b)
Ahead of the channel front
Far behind the channel front
Energy release rate:
E E 1 2
Plane strain:
3 4
No mismatch: = = 0;
-1
1 -0.25

Stiff film on compliant substrate: > 0;
Compliant film on stiff substrate: < 0;
If f = s = 0.5, = 0; If f = s = 1/3, = /4;
Film
h
0
Substrate
Three-dimensional analysis is required to determine the shape of the channel front (Nakamura and Kamath, 1992).
The crack reaches a steady state when the length exceeds a few times the film thickness.
Both and change signs when the materials are switched.
Partially cracked films
Film Substrate

Stress intensity factor at the crack tip:
a K I F , , h h
Channeling: crack(s) growing parallel to the interface, assuming a constant depth in the perpendicular direction.
Elastic mismatch
For elastic films and substrates, the crack behavior depends on the elastic mismatch between them.
EM 397: Thin Film Mechanics
VIII. Film Cracking under Tension