Cracking in an Elastic Film on a Power-law Creep Underlayer Jim Liang, Zhen Zhang, Jean Prévost,...
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Cracking in an Elastic Film on a Power-law Creep Underlayer Jim Liang, Zhen Zhang , Jean Prévost, Zhigang Suo Scaling law for a stationary crack The crack starts to advance when the stress inte nsity factor K attains a threshold value K th The crack initiation time is obtained by equating the K to K th The stress intensity factor scales with the initial stress and time as 1 2 1 1 2 1 3 ~ n n n t K The time needed for the crack to initiate its gro wth t I scales with the film initial stress as 1 3 ~ n I t 1 1 1 n n n n lt EHh A t l K (,) n where 2-D shear-lag model Rigid substrate x u v x u x u v E 2 1 Elastic thin film 0 h x , , , U E r t K t r u Field around crack Field around crack t u t u t u H A n n n 2 1 / 1 Power-law creep underlayer n ( ) 1 0.7303 2 0.6687 3 0.6368 4 0.6132 5 0.6011 n ( ) 1 1.0526 2 0.9109 3 0.8380 4 0.7936 5 0.7636 0.3 0.3 Scaling law for a crack advancing in steady-state Many brittle solids are susceptible to subcritica l crack growth The stress intensity factor and the crack velocity in the steady state is determined by the intersection of the two V-K curves The atomic bonds do not break when K < K th , and br eak instantaneously when K K c The atomic bonds break at a finite rate when K t <K< K c, and crack velocity V ) ( K F dt da 1 1 n n L h EH A V K L (,) n where Calculated by X-FEM Calculated by X-FEM
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Transcript of Cracking in an Elastic Film on a Power-law Creep Underlayer Jim Liang, Zhen Zhang, Jean Prévost,...
Cracking in an Elastic Film on a Power-law Creep UnderlayerJim Liang, Zhen Zhang, Jean Prévost, Zhigang Suo
Scaling law for a stationary crackThe crack starts to advance when the stress intensity factor K attains a threshold value Kth
The crack initiation time is obtained by equating the K to Kth
The stress intensity factor scales with the initial stress and time as 12
1
12
13
~
nn
n
tK
The time needed for the crack to initiate its growth tI scales with the film initial stress as 13~ n
It
1
1 1n n n nl t EHh A t
lK ( , )n where
2-D shear-lag model
Rigid substrate
x
uv
x
u
x
uvE
2
1
Elastic thin film
0
hx
,,, UE
rtKtru
Field around crackField around crack
t
u
t
u
t
u
H
A n
n
n
2
1
/1
Power-law creep underlayer
n ( )
1 0.7303
2 0.6687
3 0.6368
4 0.6132
5 0.6011
n ( )1 1.0526
2 0.9109
3 0.8380
4 0.7936
5 0.7636
0.3
0.3 Scaling law for a crack advancing in steady-state
Many brittle solids are susceptible to subcritical crack growth
The stress intensity factor and the crack velocity in the steady state is determined by the intersection of the two V-K curves
The atomic bonds do not break when K < Kth, and break instantaneously when K Kc
The atomic bonds break at a finite rate when Kt<K<Kc,and crack velocity V
)(KFdt
da
11 n nL h EH A V
K L( , )n where
Calculated by X-FEM
Calculated by X-FEM
Numerical results by X-FEM
A stationary crack, length 2a, is in the blanket film. The dimensionless ratio l/a indicates the time. Initially, l/a=0, the underlayer has not creep.
Shear stresses at the film/underlayer interface
l/a = 0.0
l/a = 2.15
l/a = 13.6
l/a = 17.1
After a short time, l/a=2.15 the crack opens, generating a region of high equivalent shear stress
After a long time, l/a=13.6, the crack approaches the equilibrium opening, the flow of the underlayer slows down, and the equivalent shear stress around the crack decreases. Far away from the crack, the film remains undisturbed. In between, stress relaxation is still occurring.
The crack tip appears to have created a complex flow pattern that generated two regions of relatively slow flow.
Normalized Time, t/tm
Norm
alized Stress Intensity Factor, K
/( l
m
½)
n=1
2
34
5
Semi-infinite stationary crack in a blanket film
Confirmation of equation by X-FEM. 12
112
13
,
nnnn
n
tAHhEnK
Finite stationary crack in a blanket film
Normalized Time
1 1
1
n n n n
n
l EHh A t
a a
Norm
alized Stress Intensity F
actor, K/
[(a)
½]
1
1 2 1, n n n nK n v EHh A t
K a
n = 1
n = 2
n = 3
n = 4
In a short time, l/a 0, the underlayer has not crept, the crack approaches a semi-infinite crack.
In a long time, l/a , the underlayer creep has affected the film over a region much larger than the crack length, so that the problem approaches that of a crack in a freestanding sheet subject to a remote stress, i.e., the Griffith crack.
If Kth > (a)1/2, the finite crack will never grow. Otherwise, the crack will initiate its growth after a delay time.
Crack advancing in a blanket film
Contact information
Zhen ZhangDivision of Engineering and Applied Science, Harvard University, Cambridge, MA 02138, USATel: 617-384-7894 E-mail:[email protected]://www.deas.harvard.edu/~zhangz
Jim LiangIntel Corp., Hillsboro, OR, USA, E-mail: [email protected]
Prof. Zhigang SuoDivision of Engineering and Applied Science, Harvard University, Cambridge, MA 02138, USATel: 617-495-3789 Fax: 617-496-0601E-mail:[email protected]://www.deas.harvard.edu/suo
Prof. Jean PrévostDepartment of Civil & Environmental Engineering, Princeton University, Princeton, NJ, USA
The time scale for the effect of the crack tip to propagate over the above length is
Normalized Time
1
1
1
1
n
nn
nn
A
tHhEl
Norm
alized Velocity
V t
c /N
ormalized C
rackE
xtension, a/
Norm
alized Stress Intensity
Factor, K
/(1/2 )
(a) n=5
(b) n=5
(c) n=5
2
0
K
nnn
n
c AHhEt
1
1
cn tV /2
0
Let crack grow when K=K0, so Introduce a length
Let V0 be the steady velocity corresponding to K0, so we get
When K=K0, the program extends the crack instantaneously by an arbitrarily specified length a.
When K<K0, the stress field evolves but the crack remains stationary.
K drops because the crack tip extends to a less relaxed part of the film. Then further stress field evolution brings K back to K0 again, the time interval t is calculated.
After a transient period, the crack attains a steady state velocity.
This process is repeated.
