Cracking in an Elastic Film on a Power-law Creep Underlayer
Jim Liang, Zhen Zhang, Jean Prévost, Zhigang Suo
2-D shear-lag model
Elastic thin film
   
Field around crack
1  v  u
u


 E

 x
2
x

 


u
v




x


   

0
x
h
K t  r
u r , , t  
U  , 
E
Calculated by
X-FEM
Power-law creep underlayer

 u  u  


 t t 
A
 1/ n
H
1 n
2n
u
t
Rigid substrate
Scaling law for a stationary crack
K  
The crack starts to advance when the stress intensity factor K attains a
threshold value Kth
where
The crack initiation time is obtained by equating the K to Kth
The stress intensity factor scales with the initial stress and time as
The time needed for the crack to initiate its growth tI scales with the film
initial stress as
l
   ( n, )
l t 
n

  EHh 
K ~
3 n 1
2  n 1
tI ~ 
t
n 1
n
A
t

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
The stress intensity factor and the crack velocity in the steady
state is determined by the intersection of the two V-K curves
1.0526
2
0.9109
3
0.8380
4
0.7936
5
0.7636
Calculated by X-FEM
 3 n 1
L
   ( n, )
L h
EH


da
 F (K )
dt
1
1
2  n 1
K  
where
 (   0.3 )
1
n 1
Scaling law for a crack advancing in steady-state
Many brittle solids are susceptible to subcritical crack growth
n
1
A nV 

1
n
n
 (   0.3 )
1
0.7303
2
0.6687
3
0.6368
4
0.6132
5
0.6011
Numerical results by X-FEM
Shear stresses at the film/underlayer interface
l/a =
13.6
l/a =
17.1
n=1
2
3
4
5
Normalized Time, t/tm
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.
Confirmation of equation K   n, 
The crack tip appears to have created a complex flow
pattern that generated two regions of relatively slow flow.
E Hh A t 
n
n
1
2  n 1
n=1
n=2
n=3
n=4
K    n, v    EHh  A t 
n
l
 
a
n 1
n1  n
1
2 n1
EHh n n 1 A nt

a n 1
by X-FEM.
In a short time, l/a 0, the underlayer has not crept, the crack
approaches a semi-infinite crack.
Prof. Jean Prévost
Jim Liang
K  a
Normalized Time
Contact information
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.
3 n 1
2  n 1
Finite stationary crack in a blanket film
Normalized Stress Intensity Factor,
K/[(a)½]
l/a =
2.15
Normalized Stress Intensity Factor,
K/( lm ½)
l/a =
0.0
Semi-infinite stationary crack in a blanket film
Intel Corp., Hillsboro, OR, USA,
E-mail: jim.liang@intel.com
Department of Civil & Environmental
Engineering, Princeton University,
Princeton, NJ, USA
Zhen Zhang
Prof. Zhigang Suo
Division of Engineering and Applied
Science, Harvard University,
Cambridge, MA 02138, USA
Tel: 617-384-7894
E-mail:zzhang@fas.harvard.edu
http://www.deas.harvard.edu/~zhangz
Division of Engineering and Applied
Science, Harvard University, Cambridge,
MA 02138, USA
Tel: 617-495-3789
Fax: 617-496-0601
E-mail:suo@deas.harvard.edu
http://www.deas.harvard.edu/suo
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

tc 
n n 1  n
E Hh  A
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.
(c)
n=5
V0    / tc
2n
Normalized Velocity
V tc /
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.
(b)
n=5
n 1
Normalized Crack
Extension, a/
The time scale for the effect of the crack tip to
propagate over the above length is
When K<K0, the stress field evolves but the crack
remains stationary.
(a)
n=5
 K0 


 
2
Normalized Stress Intensity
Factor, K/(1/2 )
Let crack grow when K=K0, so Introduce a length
This process is repeated.
After a transient period, the crack attains a steady
state velocity.
Normalized Time
l   n1 E Hh n t 

 
n n 1
  A

1
n 1