Cracking%20in%20an%20Elastic%20Film%20on%20a%20Power-law%20Creep%20Underlayer

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Cracking in an Elastic Film on a Power-law Creep
Underlayer Jim Liang, Zhen Zhang, Jean Prévost,
Zhigang Suo
2-D shear-lag model
Calculated by X-FEM
Rigid substrate
n ? ( )
1 1.0526
2 0.9109
3 0.8380
4 0.7936
5 0.7636
Scaling law for a stationary crack

• The 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

Calculated by X-FEM

• The time needed for the crack to initiate its
  growth tI scales with the film initial stress as

n ? ( )
1 0.7303
2 0.6687
3 0.6368
4 0.6132
5 0.6011
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Numerical results by X-FEM
Finite stationary crack in a blanket film
Semi-infinite stationary crack in a blanket film
Normalized Stress Intensity Factor, K/(? lm ½)
Normalized Stress Intensity Factor, K/?(?a)½
Normalized Time, t/tm

• In a short time, l/a? 0, the underlayer has not
  crept, the crack approaches a semi-infinite crack.

Contact information

• 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 gt ?(?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

• Let crack grow when KK0, so Introduce a length

• When KltK0, the stress field evolves but the crack
  remains stationary.

• When KK0, the program extends the crack
  instantaneously by an arbitrarily specified
  length ?a.

• The time scale for the effect of the crack tip to
  propagate over the above length is

• 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.

• This process is repeated.

• Let V0 be the steady velocity corresponding to
  K0, so we get

• After a transient period, the crack attains a
  steady state velocity.