1. 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 intensity 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  The time needed for the crack to initiate its growth t I scales with the film initial stress as where 2-D shear-lag model Rigid substrate Elastic thin film Field around crack Power-law creep underlayer n  ( ) 10.7303 20.6687 30.6368 40.6132 50.6011 n  ( ) 11.0526 20.9109 30.8380 40.7936 50.7636 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 < K th, and break instantaneously when K K c  The atomic bonds break at a finite rate when K t <K<K c, and crack velocity V where Calculated by X-FEM

2. 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/t m Normalized Stress Intensity Factor, K /(  l m ½) n=1 2 3 4 5 Semi-infinite stationary crack in a blanket film  Confirmation of equation by X-FEM. Finite stationary crack in a blanket film Normalized Time Normalized Stress Intensity Factor, K /[  (  a )½] 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 K th >  (  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 Zhang 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 Jim Liang Intel Corp., Hillsboro, OR, USA, E-mail: jim.liang@intel.com Prof. Zhigang Suo 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 Prof. Jean Prévost Department 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 Normalized Velocity V t c /  Normalized Crack Extension,  a/  Normalized Stress Intensity Factor, K/(  1/2 ) (a)n=5 (b)n=5 (c)n=5  Let crack grow when K=K 0, so Introduce a length  Let V 0 be the steady velocity corresponding to K 0, so we get  When K=K 0, the program extends the crack instantaneously by an arbitrarily specified length  a.  When K<K 0, 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 K 0 again, the time interval  t is calculated.  After a transient period, the crack attains a steady state velocity.  This process is repeated.