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

1. Elastic thin film Field around crack Power-law creep underlayer where Scaling law for a crack advancing in steady-state Many brittle solids are susceptible to subcritical crack growth The atomic bonds do not break when K < Kth, and break instantaneously when K Kc where 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 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 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 tIscales with the film initial stress as

2. Normalized Stress Intensity Factor, K/(1/2 ) l/a = 0.0 l/a = 2.15 l/a = 13.6 l/a = 17.1 Normalized Crack Extension, a/ Shear stresses at the film/underlayer interface Normalized Velocity V tc / n = 1 n=1 n = 2 2 3 n = 3 4 5 n = 4 Normalized Time 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 by X-FEM. After a short time, l/a=2.15 the crack opens, generating a region of high equivalent shear stress Prof. Jean Prévost Department of Civil & Environmental Engineering, Princeton University, Princeton, NJ, USA Jim Liang Intel Corp., Hillsboro, OR, USA, E-mail: jim.liang@intel.com 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. 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 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 The crack tip appears to have created a complex flow pattern that generated two regions of relatively slow flow. (a) n=5 (b) n=5 (c) n=5 Normalized Time 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/( lm½) 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 > (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 K=K0, so Introduce a length When K<K0, the stress field evolves but the crack remains stationary. When K=K0, 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.