1. Dipartimento di Ingegneria Strutturale Simulation of fracture phenomena in polycristalline microsystems by a domain decomposition approach Federica Confalonieri, Giuseppe Cocchetti, Aldo Ghisi, Alberto Corigliano

2. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 Outline 2 1.Reference problem 2.Gravouil-Combescure’s algorithm 3.Proposed algorithm 4.Elastic-damage interface law 5.Numerical examples 6.Closing remarks

3. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 Reference problem 3 Analysis of the mechanical response under impact dynamics and crack propagation Engineering motivation: failure of polysilicon inertial MEMS sensors exposed to accidental drops and shocks

4. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 Reference problem 4 Simulation of polysilicon MEMS at the micro-scale level The behaviour of the structural parts composing a micro-system is simulated The grain morphology has to be properly described Heterogeneities and defects strongly influence the micro-structural behaviour Macro-scale (mm) Package Die Sensor Meso-scale (micron) Micro-scale (sub- micron) Polysilicon film

5. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 Problem formulation 5 Weak form of equilibrium Semi-discretized equations

6. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 High computational burden: very refined spatial discretization small explicit time steps Limits of a traditional monolithic FE simulation 6 Numerical strategy: Voronoi tessellation algorithm for the creation of a virtual polycristalline solid 3D monolithic finite element code: Implicit/explicit algorithm for the solution of the semi-discretized equations of motion Automatic procedure for the introduction of zero-thickness cohesive elements [Corigliano et al., 2007] [Corigliano et al., 2008] [Mariani et al., 2011] Domain decomposition approach

7. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 Domain decomposition approach 7 The grain structure of the polysilicon is well suited to a decomposition into subdomains. Each subdomain corresponds to a single grain or to a set of grains.

8. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 Gravouil-Combescure’s algorithm 8 General scheme Subdivision in N subdomains Dynamic solution on each sub-domain Subdomain coupling through interface condition Governing equations Equilibrium Continuity of velocities at interface

9. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 Gravouil-Combescure’s algorithm 9 Dynamic equilibrium solved on each sub- domain considered isolated and subject to external actions only. “unconstrained problem” F ext “constrained problem” Correction of the “free” solution to take into account interface interactions. Λ Condensed interface problem

10. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 Proposed algorithm 10

11. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 Proposed algorithm 11

12. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 Fracture propagation 12 Crack propagation is allowed both inside and along grain boundaries through a cohesive approach. An algorithm able to introduce dynamically cohesive elements is used: 6-node triangular cohesive elements are introduced between 10-nodes tetraedral elements. Softening traction t –separation [ u ] law at grain boundaries and within grains is assumed.

13. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 Material properties 13 Matrix of elastic moduli for single-crystal Si (cfc symmetry) [Brantley, 1973 ] Polysilicon is assumed to feature: one axis of elastic symmetry aligned with epitaxial growth direction x 3 random orientation of other two elastic symmetry directions in the x 1 - x 2 plane Each grain is treated as a continuum and is assumed to be elastic anisotropic, since each grain has its own crystal orientation. The intra-granular constitutive behaviour can be described by an orthotropic elastic law, as a result of the cubic-symmetry of face-centered mono-silicon. Reference value for nominal tensile strength:  c = 2 ÷ 4 GPa

14. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 Numerical examples 14

15. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 Numerical examples 15 Reaction – displacement jump

16. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 16 Numerical examples

17. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 Numerical examples 17 Number of nodes67845 Number of elements44265

18. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 18 Numerical examples Elastic analysis Monolithic solution367 s Proposed algorithm (1 grain = 1 subdomain) 187 s- 49,1 %

19. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 19 Numerical examples Fracture simulation Monolithic solution Proposed algorithm (1 grain = 1 subdomain)

20. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 Closing remarks 20 Advantages of the procedure: Fractures can propagate both in the grains and on the intergranular surfaces Efficient handling of the implicit/explicit numerical technique Reduction of the computational burden Future developments: Parallel computing Optimization of the decomposition into subdomains

21. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 Thanks for your attention! 21

22. F. Confalonieri, G. Cocchetti, A. Ghisi, A. Corigliano-AIMETA 2011 References 22 [1] Corigliano A., Cacchione F., Frangi A. and Zerbini S., "Numerical simulation of impact-induced rupture in polisylicon MEMS", Sensors letters, 6, 1-8 (2007) [2] Corigliano A., Cacchione F., Frangi A. and Zerbini S., “Numerical modelling of impact rupture in polysilicon microsystems”, Computational Mechanics, 42, 251-259 (2008) [3] Mariani S., Ghisi A., Fachin F., Cacchione F., Corigliano A. and Zerbini S., “A three-scale FE approach to reliability analysis of MEMS sensors subject to impacts”, Meccanica, 43, 469-483 (2008) [4] Corigliano A., Ghisi A., Langfelder G., Longoni A., Zaraga F. and Merassi A., “A microsystem for the fracture characterization of polysilicon at the micro-scale”, European Journal of Mechanics Solids- A/Solids, 30, 127-136 (2011) [5] Mariani S., Martini R., Ghisi A, Corigliano A. and Simoni B., “Monte Carlo simulation of micro-cracking in polysilicon MEMS exposed to shocks”, International Journal of Fracture, 167, 83-101 (2011) [6] Gravouil A. and Combescure A., "Multi-time-step explicit-implicit method for non-linear structural dynamics", International Journal for Numerical Methods in Engineering, 50, 199-225 (2001) [7] Mahjoubi N., Gravouil A. and Combescure A., "Coupling subdomains with heterogeneous time integrators and incompatible time steps", Computational Mechanics, 44, 825-843 (2009) [8] Farhat C. and Roux F.X., “A method for finite element tearing and interconnencting and its parallel solution algorithm”, International Journal for Numerical Methods in Engineering, 32, 1205-1227 (1991) [9] Confalonieri F., Cocchetti G. and Corigliano A., "A domain decomposition approach for elastic solids with damageable interfaces", XXV GIMC conference, Siracusa (2011) [10] Brantley B.A., "Calculated elastic constants for stress problems associated with semiconductor devices", Journal of Applied Physics, 44, 534-535 (1973) [11] Camacho G.T. and Ortiz M.,"Computational modelling of impact damage in brittle materials", International Journal of Solids and Structures, 33, 20-22 (1996) [12] Pandolfi A. and Ortiz M., "Finite-deformation irreversible cohesive elements for three-dimensional crack- propagation analysis", International Journal for Numerical Methods in Engineering, 44, 1267-1282 (1999)