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Simulation of fracture phenomena in polycristalline microsystems by a

Simulation of fracture phenomena in polycristalline microsystems by a domain decomposition approach Federica Confalonieri , Giuseppe Cocchetti , Aldo Ghisi , Alberto Corigliano. Outline. Reference problem Gravouil-Combescure’s algorithm Proposed algorithm

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Simulation of fracture phenomena in polycristalline microsystems by a

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

  2. Outline • Reference problem • Gravouil-Combescure’s algorithm • Proposed algorithm • Elastic-damage interface law • Numerical examples • Closing remarks

  3. Reference problem 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. Reference problem Polysilicon film Micro-scale (sub-micron) 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 Sensor Meso-scale (micron) Die Macro-scale (mm) Package

  5. Problem formulation • Weak form of equilibrium • Semi-discretized equations

  6. Limits of a traditional monolithic FE simulation Numerical strategy: Voronoi tessellation algorithm for the creation of a virtual polycristalline solid [Corigliano et al., 2007] [Corigliano et al., 2008] [Mariani et al., 2011] 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 High computational burden: • very refined spatial discretization • small explicit time steps Domain decompositionapproach

  7. Domain decomposition approach • 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. Gravouil-Combescure’s algorithm • General scheme Subdivision in N subdomains Dynamicsolution on each sub-domain Subdomaincouplingthrough interface condition • Governing equations • Equilibrium • Continuity of velocities at interface

  9. Gravouil-Combescure’s algorithm “unconstrainedproblem” Fext Dynamicequilibriumsolved on each sub-domain consideredisolated and subject to externalactionsonly. “constrainedproblem” Correction of the “free” solution to take into account interfaceinteractions. Λ Condensedinterfaceproblem

  10. Proposed algorithm

  11. Proposed algorithm

  12. Fracture propagation 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 tetraedralelements. Softening traction t –separation [u] law at grain boundaries and within grains is assumed.

  13. Material properties 13 • Polysiliconis assumed to feature: • one axis of elastic symmetry aligned with epitaxial growth direction x3 • random orientation of other two elastic symmetry directions in the x1- x2plane Matrix of elastic moduli for single-crystal Si (cfc symmetry) [Brantley, 1973 ] • 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-centeredmono-silicon. Reference valuefor nominal tensile strength: sc= 2 ÷ 4 GPa

  14. Numerical examples

  15. Numerical examples Reaction – displacement jump

  16. Numerical examples

  17. Numerical examples 17

  18. Numerical examples

  19. Numerical examples

  20. Closing remarks • 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. Thanks for your attention!

  22. References • [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] CoriglianoA., Cacchione F., Frangi A. and Zerbini S., “Numerical modelling of impact rupture in polysilicon microsystems”, Computational Mechanics, 42, 251-259 (2008) • [3] MarianiS., 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] CoriglianoA., 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] MarianiS., 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] GravouilA. 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., GravouilA. and CombescureA., "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.,"Computationalmodelling 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)

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