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$: Dipartimento di Ingegneria Strutturale Simulation of fracture phenomena in polycristalline microsystems by a domain decomposition approach Federica Confalonieri,$
Dipartimento di Ingegneria Strutturale

Simulation of fracture phenomena in polycristalline microsystems by a domain decomposition approach

Federica Confalonieri, Giuseppe Cocchetti, Aldo Ghisi, Alberto Corigliano

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


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


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


Problem formulation 5

• Weak form of equilibrium

u NUε = BUσ = dBU

inttint int

[u] = B U

t = k B U

• Semi-discretized equations

( )

, on f

u

d d d d dS

u u σ ε u ε u t u u b u f u 0

u u 0

cohes_intint ext MU + F U + F U = F


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


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.


Gravouil-Combescure’s algorithm 8

• General scheme

Subdivision in N subdomains

Dynamic solution on each sub-domain

Subdomain coupling through interface condition

k=1,N k k k kext intMU KU F F

1

N

k

k kC U 0

• Governing equations• Equilibrium

• Continuity of velocities at interface


Gravouil-Combescure’s algorithm 9

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U U U

U U U

U U U

11 1free free n

k k k kn n ext M U K U F

Dynamic equilibrium solved on each sub-domain considered isolated and subject to external actions only.

“unconstrained problem”Fext

“constrained problem”Correction of the “free” solution to take into account interface interactions.

1 1free free

k kn n

Tk k kn+1M U K U C Λ

1

0N

i i

i

C U

Λ

1 11

free

Nk k

n nk

HΛ C UCondensed interface problem


Proposed algorithm 10


Proposed algorithm 11


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.


Material properties 13

79.600000079.600000079.6000000165.763.963.900063.9165.763.900063.963.9165.7

GPa

D

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 x3

• random orientation of other two elastic symmetry directions in the x1- x2 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: sc = 2 ÷ 4 GPa


Numerical examples 14



Reaction – displacement jump


16Numerical examples



Number of nodes 67845

Number of elements 44265



Elastic analysis

Monolithic solution 367 s

Proposed algorithm(1 grain = 1 subdomain) 187 s - 49,1 %



Fracture simulation

Monolithic solution

Proposed algorithm(1 grain = 1 subdomain)


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


Thanks for your attention!

21


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)

dipartimento di ingegneria strutturale simulation of fracture phenomena in polycristalline...

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