Vol. 6, No. 7-8, 2011

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ISSN: 1559-3959
Perturbation stochastic finite element-based homogenization of polycrystalline materials

Severine Lepage, Fernando V. Stump, Isaiah H. Kim and Philippe H. Geubelle

Vol. 6 (2011), No. 7-8, 1153–1170
Abstract
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This work presents a study of the influence on the macroscopic (homogenized) elastic properties of polycrystalline materials induced by uncertainties in the material texture and microstructure geometry. Since many microelectromechanical systems are made of materials deposited as thin films with 111fiber texture, we study the variance of the homogenized elastic properties of the material around its nominal 111 texture. To perform this analysis, the perturbation stochastic finite element method (PSFEM) is coupled to the mathematical theory of homogenization leading to a second-order perturbation-based homogenization method. This method is able to evaluate the mean and variance of a given homogenized property as a function of the grain property uncertainty. The multiscale formulation is implemented in a plane-stress linear elastic finite element framework based on a multigrain periodic unit cell generated by Voronoi tessellation. This perturbation-based homogenization method is verified against Monte Carlo simulations, showing its effectiveness and limitations. Then, through applications, it is evaluated how different levels of uncertainty in grains induce uncertainty in the macroscopic elastic properties of the polycrystalline material. In particular, the influence of the unit cell is studied. Finally, by coupling the PSFEM with the Monte Carlo method, the effects on the macroscopic properties of uncertainty of both the geometry and orientation of the grains is estimated.

Keywords
perturbation stochastic finite element, homogenization, Monte Carlo method, polycrystalline material
Milestones
Received: 6 April 2010
Revised: 5 October 2010
Accepted: 12 November 2010
Published: 21 December 2011

Proposed: Adair R. Aguiar
Authors
Severine Lepage
Beckman Institute for Advanced Science and Technology
University of Illinois at Urbana–Champaign
405 N. Mathews Av.
Urbana, IL 61801
United States
Present address:
e-Xstream engineering (L) sarl
ZI Bommelscheuer
L-4940 Bascharage
Luxemburg
Fernando V. Stump
Department of Mechanical Science and Engineering
University of Illinois at Urbana–Champaign
1206 W. Green St.
Urbana, IL 61801
United States
Isaiah H. Kim
Department of Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
1206 W. Green St.
Urbana, IL, 61801
United States
Philippe H. Geubelle
Department of Aerospace Engineering
University of Illinois at Urbana-Champaign
104 S. Wright St.
Urbana, IL 61801
United States