Probabilistic Multi-Scale Modeling of Materials

Abstract

Process conditions of MEMS and nano-scale materials often introduce texture in the material that manifests itself as anisotropy in the mechanical properties. However, due to random orientations and shapes of the crystals, the materials also exhibit inhomogeneity at the microscopic level. These random variations in the material characteristics can have significant influence on the mechanical response of MEMS devices or nano-materials. An accurate assessment of the effects of such uncertainties is paramount to the successful implementation of such materials in many practical applications. Materials can be viewed as either an aggregate of discrete constituents or as a homogeneous continuum. While the former viewpoint is used primarily in the materials science domain, the latter is often used for practical applications. To goal of this work is to combine these two viewpoints using a probabilistic multi-scale model. Using an appropriate micro-scale model, the multi-scale modeling framework can be applied to represent any physical property of the material, such as structural stiffness, strength and thermal conductivity. In this paper we focus on a probabilistic model of the elastic stiffness of heterogeneous materials. The basic conjecture of the multi-scale modeling approach is that the variability of the material properties, which is observed at the macro-scale (i.e. observed directly in the application), is a result of variations in both the micro-geometry and in the micro-constitutive properties. The presented stochastic homogenization theory blends the discrete micro-material models with a continuum constitutive model. The probabilistic properties are derived using the locally averaged random field theory and they follow directly from the mechanical interactions of the randomly shaped or randomly oriented micro- constituents in the material. The resulting continuous random field model of the material properties can readily be used in a stochastic finite element analysis. Traditional, deterministic homogenization procedures can accurately describe the average macroscopic behavior of materials. However, a probabilistic analysis based on a straightforward randomization of a deterministic model (i.e. using the same constitutive equations but with random parameters) often under-predicts the true uncertainty of a structural response. The stochastic properties of the micromechanically-consistent random field are quite different from those obtained using a straightforward randomization of deterministic material models. It is shown that a continuum random field model based on one (E only) or two (E and v) random fields cannot be consistent with a lattice-type micro-mechanical model. A structural analysis where only E is considered a random field will underestimate the variability of the response. The proposed multi-scale model combines the advantages of the discrete and homogeneous continuum approaches. This approach preserves the computational efficiency of the continuum model while maintaining the micro-mechanical detail available in a discrete model.