Micromechanics as a basis of stochastic finite elements and differences: An overview M Ostoja-Starzewski Department of Materials Science and Mechanics Michigan State University, East Lansing, MI 48824-1226 A generalization of conventional deterministic finite element and difference methods to deal with spatial material fluctuations hinges on the problem of determination of stochastic constitutive laws. This problem is analyzed here through a paradigm of micromechanics of elastic polycrystals and matrix-inclusion composites. Passage to a sought-for random meso-continuum is based on a scale dependent window playing the role of a Representative Volume Element (RVE). It turns out that the microstructure cannot be uniquely approximated by a random field of stiffness with continuous realizations, but, rather, two random continuum fields may be introduced to bound the material response from above and from below. Since the RVE corresponds to a single finite element, or finite difference cell, not infinitely larger than the crystal size, these two random fields are to be used to bound the solution of a given boundary value problem at a given scale of resolution. The window- based random continuum formulation is also employed in analysis of rigid perfectly-plastic mate­ rials, whereby the classical method of slip-lines is generalized to a stochastic finite difference scheme. The present paper is complemented by a comparison of this methodology to other existing stochastic solution methods. 1. INTRODUCTION a locally isotropic form The necessity to account for random effects in determining a.. = X(x, co) 5..e +2u(x, co)e.. n ^ the response of a mechanical system is due, in general, to »J iJ kk 'J {Li> three different sources: random external forcing, random is adopted by simply postulating one or both elastic constants, boundary conditions, and random material parameters. In the such as the Young's modulus, to be a random field. last fifteen years the powerful finite element method has While the effort in SFE has been on the development of effi­ undergone various new developments to incorporate these cient numerical methods for solution of boundary value prob­ random effects, and is now termed Stochastic Finite Elements lems, the above model - equations (1.1-3) - lacks a connection (SFE), see e.g. (Contreras, 1980; Benaroya and Rehak, 1988). to the material microstructure. It is the determination of that In this paper we focus only on the type of SFE problems missing micromechanics link, which forms the main objective of which deal with randomness stemming from fluctuations in this paper. Additionally, our methodolgy may also be applied to material properties. Most of the past research in that area other than elastic microstructures, and used in solution of ran­ concerned linear elastic structural responses and relied on a dom media problems by finite differences. A very closely related straightforward generalization of Hooke's law, that is issue of specification of continuum random fields approximating elastic microstructures is studied by Ostoja-Starzewski (1993b). p = C(x, cu)e (11) The paper's outline is as follows. In Section 2 we describe the In equation (1.1) x stands for a location within the body passage from the level of a linear elastic microstructure to that of domain, co is an index from the sample space Q, and C (x, co) two random meso-continuum models Cs (x, co), where 5 indi­ is a continuous realization of a random tensor field of stiff­ cates the scale dependence. Next follows a stochastic variational nesses. Part of the assumption (1.1) is the in vertibility of such formulation of finite elements - in both dispalcement and force a constitutive law, that is approaches - which illustrates the role of these meso-continuum models in bounding the actual response. The micromechanics e = S (x, co)g S (x, co) = C*""1 (x, co) (1.2) approach is employed in Section 3, which focuses on rigid—per­ fectly plastic materials with random fluctuations in the yield whereby e and g in (1.1) and (1.2)| are uniform fields functions. It follows here that the method of slip-lines - well applied to a hypothetical and unspecified Representative Vol­ known from the deterministic homogenous media problems - is ume Element (RVE) of a random medium. In fact, typically, now to be generalized to stochastic finite differences. Section 4 part of "MECHANICS PAN-AMERICA 1993" edited by MRM Crespo da Silva and CEN Mazzilli ASME Book No AMR134 Appl Mech Rev vol 46, no 11, part 2, November 1993 S136 © 1993 American Society of Mechanical Engineers Downloaded 22 Oct 2009 to 130.126.179.16. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Appl Mech Rev vol 46, no 11 part 2 Nov 1993 Ostoja-Starzewski: Micromechanics as basis of stochastic FEs S137 is devoted to comparisons with other existing approaches. process to ensure that there is no overlap of inclusions shaped Thus, we briefly review the classical SFE methods for elastic as round disks. We assume the disks to be occupied by a structures and discuss their applicability in micromechanics- homogeneous isotropic continuum of one kind, while the based analyses. In the area of plasticity, we discuss the rela­ matrix by a continuum of another. tion of our formulation of Section 3 to a recent study of Nor- In case of both models we assume all the phases to satisfy dgren(1992). the so-called ellipticity conditions: 3a, (3 > 0 such that for any £ the following inequalities hold for all the phases 2. ELASTIC MEDIA PROBLEMS ocee<eCe<Pee (2.1) Thus, we have two realistic ergodic media models without 2.1 Random medium model holes and rigid inclusions described by random fields C = Fundamental role in our formulation is played by the concept (C(x, co); xe B;coe Q.} with piecewise-constant realiza­ of a random medium (or random microstructure), which, as is tions. This piecewise-constant nature of stiffness fields is an commonly done in mechanics of random media (Willis, obstacle to employing the governing equations of continuum 1981), is taken as a family B = {B(co); co e O } of determinis­ elasticity, which require that the stiffness fields be differentia- tic media B(co), where co indicates one specimen (realization), ble. Thus, there is a need for another continuum model - one and Q. is an underlying sample (probability) space. Formally, that possibly loses some information due to a "smearing-out" Q. is equipped with a a-algebra F and a probability distribu­ procedure, but is sufficiently differentiable and grasps the tion P. In an experimental setting P may be specified by a set meso-level behavior. of stereological measurements, while in a theoretical setting P is usually specified by a chosen model of a microstructure. All 2.2 Two scale-dependent random continuum fields specimens B(co) occupy the same domain in X|, x -plane; we 2 First, with the help of Fig. 1, we introduce a square-shaped employ a two-dimensional setting (2-D) for the clarity of pre­ window of scale sentation. In the following we consider two types of the random medium B. In the first one, we take every specimen B(co) to § = t (2.2) be modeled by a realization of a Voronoi tessellation (Fig. d 1 .a), while in the second by a realization of a matrix-inclusion Equation (2.2) defines a nondimensional parameter 5, typi­ composite (Fig. 1 b). Fundamental in both cases is a planar cally greater than 1, specifying the scale L of observation space-homogeneous Poisson process of some given density. (and/or measurement) relative to a typical microscale d (i.e. In case of a Voronoi tessellation each cell, centered at a Pois­ grain size) of the material structure. 8 = 1 is the smallest scale son point x , is assumed to be occupied by a homogeneous we consider: scale of a crystal or inclusion. In view of the fact continuum governed by a stiffness tensor C(x, co) following a that the Voronoi tessellation is a random medium, the window space-homogeneous probability distribution P(C). In case of bounds a random microstructure Sg = {Bg(co); CO e D.}, where the matrix-inclusion composite, we use an inhibition Poisson Bg(co) is a single realization from a given specimen B(co). • • • FIG 1 .a) A Voronoi tessellation with an average cell size d; b) a matrix-inclusion composite with inclusions of average diameter d; in both cases a window of size L is indicated. Downloaded 22 Oct 2009 to 130.126.179.16. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm S138 MECHANICS PAN-AMERICA 1993 Appl Mech Rev 1993 Supplement A continuum-type constitutive law is obtained by postulat­ this defines a deterministic continuum B^et for a single speci­ ing the existence of an effective homogeneous continuum men B(co) Bg0n (co) of the same volume Vg (i.e. area in 2-D), whose det potential energy U, or complementary energy U*, under given C (co) = C"(co) = Cl(co) (2.10) uniform boundary conditions equals that of a microstructure whereby the window of infinite extent plays the role of an Bg(co) under the same boundary conditions. These are of two RVE of deterministic elasticity theory; in other words, it is at basic types: 8 —»<*> that the invertibility of the constitutive law is i) displacement-controlled (essential) obtained. 4. Ergodicity of the microstructure implies that u (x) = e?.x. dB s (2.3) det -,ef£ f ,0 . C (co) = C Vcoe Q (2.11) where § is a given constant tensor and dB? is the bound­ ary of J5g, where C-ef t •i s the effective response tensor (independent of CO) ii) stress-controlled (natural) of a homogeneous medium.