A new spectral framework for establishing localization relationships for elastic behavior of composites and their calibration to finite-element models

doi:10.1016/j.actamat.2008.01.017

Acta Materialia

Volume 56, Issue 10, June 2008, Pages 2272-2282

https://doi.org/10.1016/j.actamat.2008.01.017 Get rights and content

Abstract

Localization relationships aim to connect the microscale response in a composite material to the macroscale loading conditions, while taking into account the local details of the microstructure at the location of interest. Such linkages are at the core of multi-scale modeling of materials since they provide efficient scale-bridging relationships. These structure–property linkages are expressed through fourth-rank localization tensors derived from higher-order homogenization theories. This paper builds upon a recently developed spectral framework called microstructure-sensitive design that was established to formulate localization relationships for the elastic response in composite materials. The method casts existing higher-order homogenization theory into a Fourier space to achieve substantial computational advantages over other multi-scale modeling approaches. More specifically, it is demonstrated that the spectral approach transforms the localization relationship into a simple algebraic series comprising polynomials of the microstructure coefficients. A remarkable feature of this new method is that the coefficients of the polynomial expression, termed influence coefficients, are completely independent of the morphological details in a specific microstructure. Consequently, they need to be established only once. It is demonstrated in this paper that an appropriately truncated localization relationship can be obtained by calibration of the influence coefficients to the results of finite-element models. These, and other, salient features of the proposed spectral framework are first theoretically established, and then demonstrated with a simple case study.

Introduction

Generalized composite theories for effective elastic response of heterogeneous materials are well established in literature, especially for linear properties such as conductivity and elastic stiffness [1], [2], [3], [4]. Inherent to these theories is the concept of a scale-bridging localization tensor that relates the local fields of interest at the microscale to the macroscale (typically averaged) fields. For example, the fourth-rank localization tensor for elastic deformation, a, relates the local elastic strain at any location of interest in the microstructure to the macroscale strain imposed on the composite as $ε (x) = a (x) 〈 ε (x) 〉$ $\begin{matrix} a (x) = (I - 〈 Γ^{r} (x, x^{'}) C^{'} (x^{'}) 〉 \\ + 〈 Γ^{r} (x, x^{'}) C^{'} (x^{'}) Γ^{r} (x^{'}, x^{″}) C^{'} (x^{″}) 〉 - \dots) \end{matrix}$ In Eqs. (1), (2), I is the fourth-rank identity tensor, C′(x) is the deviation in the local elastic stiffness at spatial location x with respect to that of a selected reference medium, Γ^r is a symmetrized derivative of the Green’s function defined using the elastic properties of the selected reference media, and 〈〉 brackets denote ensemble averages over representative volume elements (RVEs).

The evaluation of the terms in the series expression in Eq. (2) requires knowledge of higher-order spatial correlations of local states in the microstructure (also referred to as the n-point statistics of the microstructure [4], [5], [6], [7]). Local states are defined as a combination of measurable microstructural parameters (such as phase, lattice orientation, composition), and their correlations contain quantitative information regarding their spatial distribution. The series terms in Eq. (2) correspond to a hierarchy of local statistics of the microstructure. More explicitly, the first 〈〉 term in Eq. (2) captures the contribution to the tensor a(x) from a particular local state at point x′ in the material. By evaluating this term at different x′ in the material and taking an average, we capture the contribution from the local two-point statistics of the microstructure (these are not the complete set of two-point statistics because the point x is fixed and only x′ is allowed to vary). In a very similar manner, the second 〈〉 term in Eq. (2) reflects the contribution of two particular local states at points x′ and x″, respectively, on a(x). Consequently, this term captures the contributions from the local three-point statistics of the microstructure. Prior work in literature has primarily used Eq. (2) as an intermediate step in arriving at the macroscale effective elastic stiffness of the composite material [1], [5], [8]. In this work, we focus our attention on using Eq. (2) to formulate an efficient alternative formulation that can facilitate rapid computation of the localization tensor for any composite microstructure of interest.

There exist two main difficulties in the computation of the localization tensor defined in Eq. (2). The first difficulty stems from the evaluation of the ensemble averages that are in fact convolution integrals whose integrands exhibit singularities (also known as the principal value problem). The second difficulty is that the accuracy of the solutions obtained is quite sensitive to the selection of the reference medium [8]. It should also be noted that the expression of the localization tensor in Eq. (2) does not lend itself to a scheme where some of the calculations performed for one microstructure may be efficiently carried forward to the calculations for a different microstructure. In other words, any changes in the microstructure would force one to re-evaluate almost all of the terms in the series expansion.

In this paper, we present a new mathematical framework to cast Eq. (2) into a computationally efficient, potentially invertible, scale-bridging linkage that is especially suited for multi-scale design and analyses of composite microstructures. These scale-bridging relationships are expected to capture the salient aspects of the microstructure–property linkages at a given length scale, and to communicate them to a higher length scale. It is therefore important to recognize that our focus is not so much on accurately capturing all of the finest details of these linkages. We remind the reader that micromechanical finite-element modeling approaches are better suited for capturing the intricate details of the microscale response. Our approach here is to provide computationally efficient scale-bridging relations through series expansions that consider at least the local three-point statistics of the microstructure.

Recently, together with our collaborators, we have developed a new spectral framework called microstructure-sensitive design (MSD) for efficient description of invertible microstructure–property linkages in composite materials [9], [10], [11], [12]. A central element of this new framework is the transformation of the generalized composite theories into an efficient spectral (Fourier) space [8], [13]. The main advantages of this new framework have been demonstrated through several design case studies, where the goal was to identify the class of microstructures that are theoretically predicted to meet or exceed a set of designer-specified effective property or performance criteria [14], [15], [16]. Most of the prior work in MSD has focused on effective macroscale properties of the composite, using either the elementary bounding theories [14] or the second-order perturbation estimates [8], [13]. In a recent paper, we have explored a first-order spectral description of the localization tensor [17] for elastic deformation of polycrystalline materials. In this paper, we develop a much more robust approach for higher-order spectral description of the localization tensor for elastic deformations that is applicable to all composite materials. In parallel work, we have used the same spectral method to describe localized properties of polycrystalline materials [18].

Section snippets

Spectral framework for the localization tensor

In prior work [13], we have introduced the concept of a microstructure function, M(x,h), that defines the probability density associated with finding the local state h in the neighborhood of a spatial location of interest, x, as $M (x, h) d h = {\frac{V_{h}}{V}|}_{x}$ In this definition, the local state h is an element of the continuous local state space H with an associated invariant measure dh. V is the volume of the neighborhood used in the definition of the microstructure function and is typically dictated either by

Case study: composite material with two isotropic constituents

In this first foray into the higher-order spectral representation of the localization tensors, we report a relatively simple case study with the goal of validating the main concepts underlying the spectral framework described in Section 2. More specifically, we have investigated a class of composite materials with two constituent isotropic phases (labeled α and β). We have considered two different cases of contrast in the local elastic properties (see Table 1). For each composite material

Conclusions

This study has demonstrated that the elastic localization tensors can be cast as a series of simple algebraic polynomials of microstructure parameters in an appropriately selected Fourier space. The coefficients in these polynomial expressions, called influence coefficients, are found to be completely independent of the morphological details of the microstructure. The main advantage of this formulation is that the influence coefficients need only be computed once for a given material system.

Acknowledgement

The authors acknowledge financial support for this work from the DARPA-ONR Dynamic 3D Digital Structure project, Award No. N000140510504 (Program Manager: Dr Julie Christodoulou).

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A new spectral framework for establishing localization relationships for elastic behavior of composites and their calibration to finite-element models

Abstract

Introduction

Section snippets

Spectral framework for the localization tensor

Case study: composite material with two isotropic constituents

Conclusions

Acknowledgement

Prog Mater Sci

J Mech Phys Solids

J Mech Phys Solids

Acta Mater

J Mech Phys Solids

Int J Plast

J Mech Phys Solids

Acta Mater

Int J Plast

J Mech Phys Solids

Comp. Mater. Sci.

The theory of composites