Hashin-Shtrikman Variational Principles Research Articles

A variance reduction strategy for numerical random homogenization based on the equivalent inclusion method

Using the equivalent inclusion method (a method strongly related to the Hashin–Shtrikman variational principle) as a surrogate model, we propose a variance reduction strategy for the numerical homogenization of random composites made of “inclusions” (or rather inhomogeneities) embedded in a homogeneous matrix. The efficiency of this strategy is demonstrated within the framework of two-dimensional, linear conductivity. Significant computational gains vs full-field simulations are obtained even for high contrast values. We also show that our strategy allows to investigate the influence of parameters of the microstructure on the macroscopic response. A specific attention is given to the computational cost of the surrogate model. In particular, an inexpensive approximation of the so-called influence tensors (that are used to compute the surrogate model) is proposed.

TFA and HS based homogenization techniques for nonlinear composites

The paper presents two reduced order homogenization techniques for studying the response of nonlinear composite materials. The first approach is based on Transformation Field Analysis, which considers the presence of eigenstrains to account for inelastic strains, while the second approach is derived from Hashin–Shtrikman variational principle, which introduces eigenstresses, namely polarization stresses, on a homogeneous elastic reference material to account for the heterogeneity and the inelastic response. In particular, the case of elasto-plastic periodic composites is investigated. A Unit Cell is identified and divided in subsets. In each subset, the eigenstrains or eigenstresses are assumed uniform. A very effective technique is derived for the Hashin–Shtrikman approach updating the elastic reference material properties during the inelastic strain evolution. Numerical procedures are implemented to derive the nonlinear response of composites with plastic constituents. Several numerical applications are carried out to assess the effectiveness of the two presented reduced order models. A deep investigation on the differences and similarities of the two approaches is presented, proving their equivalence under particular circumstances. Simple and complex loading histories are considered, comparing the results with a finite element solution, considered as the reference solution.

Hashin–Shtrikman bounds of periodic linear elastic media with cubic symmetry

Based on the Hashin–Shtrikman variational principle, novel bounds on the effective shear moduli of a two-phase periodic composite are derived. The composite constituents are assumed to be isotropic, while the microstructure is assumed to exhibit cubic symmetry. A solution of the subsidiary boundary value problem is expressed as a double contraction of a fourth-order cubic tensor with the applied macroscopic strain. The bounds for cubic shear moduli are new, while the bounds for the bulk modulus are equal to the classical ones. The new bounds are verified for composites with the cubic, frame, octet and cubic + octet structures. It is shown that they are nearly attained for the cubic, octet and cubic + octet structures.

Negligible microstructure in a class of porous nonlinear materials via variational bounds

This work is devoted to the estimation of the effective energy density of porous nonlinear materials by means of variational bounds. Because of the unavailability of an improved upper bound, attention is turned to the improved bound obtained by considering constant nonzero polarization fields in the formal lower bound, which follows from the generalization to nonlinear behavior of the Hashin–Shtrikman variational principles. A particular class of nonlinearity is considered which is relevant in different physical situations. Also, 11 different microstructures are considered. Several computational experiments performed show that the elementary upper bound and the improved lower bound are indistinguishable, suggesting that nonlinearity dominates over the microstructural effects. In other words, at least for the nonlinearity considered here, the influence of microstructure on the effective behavior is negligible. So, in this case, there is no need to search for an improved upper bound, as the elementary one provides a simple but accurate estimate of the effective energy density. Copyright © 2017 John Wiley & Sons, Ltd.

Estimations and bounds of the effective conductivity of composites with anisotropic inclusions and general imperfect interfaces

The present work aims to determine the effective thermal conductivity of composites made of an isotropic matrix phase in which circular or spherical inhomogeneities are embedded. The inhomogeneity phases can be anisotropic and the interface between the inhomogeneity and matrix phase can be modeled by a general thermal imperfect interface model across which both the temperature and normal heat flux across can suffer a discontinuity. To achieve this objective, we derive first a unified and exact solution for the thermal fields of the inhomogeneity problem consisting of a spherical or circular anisotropic inhomogeneity inserted via a general thermal imperfect interface into an infinity isotropic matrix medium subjected to a remote uniform loading at its external surface. Unlike the relevant results in elasticity, the intensity and heat flux fields inside circular and spherical inhomogeneities are shown to remain uniform even in the presence of the general thermal imperfect interface and anisotropy of inhomogeneity. Next, with the help of the foregoing solution results for the heterogeneity problem, the differential scheme is extended to predicting the effective thermal conductivity of composites with taking into account the imperfect interfaces between the constituent phases. Finally, the minimum potential and complementary energy principles and the morphologically representative pattern approach based on the Hashin–Shtrikman variational principles and the variational polarization principles are applied to such inhomogeneous materials and to bracketing their effective thermal properties. By constructing trial appropriate intensity and heat flux fields, the first- and second-order upper and lower bounds are obtained for the effective thermal conductivity of multiphase materials consisting of spherical or circular inhomogeneities embedded in a matrix. The estimations obtained by the differential scheme for the effective conductivity are shown to comply with the first- and second-order upper and lower bounds. Numerical results are provided to illustrate the dependence of the effective conductivity on the sizes of inhomogeneities and to compare the estimations with the relevant upper and lower bounds.

Analytical and semi-analytical modeling of effective moduli bounds: Application to transversely isotropic piezoelectric materials

In this article, analytical and semi-analytical models of upper and lower bounds for the effective moduli of transversely isotropic piezoelectric heterogeneous materials based on the generalized Hashin–Shtrikman variational principle are presented. Compact matrix formulations are used to derive closed-form bound expressions for coupled and uncoupled effective moduli. Analytical models are given for some uncoupled coefficients and simplified formulations for the others. For more narrow bounds, downstream and upstream bounds are developed based on an incremental procedure. Numerical predictions are performed based on the developed methodological approaches, and the obtained results showed the applicability and effectiveness of the proposed models for transversely isotropic elastic and piezoelectric composite materials with ellipsoidal reinforcements of different types and shapes.

Combining FFT methods and standard variational principles to compute bounds and estimates for the properties of elastic composites

In this paper, we develop a computational approach based on variational principles combined with FFT technique to determine the effective elastic properties of composite materials. The unit cell problem of the composite is recast in a weak and integral form by making use of classic variational approaches, based on the strain and the stress potential, and the Hashin–Shtrikman variational principles. The problem is discretized with Fourier series and the stationary point is computed numerically by means of an iterative scheme. These algorithms use a representation of the local elastic tensor on the double grid (twice the size of the discretization) which is introduced by the integral formulation; it advantageously accounts for the exact geometry of the microstructure and allows the derivation of rigorous bounds of the effective elasticity coefficients. In the second part of the paper, we establish a hierarchy between the different FFT solutions: the strain, the stress based solutions coming from the classic variational formulations and the polarization based solution derived from the Hashin–Shtrikman principle, which depends on the choice of the elastic moduli of the reference material. It is proved that the strain and the stress based solutions deliver optimal bounds since they are always better than those obtained from the Hashin–Shtrikman variational principle when the same discretization is used. Alternatively, when the elastic moduli of the reference material is negative, the polarization based solution provides an estimate of the effective properties which is comprised between the strain based upper bound and the stress based lower bound.

Microstructure-statistics-property relations of anisotropic polydisperse particulate composites using tomography

In this paper, a systematic method is presented for developing microstructure-statistics-property relations of anisotropic polydisperse particulate composites using microcomputer tomography (micro-CT). Micro-CT is used to obtain a detailed three-dimensional representation of polydisperse microstructures, and an image processing pipeline is developed for identifying particles. In this work, particles are modeled as idealized shapes in order to guide the image processing steps and to provide a description of the discrete micro-CT data set in continuous Euclidean space. n-point probability functions used to describe the morphology of mixtures are calculated directly from real microstructures. The statistical descriptors are employed in the Hashin-Shtrikman variational principle to compute overall anisotropic bounds and self-consistent estimates of the thermal-conductivity tensor. We make no assumptions of statistical isotropy nor ellipsoidal symmetry, and the statistical description is obtained directly from micro-CT data. Various mixtures consisting of polydisperse ellipsoidal and spherical particles are prepared and studied to show how the morphology impacts the overall anisotropic thermal-conductivity tensor.

On energy formulations of electrostatics for continuum media

In this paper we present a unified energy formulation of electrostatics for continuum media including conductors, dielectrics, ferroelectrics, etc. The effects of boundary devices, external charges and polarizations are taken into account in formulating the free energy. We further explore the relations between various energy formulations including Landau–Ginzburg–Devonshire's formulation, Toupin's formulation, Ericksen's formulation, formulations with respect to a change of comparison or background medium, a formulation without introducing the quantity of polarization, and the Hashin–Shtrikman variational principle. Finally, we apply the formulation to calculate the effective free energy of a polarizable body in the proximity of a conductor, and to estimate the effective properties of nonlinear dielectric composites. We expect that this formulation and clarification will provide a solid ground for addressing electro-elastics of deformable bodies.

Variational bounds of the effective moduli of piezoelectric composites

The classical Hashin-Shtrikman variational principle was re-generalized to the heterogeneous piezoelectric materials. The auxiliary problem is very much simplified by selecting the reference medium as a linearly isotropic elastic medium. The electromechanical fields in the inhomogeneous piezoelectrics are simulated by introducing into the homogeneous reference medium certain eigenstresses and eigen electric fields. A closed-form solution can be obtained for the disturbance fields, which is convenient for the manipulation of the energy functional. As an application, a two-phase piezoelectric composite with nonpiezoelectric matrix is considered. Expressions of upper and lower bounds for the overall electromechanical moduli of the composite can be developed. These bounds are shown better than the Voigt-Reuss type ones.

Bounds for the effective properties of heterogeneous plates

Abstract This paper presents new bounds for heterogeneous plates which are similar to the well-known Hashin–Shtrikman bounds, but take into account plate boundary conditions. The Hashin–Shtrikman variational principle is used with a self-adjoint Green-operator with traction-free boundary conditions proposed by the authors. This variational formulation enables to derive lower and upper bounds for the effective in-plane and out-of-plane elastic properties of the plate. Two applications of the general theory are considered: first, in-plane invariant polarization fields are used to recover the “first-order” bounds proposed by Kolpakov [Kolpakov, A.G., 1999. Variational principles for stiffnesses of a non-homogeneous plate. J. Meth. Phys. Solids 47, 2075–2092] for general heterogeneous plates; next, “second-order bounds” for n -phase plates whose constituents are statistically homogeneous in the in-plane directions are obtained. The results related to a two-phase material made of elastic isotropic materials are shown. The “second-order” bounds for the plate elastic properties are compared with the plate properties of homogeneous plates made of materials having an elasticity tensor computed from “second-order” Hashin–Shtrikman bounds in an infinite domain.

Stochastic Modeling of Chaotic Masonry via Mesostructural Characterization

The purpose of this study is to explore three numerical approaches to the elastic homogenization of disordered masonry structures with moderate meso/macro-lengthscale ratio. The methods investigated include a representative of perturbation methods, the Karhunen-Loeve expansion technique coupled with Monte-Carlo simulations and a solver based on the Hashin-Shtrikman variational principles. In all cases, parameters of the underlying random field of material properties are directly derived from image analysis of a real-world structure. Added value as well as limitations of individual schemes are illustrated by a case study of an irregular masonry panel.

Microstructure-based modeling of elastic functionally graded materials: One-dimensional case

Functionally graded materials (FGMs) are two-phase composites with continuously changing microstructure adapted to performance requirements. Traditionally, the overall behavior of FGMs has been determined using local averaging techniques or a given smooth variation of material properties. Although these models are computationally efficient, their validity and accuracy remain questionable, since a link with the underlying microstructure (including its randomness) is not clear. In this paper, we propose a modeling strategy for the linear elastic analysis of FGMs systematically based on a realistic microstructural model. The overall response of FGMs is addressed in the framework of stochastic Hashin-Shtrikman variational principles. To allow for the analysis of finite bodies, recently introduced discretization schemes based on the Finite Element Method and the Boundary Element Method are employed to obtain statistics of local fields. Representative numerical examples are presented to compare the performance and accuracy of both schemes. To gain insight into similarities and differences between these methods and to minimize technicalities, the analysis is performed in the one-dimensional setting.

Upper and lower stiffness bounds for porous anisotropic rocks

SUMMARY We derive double inequalities providing the bounds for components of the effective stiffness tensor of a two-phase, porous-cracked medium with aligned ellipsoidal inclusions. The bounds are derived on the basis of the Hashin–Shtrikman variational principle, and the conditions for positive semi-definiteness of quadratic forms. Inequalities are presented for isotropic, cubic, hexagonal and orthorhombic overall symmetries. The results obtained for orthorhombic symmetry are valid for the general determination of transport properties (effective permeability, thermal and electrical conductivity). We conclude that inequalities for diagonal components of the effective tensor have the form of bounds, whereas in general these bounds do not exist for off-diagonal components. One important implication of this is that Voigt–Reuss averages do not provide the upper and lower bounds for off-diagonal components of the effective tensor, as is sometimes assumed. We also present numerical results for stiffness bounds obtained by modelling various shales with isotropic, transverse isotropic and orthorhombic overall symmetries.

The Composite Eshelby Tensors and their applications to homogenization

In recent studies, the exact solutions of the Eshelby tensors for a spherical inclusion in a finite, spherical domain have been obtained for both the Dirichlet- and Neumann boundary value problems, and they have been further applied to the homogenization of composite materials [15], [16]. The present work is an extension to a more general boundary condition, which allows for the continuity of both the displacement and traction field across the interface between RVE (representative volume element) and surrounding composite. A new class of Eshelby tensors is obtained, which depend explicitly on the material properties of the composite, and are therefore termed “the Composite Eshelby Tensors”. These include the Dirichlet- and the Neumann-Eshelby tensors as special cases. We apply the new Eshelby tensors to the homogenization of composite materials, and it is shown that several classical homogenization methods can be unified under a novel method termed the “Dual Eigenstrain Method”. We further propose a modified Hashin-Shtrikman variational principle, and show that the corresponding modified Hashin-Shtrikman bounds, like the Composite Eshelby Tensors, depend explicitly on the composite properties.

Estimation of very narrow bounds to the behavior of nonlinear incompressible elastic composites

Variational bounds for the effective behavior of nonlinear composites are improved by incorporating more-detailed morphological information. Such bounds, which are obtained from the generalized Hashin–Shtrikman variational principles, make use of a reference material with the same microstructure as the nonlinear composite. The geometrical information is contained in the effective properties of the reference material, which are explicitly present in the analytical formulae of the nonlinear bounds. In this paper, the variational approach is combined with estimates for the effective properties of the reference composite via the asymptotic homogenization method (AHM), and applied to a hexagonally periodic fiber-reinforced incompressible nonlinear elastic composite, significantly improving some recent results.

Hashin–Shtrikman Based FE Analysis of the Elastic Behaviour of Finite Random Composite Bodies

This paper demonstrates the effect of non-local interactions in an edge-cracked body made of composite material, in the case that the scale of the body is not large in comparison with the scale of the microstructure, as can occur, for example, in laboratory testing of concrete specimens. Not only are there significant boundary layers in the vicinity of all of the specimen boundaries, but the small size of the body permits their interaction, to alter the value of the mean field globally relative to the value that would be predicted by the use of “ordinary” homogenization, under conditions of “displacement control”. Such an effect would be present even for an uncracked body but the presence of the crack induces strong gradients in the mean field which significantly enhance this effect. The mean strain component \(\langle {e_{22}}\rangle\) tends to become concentrated around the line ahead of the crack in comparison with the prediction of homogenization; there is, in addition, an enhancement of the stress and strain concentrations in the harder phase. The method of analysis is a finite-element implementation of a variational formulation related to the Hashin–Shtrikman variational principle, developed in the context of non-local effective response by the present authors. The present calculations are considered to be more accurate than some previously reported (Luciano and Willis, Journal of the Mechanics and Physics of Solids 53, 1505–1522, 2005) and the conclusions differ correspondingly.

Bounds and self-consistent estimates for elastic constants of random polycrystals with hexagonal, trigonal, and tetragonal symmetries

Peselnick, Meister, and Watt have developed rigorous methods for bounding elastic constants of random polycrystals based on the Hashin–Shtrikman variational principles. In particular, a fairly complex set of equations that amounts to an algorithm has been presented previously for finding the bounds on effective elastic moduli for polycrystals having hexagonal, trigonal, and tetragonal symmetries. A more analytical approach developed here, although based on the same ideas, results in a new set of compact formulas for all the cases considered. Once these formulas have been established, it is then straightforward to perform what could be considered an analytic continuation of the formulas (into the region of parameter space between the bounds) that can subsequently be used to provide self-consistent estimates for the elastic constants in all cases. This approach is very similar in spirit but differs in its details from earlier work of Willis, showing how Hashin–Shtrikman bounds and certain classes of self-consistent estimates may be related. These self-consistent estimates always lie within the bounds for physical choices of the crystal elastic constants and for all the choices of crystal symmetry considered. For cubic symmetry, the present method reproduces the self-consistent estimates obtained earlier by various authors, but the formulas for both bounds and estimates are generated in a more symmetric form. Numerical values of the estimates obtained this way are also very comparable to those found by the Gubernatis and Krumhansl coherent potential approximation (or CPA), but do not require computations of scattering coefficients.

Variational principles for nonlinear piezoelectric materials

In the present paper, we consider the behavior of nonlinear piezoelectric materials by generalization for this case of the Hashin-Shtrikman variational principles. The new general formulation used here differs from others, because, it gives the possibility to evaluate the upper and lower Hashin-Shtrikman bounds for specific physical nonlinearities of piezoelectric materials. Geometrical nonlinearities are not considered.

Extended Hashin-Shtrikman Variational Principles

Internal parameters, eigenstrains, or eigenstresses, arise in functionally graded materials, which are typically present in particulate, layered, or rock bodies. These parameters may be realized in different ways, e.g., by prestressing, temperature changes, effects of wetting, swelling, they may also represent inelastic strains, etc. In order to clarify the use of eigenparameters (eigenstrains or eigenstresses) in physical description, the classical formulation of elasticity is presented, and the two most important Lagrange's and Castigliano's variational principles are formulated in the sequel. Then the classical Hashin-Shtrikman principles are recalled and the involvement of eigenparameters is studied in more detail.