Bounds on the overall properties of composites with ellipsoidal inclusions

A new model is put forward to bound the effective elastic moduli of composites with ellipsoidal inclusions. In the present paper, transition layer for each ellipsoidal inclusion is introduced to make the trial displacement field for the upper bound and the trial stress field for the lower bound satisfy the continuous interface conditions which are absolutely necessary for the application of variational principles. According to the principles of minimum potential energy and minimum complementary energy, the upper and lower bounds on the effective elastic moduli of composites with ellipsoidal inclusions are rigorously derived. The effects of the distribution and geometric parameters of ellipsoidal inclusions on the bounds of the effective elastic moduli are analyzed in details. The present upper and lower bounds are still finite when the bulk and shear moduli of ellipsoidal inclusions tend to infinity and zero, respectively. It should be mentioned that the present method is simple and needs not calculate the complex integrals of multi-point correlation functions. Meanwhile, the present paper provides an entirely different way to bound the effective elastic moduli of composites with ellipsoidal inclusions, which can be developed to obtain a series of bounds by taking different trial displacement and stress fields.     

Dynamic stability analysis of an elastic composite material having a negative-stiffness phase

The rigorous classical bounds of elastic composite materials theory provide limits on the achievable composite stiffnesses in terms of the properties and arrangements of the composite's constituents. These bounds result from the assumption, presumably made for stability reasons, that each constituent material must have positive-definite elastic moduli. If this assumption is relaxed, recently published elasticity analyses and experimental measurements show these bounds can be greatly exceeded, resulting in new materials of enormous potential.The key question is whether a composite material having a non-positive-definite constituent can be stable overall in the practically useful situation of applied traction boundary conditions. Drugan [2007. Elastic composite materials having a negative-stiffness phase can be stable. Phys. Rev. Lett. 98 (5), article no. 055502] first proved the answer is yes, by applying the energy criterion of elastic stability to the basic two- and three-dimensional composites consisting of a cylinder or sphere having non-positive-definite (but strongly elliptic) moduli with a thin positive-definite coating and proving overall stability provided the coating is sufficiently stiff.Here, we perform a complete and direct dynamic stability analysis of the plane strain fundamental elastic composite consisting of a circular cylinder of non-positive-definite material firmly bonded to a positive-definite concentric coating, for the full range of coating thicknesses (i.e., volume fractions). We determine quantitatively the full permissible range of inclusion and coating moduli, as a function of coating thickness, for which the overall composite is stable under dead traction boundary conditions. Among the results, we show that in the thin-coating case, the present dynamic stability analysis leads to precisely the same analytical stability requirements as those derived via the energy criterion by Drugan [2007. Elastic composite materials having a negative-stiffness phase can be stable. Phys. Rev. Lett. 98 (5), article no. 055502], and we derive new analytical stability requirements that are valid for a wider range of coating thickness. At the other extreme, we show that in the case of very thick coatings (corresponding to the dilute case of a matrix-inclusion composite), even an inclusion with merely strongly elliptic moduli can be stabilized by a positive-definite matrix satisfying weak requirements, for which we derive analytical expressions. Overall, our results show that surprisingly weak restrictions on the moduli and thickness of the positive-definite coating are sufficient to stabilize a non-positive-definite inclusion, even one whose moduli are merely strongly elliptic. These results legitimize expanding the search for novel materials with extreme properties to those incorporating a non-positive-definite constituent, and they provide quantitative restrictions on the constituent materials’ moduli and volume fractions, for the geometry examined here, that ensure overall stability of such composite materials.     

New variational principles in plasticity and their application to composite materials

I , a variational method for bounding the effective properties of nonlinear composites with isotropic phases, proposed recently by ponte castañeda (J. Mech. Phys. Solids 39, 45, 1991), is given full variational principle status. Two dual versions of the new variational principle are presented and their equivalence to each other, and to the classical variational principles, is demonstrated. The variational principles are used to determine bounds and estimates for the effective energy functions of nonlinear composites with prescribed volume fractions in the context of the deformation theory of plasticity. The classical bounds of Voigt and Reuss for completely anisotropic composites are recovered from the new variational principles and are given alternative, simpler forms. Also, use of a novel identity allows the determination of simpler forms for nonlinear Hashin-Shtrikman bounds, and estimates, for isotropic, particle-reinforced composites, as well as for transversely isotropic, fiber-reinforced composites. Additionally, third-order bounds of the Beran type are determined for the first time for nonlinear composites. The question of the optimality of these bounds is discussed briefly.     

The asymptotic homogenization elasticity tensor properties for composites with material discontinuities

The classical asymptotic homogenization approach for linear elastic composites with discontinuous material properties is considered as a starting point. The sharp length scale separation between the fine periodic structure and the whole material formally leads to anisotropic elastic-type balance equations on the coarse scale, where the arising fourth rank operator is to be computed solving single periodic cell problems on the fine scale. After revisiting the derivation of the problem, which here explicitly points out how the discontinuity in the individual constituents’ elastic coefficients translates into stress jump interface conditions for the cell problems, we prove that the gradient of the cell problem solution is minor symmetric and that its cell average is zero. This property holds for perfect interfaces only (i.e., when the elastic displacement is continuous across the composite’s interface) and can be used to assess the accuracy of the computed numerical solutions. These facts are further exploited, together with the individual constituents’ elastic coefficients and the specific form of the cell problems, to prove a theorem that characterizes the fourth rank operator appearing in the coarse-scale elastic-type balance equations as a composite material effective elasticity tensor. We both recover known facts, such as minor and major symmetries and positive definiteness, and establish new facts concerning the Voigt and Reuss bounds. The latter are shown for the first time without assuming any equivalence between coarse and fine-scale energies (Hill’s condition), which, in contrast to the case of representative volume elements, does not identically hold in the context of asymptotic homogenization. We conclude with instructive three-dimensional numerical simulations of a soft elastic matrix with an embedded cubic stiffer inclusion to show the profile of the physically relevant elastic moduli (Young’s and shear moduli) and Poisson’s ratio at increasing (up to 100 %) inclusion’s volume fraction, thus providing a proxy for the design of artificial elastic composites.     

A variational approach to the theory of the elastic behaviour of polycrystals

Variational principles for anisotropic and nonhomogeneous elasticity, established by the authors in a previous paper, have been applied to the derivation of lower and upper bounds for the elastic moduli of polycrystals in terms of the moduli of the constituting crystals. The results hold for arbitrary crystal shapes. Explicit results tor cubic polycrystals showed that the present bounds are a considerable improvement of the well-known Voigt and Reuss bounds. Good agreement with experimental results has been obtained.     

A micromechanics-based nonlocal constitutive equation incorporating three-point statistics for random linear elastic composite materials

A variational formulation employing the minimum potential and complementary energy principles is used to derive a micromechanics-based nonlocal constitutive equation for random linear elastic composite materials, relating ensemble averages of stress and strain in the most general situation when mean fields vary spatially. All information contained in the energy principles is retained; we employ stress polarization trial fields utilizing one-point statistics so that the resulting nonlocal constitutive equation incorporates up through three-point statistics. The variational structure is developed first for arbitrary heterogeneous linear elastic materials, then for randomly inhomogeneous materials, then for general n-phase composite materials, and finally for two-phase composite materials, in which case explicit variational upper and lower bounds on the nonlocal effective modulus tensor operator are derived. For statistically uniform infinite-body composites, these bounds are determined even more explicitly in Fourier transform space. We evaluate these in detail in an example case: longitudinal shear of an aligned fiber or void composite. We determine the full permissible ranges of the terms involving two- and three-point statistics in these bounds, and thereby exhibit explicit results that encompass arbitrary isotropic in-plane phase distributions; we also develop a nonlocal “Milton parameter”, the variation of whose eigenvalues throughout the interval [0, 1] describes the full permissible range of the three-point term. Example plots of the new bounds show them to provide substantial improvement over the (two-point) Hashin–Shtrikman bounds on the nonlocal operator tensor, for all permissible values of the two- and three-point parameters. We next discuss further applications of the general nonlocal operator bounds: to any three-dimensional scalar transport problem e.g. conductivity, for which explicit results are given encompassing the full permissible ranges of the two- and three-point statistics terms for arbitrary three-dimensional isotropic phase distributions; and to general three-dimensional composites, where explicit results require future research. Finally, we show how the work just summarized, treating elastostatics, can be generalized to elastodynamics, first in general, then explicitly for the longitudinal shear example.     

Universal bounds on the electrical and elastic response of two-phase bodies and their application to bounding the volume fraction from boundary measurements

Universal bounds on the electrical and elastic response of two-phase (and multiphase) ellipsoidal or parallelopipedic bodies have been obtained by Nemat-Nasser and Hori. Here we show how their bounds can be improved and extended to bodies of arbitrary shape. Although our analysis is for two-phase bodies with isotropic phases it can easily be extended to multiphase bodies with anisotropic constituents. Our two-phase bounds can be used in an inverse fashion to bound the volume fractions occupied by the phases, and when the volume fraction is asymptotically small reduce to those of Capdeboscq and Vogelius, for electrical conductivity, and Capdeboscq and Kang, for elasticity. Other volume fraction bounds derived here utilize information obtained from thermal, magnetic, dielectric or elastic responses. One bound on the volume fraction can be obtained by simply immersing the body in a water filled cylinder with a piston at one end and measuring the change in water pressure when the piston is displaced by a known small amount. This bound may be particularly effective for estimating the volume of cavities in a body. We also obtain new bounds utilizing just one pair of (voltage, flux) electrical measurements at the boundary of the body.     

Apparent and effective mechanical properties of linear matrix-inclusion random composites: Improved bounds for the effective behavior

This paper is devoted to the derivation of improved bounds for the effective behavior of linear elastic matrix-inclusion composites based on a strategy which is inspired by both the works of Huet, 1990, Danielsson et al., 2007. As shown by the former author, the effective properties of random linear composites can be bounded by ensemble averages of their apparent elastic moduli defined on square (or cubic) volume elements (VEs) and computed with either affine displacement Boundary Conditions (BC) or uniform traction BC. However, in the case of a large contrast of the constituents, the discrepancy between the upper and lower bounds remains significant, even for large values of the VE size. This occurs because the contribution to the total potential (or complementary) energy of the particles (or pores) which intersect the edges of the VE becomes unphysically very large when uniform BC are directly applied to the particles. To avoid such limitations, we considerer non-square (or non-cubic) VEs consisting in simply connex assemblages of cells, each cell being composed of an inclusion surrounded by the matrix, thus forbidding any direct application of BC to the particles. Such VEs are generated by extending the scheme proposed by Danielsson et al. (2007) in the context of periodic random microstructures to fully random microstructures. By applying the classical energy bounding theorems to the non-square VEs, new bounds for the effective behavior are derived. Their application to a two-phase composite composed of an isotropic matrix and aligned identical fibers randomly distributed in the transverse plane leads to sharper bounds which converge quickly with the VE size, even for infinite contrasts.     

An exact solution for micromechanical analysis of periodically layered composites

This paper presents an exact solution for effective properties and local fields of periodic layered composites, obtained based on variational asymptotic method for unit cell homogenization, a recently developed micromechanics modeling framework. The layered composites could be made of multiple general anisotropic layers. This solution reproduces some published results when specialized to layered composites made of two orthotropic or isotropic layers and is located within the correctly interpreted bounds of the rules of mixtures according to Voigt and Reuss. It is also emphasized in this paper that the rules of mixtures according to Voigt and Reuss are absolute upper and lower bounds for the effective properties of layered composites if the bounds are interpreted in terms of the stiffness matrix or the fourth-order elasticity tensor. The confusion in a few recent papers regarding effective Young's modulus of layered isotropic composites could exceed the Voigt and Reuss estimates is clarified.     

Bounds on the non-local effective elastic properties of composites

The paper deals with the effective linear elastic behaviour of random media subjected to inhomogeneous mean fields. The effective constitutive laws are known to be non-local. Therefore, the effective elastic moduli show dispersion, i.e¹ they depend on the “wave vector” k of the mean field. In this paper the well-known Hashin-Shtrikman bounds (1962) for the Lamé parameters of isotropic multi-phase mixtures are generalized to inhomogeneous mean fields k ≠ 0. The bounds involve two-point correlations of random elastic moduli. In the limit k → ∞ the bounds converge to the exact result. The interest is focussed on composites with cell structures and on binary mixtures. To illustrate the results, numerical evaluations are carried out for a binary cell material composed of nearly spherical grains of equal size.     

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.     

On the stiffness of materials containing a disordered array of microscopic holes or hard inclusions

We study the macroscopic mechanical behavior of materials with microscopic holes or hard inclusions. Specifically, we deal with the effective elastic moduli of composites whose microgeometry consists of either soft or hard isolated inclusions surrounded by an elastic matrix. We approach this problem by taking the stiffness of the inclusion phase to be a complex variable, which we eventually evaluate at the soft or hard limits. Our main result states that there is a certain class of non-physical, negative-definite values of the elastic moduli of the inclusion phase for which the effective tensor does not have infinities or become otherwise singular.We present applications of this result to the estimation of effective moduli and to homogenization theorems. The first application involves using complexanalytic methods to obtain rigorous and accurate bounds on the effective moduli of the high-contrast composites under consideration. We also discuss the variational estimates of Rubenfeld & Keller, which yield a complementary set of bounds on these moduli. The best bounds are given by a combination of the analytical and variational results. As a second application, we show that certain known theorems of homogenization for materials with holes are simple consequences of our main result, and in this connection we establish corresponding new theorems for materials with hard inclusions. While our rederivation of the homogenization theorems for materials with holes can be closely related to other known constructions, it appears that certain elements provided by our main result are essential in the proof of homogenization for the hard-inclusion case.     

Homogenization methods to approximate the effective response of random fibre-reinforced Composites

In this article a fibre-reinforced composite material is modelled via an approach employing a representative volume element with periodic boundary conditions. The effective elastic moduli of the material are thus derived. In particular, the method of asymptotic homogenization is used where a finite number of fibres are randomly distributed within the representative periodic cell. The study focuses on the efficacy of such an approach in representing a macroscopically random (hence transversely isotropic) material. Of particular importance is the sensitivity of the method to cell shape, and how this choice affects the resulting (configurationally averaged) elastic moduli. The averaging method is shown to yield results that lie within the Hashin–Shtrikman variational bounds for fibre-reinforced media and compares well with the multiple scattering and (classical) self-consistent approximations with a deviation from the latter in the larger volume fraction cases. Results also compare favourably with well-known experimental data from the literature.     

Material instability-induced extreme damping in composites: A computational study

We investigate the effective viscoelastic performance of particle-reinforced composite materials whose particulate phase undergoes a material instability resulting in temporarily non-positive-definite elastic moduli. Recent experiments have shown that phase transitions in geometrically-constrained composite phases (such as in particles embedded in a stiff matrix) can lead to stable non-positive-definite elastic moduli, and they hinted at strong damping increases that can be achieved from such metastable composite phases. All previous theoretical efforts to explain such phenomena have used simplistic one-dimensional models or they were based on composite bounds and specific two-phase solids. Here, we study particle–matrix composites with periodic randomized particle dispersion. A finite element discretization is used in combination with a sophisticated nonlinear solver in order to perform the numerous calculations in a feasible amount of computing time. Our computational analysis shows that stable non-positive-definite inclusion moduli can indeed lead to extreme damping increases (i.e. greatly exceeding the intrinsic damping of each composite phase) and that such extreme damping arises from a shift in microstructural mechanisms.     

Unified formulae of variational bounds for multiphase anisotropic elastic composites

Using the spherical and deviator decomposition of the polarization and strain tensors, we present a general algorithm for the calculation of variational bounds of dimension d for any type of anisotropic linear elastic composite as a function of the properties of the comparison body. This procedure is applied in order to obtain analytical expressions of bounds for multiphase, linear elastic composites with cubic symmetry where the geometric shapes of the inclusions are arbitrary. For the validation, it can be proved that for the isotropic particular case, the bounds coincide with those recently reported by Gibiansky and Sigmund. On the other hand, based on this general procedure some, classical bounds reported by Hashin for transversely isotropic composites, are reproduced. Numerical calculations and some comparisons with other models and experimental data are shown.     

A micromechanically based couple-stress model of an elastic orthotropic two-phase composite

We determine couple-stress moduli and characteristic lengths of a two-dimensional matrix-inclusion composite, with inclusions arranged in a periodic square array and both constituents linear elastic of Cauchy type. In the analysis we replace this composite by a homogeneous planar, orthotropic, couple-stress continuum. A generalization of the original Mindlin's (1963) derivation of field equations for such a continuum results in two (not just one!) characteristic lengths. We evaluate the couple-stress properties from the response of a unit cell under several types of boundary conditions: displacement, displacement-periodic, periodic and mixed, and traction controlled. In the parametric study we vary the stiffness ratio of both phases to cover a range of different media from nearly porous materials through composites with very stiff inclusions. We find that the aforementioned boundary conditions result in hierarchies of orthotropic couple-stress moduli, whereas both characteristic lengths are fairly insensitive to boundary conditions, and fall between 0.12% and 0.22% of the unit cell size for the inclusions' volume fraction of 18%.     

On thermoelastostatics of composites with nonlocal properties of constituents II. Estimation of effective material and field parameters

One considers a linear thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated heterogeneities. It is assumed that the stress–strain constitutive relations of constituents are described by the nonlocal integral operators, whereas the equilibrium and compatibility equations remain unaltered as in classical local elasticity. The general integral equations connecting the stress and strain fields in the point being considered and the surrounding points are obtained. The method is based on a centering procedure of subtraction from both sides of a known initial integral equation their statistical averages obtained without any auxiliary assumptions such as, e.g., effective field hypothesis implicitly exploited in the known centering methods. In a simplified case of using of the effective field hypothesis for analyzing composites with one sort of heterogeneities, one proves that the effective moduli explicitly depend on both the strain and stress concentrator factor for one heterogeneity inside the infinite matrix and does not directly depend on the elastic properties (local or nonlocal) of heterogeneities. In such a case, the Levin’s (1967) formula in micromechanics of composites with locally elastic constituents is generalized to their nonlocal counterpart. A solution of a volume integral equation for one heterogeneity subjected to inhomogeneous remote loading inside an infinite matrix is proposed by the iteration method. The operator representation of this solution is incorporated into the new general integral equation of micromechanics without exploiting of basic hypotheses of classical micromechanics such as both the effective field hypothesis and “ellipsoidal symmetry” assumption. Quantitative estimations of results obtained by the abandonment of the effective field hypothesis are presented.     

Mathematical study of boundary-value problems within the framework of Steigmann–Ogden model of surface elasticity

Mathematical questions pertaining to linear problems of equilibrium dynamics and vibrations of elastic bodies with surface stresses are studied. We extend our earlier results on existence of weak solutions within the Gurtin–Murdoch model to the Steigmann–Ogden model of surface elasticity using techniques from the theory of Sobolev’s spaces and methods of functional analysis. The Steigmann–Ogden model accounts for the bending stiffness of the surface film; it is a generalization of the Gurtin–Murdoch model. Weak setups of the problems, based on variational principles formulated, are employed. Some uniqueness-existence theorems for weak solutions of static and dynamic problems are proved in energy spaces via functional analytic methods. On the boundary surface, solutions to the problems under consideration are smoother than those for the corresponding problems of classical linear elasticity and those described by the Gurtin–Murdoch model. The weak setups of eigenvalue problems for elastic bodies with surface stresses are based on the Rayleigh and Courant variational principles. For the problems based on the Steigmann–Ogden model, certain spectral properties are established. In particular, bounds are placed on the eigenfrequencies of an elastic body with surface stresses; these demonstrate the increase in the body rigidity and the eigenfrequencies compared with the situation where the surface stresses are neglected.     

Tailored heterogeneity increases overall stability regime of composites having a negative-stiffness inclusion

Recent theoretical and experimental results have shown the possibility of enormous increases in composite material overall elastic stiffness, damping, thermal expansion, piezoelectricity, etc., when the composite contains a tuned non-positive-definite (i.e., negative stiffness) constituent. For such composite materials to have practical utility, they must be stable. Recent research has shown they can be, for a limited range of constituent negative stiffness. This research has treated linear elastic composite materials with homogeneous phases, via the energy method and full dynamic stability analyses.In the present work, we first show how to analyze the composites previously treated by the comprehensive but simpler static stability approach, obtaining closed-form results. We then employ this approach to show that permitting heterogeneity of the positive-definite phase can substantially increase the range of constituent negative stiffness while maintaining overall composite stability. We first treat the positive-definite phase heterogeneity as piecewise homogeneous, and then treat it as continuously-varying. In the continuously-varying heterogeneity case, we seek the radially optimal distribution of the elastic moduli in the coatings, under constant coating average moduli constraint, to permit the most negative possible inclusion stiffness while maintaining overall composite stability. This is accomplished for three coating cases: constant bulk modulus but arbitrarily radially-varying shear modulus; constant shear modulus but arbitrarily radially-varying bulk modulus; and both moduli arbitrarily radially varying. We find the optimal coatings to be: a heterogeneous one with shear modulus being a specific continuously decreasing function of radius for the first case; a homogeneous one for the second case; and a heterogeneous one with both moduli being either Dirac-delta or Heaviside-step decreasing functions of radius for the last case (if the coating moduli are unrestricted in magnitude or have upper limits, respectively). The results show a substantial increase in the permissible inclusion negative stiffness range is provided by coating heterogeneity, while maintaining overall composite stability. Such an increased range of constituent negative stiffness provides an enlarged tuning parameter range for the development of novel, high-performance composite materials.