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This paper is concerned with the estimation of the effective transport characteristics of fluid-saturated porous media via rigorous microstructure-property relations. We are particularly interested in predicting the formation factor F , mean survival time τ, principal NMR (diffusion) relaxation time T1, principal viscous relaxation time Θ1, and fluid permeability k. To do so, we employ rigorous methods to estimate the fluid permeability and these other transport properties of “hyperuniform” and nonhyperuniform models of porous media from microstructural information. Disordered hyperuniform materials are exotic amorphous states of matter that have attracted great attention in the physical, mathematical and biological science but little is known about their fluid transport characteristics. In carrying out this investigation, we not only draw from ideas and results of the emerging field of hyperuniformity, but from homogenization theory, statistical geometry, differential equations (spectrum of Laplace and Stokes operators), and the covering and quantizer problems of discrete geometry. Among other results, we derive a Fourier representation of a classic rigorous upper bound on the fluid permeability that depends on the spectral density to infer how the permeabilities of hyperuniform porous media perform relative to those of nonhyperuniform ones. We find that the velocity fields in nonhyperuniform porous media are generally much more localized over the pore space compared to those in their hyperuniform counterparts, which has implications for their permeabilities. Rigorous bounds on transport properties suggest a new approximate formula for the fluid permeability that provides reasonably accurate permeability predictions of a certain class of hyperuniform and nonhyperuniform porous media. These comparative studies shed new light on the microstructural characteristics that determine the transport properties of general porous media. Our findings also have implications for the design of porous materials with desirable transport properties.

Heterogeneous materials are ubiquitous in nature and synthetic situations and have a wide range of important engineering applications. Accurate modeling and reconstructing three-dimensional (3D) microstructure of topologically complex materials from limited morphological information such as a two-dimensional (2D) micrograph is crucial to the assessment and prediction of effective material properties and performance under extreme conditions. Here, we extend a recently developed dilation–erosion method and employ the Yeong–Torquato stochastic reconstruction procedure to model and generate 3D austenitic–ferritic cast duplex stainless steel microstructure containing percolating filamentary ferrite phase from 2D optical micrographs of the material sample. Specifically, the ferrite phase is dilated to produce a modified target 2D microstructure and the resulting 3D reconstruction is eroded to recover the percolating ferrite filaments. The dilation–erosion reconstruction is compared with the actual 3D microstructure, obtained from serial sectioning (polishing), as well as the standard stochastic reconstructions incorporating topological connectedness information. The fact that the former can achieve the same level of accuracy as the latter suggests that the dilation–erosion procedure is tantamount to incorporating appreciably more topological and geometrical information into the reconstruction while being much more computationally efficient.

The setting of cement paste is widely understood to be caused by percolation of the links that are created by overlap of hydration products on the surfaces of reacting grains of clinker. Percolation theory predicts that the elastic modulus will increase with a certain functional form, but few attempts have been made to demonstrate this behavior quantitatively. We discuss the appropriate variables to use for this test of the theory, and show that the percolation probability is proportional to time only over a narrow time interval. We compare the measured and predicted degree of hydration at the percolation threshold, and show that the hard-core/soft-shell model strongly over-estimates the amount of hydration at the setting point. The discrepancy is attributed to agglomeration of particles in the paste, which reduces the amount of hydration needed to link the particles into an elastic network.

We present a new event-driven molecular dynamics (MD) algorithm for measuring the free energy of nearly jammed packings of spherical and non-spherical hard particles. This Bounding Cell Molecular Dynamics (BCMD) algorithm exactly calculates the free-energy of a single-occupancy cell (SOC) model in which each particle is restricted to a neighborhood of its initial position using a hard-wall bounding cell. Our MD algorithm generalizes previous ones in the literature by enabling us to study non-spherical particles as well as to measure the free-energy change during continuous irreversible transformations. Moreover, we make connections to the well-studied problem of computing the volume of convex bodies in high dimensions using random walks. We test and verify the numerical accuracy of the method by comparing against rigorous asymptotic results for the free energy of jammed and isostatic disordered packings of both hard spheres and ellipsoids, for which the free energy can be calculated directly as the volume of a high-dimensional simplex. We also compare our results to previously published Monte Carlo results for hard-sphere crystals near melting and jamming and find excellent agreement. We have successfully used the BCMD algorithm to determine the configurational and free-volume contributions to the free energy of glassy states of binary hard disks [A. Donev, F.H. Stillinger, S. Torquato, Do binary hard disks exhibit an ideal glass transition? Phys. Rev. Lett. 96 (22) (2006) 225502]. The algorithm can also be used to determine phases with locally- or globally-minimal free energy, to calculate the free-energy cost of point and extended crystal defects, or to calculate the elastic moduli of glassy or crystalline solids, among other potential applications.

The pair correlation function g 2 ( r ) provides a basic geometric descriptor for many-particle systems. It must obey two necessary conditions: (i) non-negativity for all distances r, and (ii) non-negativity of its associated structure factor S ( k ) for all k. Here we utilize an improved stochastic construction algorithm for particle configurations to establish conditions in which (i) and (ii) are also sufficient, i.e., g 2 ( r ) is in fact realizable. Two types of target pair correlation functions have been investigated in one, two, and three dimensions for hard-core particles, specifically a unit step function, and a contact δ plus step pair correlation function. Results indicate that the former target function is realizable up to a terminal density set by necessary condition (ii), at which the particle core packing fraction equals 2 - d in d dimensions. Furthermore, results are consistent with the proposition that for d > 1 the contact δ plus step function is realizable up to a terminal density due to condition (ii) at which the packing fraction of cores is ( d + 2 ) / 2 d + 1 [Torquato and Stillinger, J. Phys. Chem. B 106 (2000) 8354, 11406].

The Division of Materials Sciences and Engineering of the US Department of Energy (DOE) sponsored a workshop to identify cutting-edge research needs and opportunities, enabled by the application of theoretical and applied mechanics. The workshop also included input from biochemical, surface science, and computational disciplines, on approaching scientific issues at the nanoscale, and the linkage of atomistic-scale with nano-, meso-, and continuum-scale mechanics. This paper is a summary of the outcome of the workshop, consisting of three main sections, each put together by a team of workshop participants. Section 1 addresses research opportunities that can be realized by the application of mechanics fundamentals to the general area of self-assembly, directed self-assembly, and fluidics. Section 2 examines the role of mechanics in biological, bioinspired, and biohybrid material systems, closely relating to and complementing the material covered in Section 1. In this manner, it was made clear that mechanics plays a fundamental role in understanding the biological functions at all scales, in seeking to utilize biology and biological techniques to develop new materials and devices, and in the general area of bionanotechnology. While direct observational investigations are an essential ingredient of new discoveries and will continue to open new exciting research doors, it is the basic need for controlled experimentation and fundamentally-based modeling and computational simulations that will be truly empowered by a systematic use of the fundamentals of mechanics. Section 3 brings into focus new challenging issues in inelastic deformation and fracturing of materials that have emerged as a result of the development of nanodevices, biopolymers, and hybrid bio–abio systems. Each section begins with some introductory overview comments, and then provides illustrative examples that were presented at the workshop and which are believed to highlight the enabling research areas and, particularly, the impact that mechanics can make in enhancing the fundamental understanding that can lead to new technologies.

In this first part of a series of two papers, we present in considerable detail a collision-driven molecular dynamics algorithm for a system of non-spherical particles, within a parallelepiped simulation domain, under both periodic or hard-wall boundary conditions. The algorithm extends previous event-driven molecular dynamics algorithms for spheres, and is most efficient when applied to systems of particles with relatively small aspect ratios and with small variations in size. We present a novel partial-update near-neighbor list (NNL) algorithm that is superior to previous algorithms at high densities, without compromising the correctness of the algorithm. This efficiency of the algorithm is further increased for systems of very aspherical particles by using bounding sphere complexes (BSC). These techniques will be useful in any particle-based simulation, including Monte Carlo and time-driven molecular dynamics. Additionally, we allow for a non-vanishing rate of deformation of the boundary, which can be used to model macroscopic strain and also alleviate boundary effects for small systems. In the second part of this series of papers we specialize the algorithm to systems of ellipses and ellipsoids and present performance results for our implementation, demonstrating the practical utility of the algorithm.

We apply the algorithm presented in the first part of this series of papers to systems of hard ellipses and ellipsoids. The theoretical machinery needed to treat such particles, including the overlap potentials, is developed in full detail. We describe an algorithm for predicting the time of collision for two moving ellipses or ellipsoids. We present performance results for our implementation of the algorithm, demonstrating that for dense systems of very aspherical ellipsoids the novel techniques of using neighbor lists and bounding sphere complexes, offer as much as two orders of magnitude improvement in efficiency over direct adaptations of traditional event-driven molecular dynamics algorithms. The practical utility of the algorithm is demonstrated by presenting several interesting physical applications, including the generation of jammed packings inside spherical containers, the study of contact force chains in jammed packings, and melting the densest-known equilibrium crystals of prolate spheroids.

The determination of the maximal packing arrangements of two-dimensional, binary hard disks of radii RS and RL (with RS 6 RL) for su0ciently small RS amounts to 1nding the optimal arrangement of the small disks within a tricusp: the nonconvexcavity between three close-packed large disks. We present a particle-growth Monte Carlo algorithm for the generation of geometric packings of equi-sized hard disks within such a tricusp. The 1rst 19 members of an in1nite sequence of maximal density structures thus produced are reported. In addition, the Monte Carlo algorithm is applied to the geometric packing of disks within a 8at-sided equilateral triangle and compared to published results for that packing problem. We perform an analysis of geometric properties of the packings, e.g. packing fractions and symmetries of structures con1ned to both containers. Interestingly, we 1nd a non-monotonic increase in the packing fraction with increasing number of disks packed within both the 8at-sided triangle and tricusp. It is important to note that for disk packings within a 8at-sided equilateral triangle, this non-monotonic behavior of the packing fraction had not been reported in previously published works. For the 8at-sided equilateral triangle, local maxima occur at the triangular integers NS =1; 3; 6; 10; 15 :::; as well as NS =1 2; where NS is the number of disks in each packing. However, local maxima for packings within the tricusp exist at NS =1 ; 3; 6; 10; 18 ::: :Finally, we analyze the asymptotic approach to the upper bound on the packing fraction of the in1nite sequence of maximal structures of disks con1ned to the tricusp. c

The finite element method has been used to simulate the properties of panels with Kagome and tetragonal cores under compressive and shear loading. The simulation has been performed for two different materials: a Cu-alloy with extensive strain hardening and an Al-alloy with minimal hardening. It is shown that the Kagome core is more resistant to plastic buckling than the tetragonal core under both compression and shear. One consequence is that the Kagome structure has the greater load capacity and a deferred susceptibility to softening. Another is that the Kagome core is isotropic in shear: contrasting with the soft orientations exhibited by the tetragonal core.

Hard-particle packings have provided a rich source of outstanding theoretical problems and served as useful starting points to model the structure of granular media, liquids, living cells, glasses, and random media. The nature of “jammed” hard-particle packings is a current subject of keen interest. We demonstrate that the response of jammed hard-particle packings to global deformations cannot be described by linear elasticity (even for small particle displacements) but involves a “conical” nonlinear constitutive theory. It is the singular nature of the hard-particle potential that leads to the breakdown of linear elasticity. Interestingly, a nonlinear theory arises because the feasible particle displacements (leading to unjamming) depend critically on the local spatial arrangement of the particles, implying a directionality in the feasible strains that is absent in particle systems with soft potentials. Mathematically, the set of feasible strains has a conical structure, i.e., components of the imposed strain tensor generally obey linear inequalities. The nature of the nonlinear behavior is illustrated by analyzing several specific packings. Finally, we examine the conditions under which a packing can be considered to “incompressible” in the traditional sense.

Buildup of internal self-stresses in hyperstatic adaptive structures resists actuation. A recent paper by Guest and Hutchinson (2003) shows that periodic infinite truss structures cannot be both statically and kinematically determinate structures; therefore, a rigid infinite lattice bar framework must be hyperstatic. This paper shows that it is possible to design adaptive periodic infinite truss structures that can achieve any state of uniform strain without energy cost by actuating only a subset of the bars in a coordinated fashion. We show that actuation of only 3 bars in two dimensions or 6 bars in three dimensions per unit cell is required. A mathematical apparatus is developed and an example of such a bitriangular lattice structure is given, along with accompanying illustrations. Supporting animations can be found at the authors’ website.

Malignant brain tumors consist of a number of distinct subclonal populations. Each of these subpopulations may be characterized by its own behaviors and properties. These subpopulations arise from the constant genetic and epigenetic alteration of existing cells in the rapidly growing tumor. However, since each single-cell mutation only leads to a small number of offspring initially, very few newly arisen subpopulations survive more than a short time. The present work quantifies “emergence”, i.e. the likelihood of an isolated subpopulation surviving for an extended period of time. Only competition between clones is considered; there are no cooperative effects included. The probability that a subpopulation emerges under these conditions is found to be a sigmoidal function of the degree of change in cell division rates. This function has a non-zero value for mutations which confer no advantage in growth rate, which represents the emergence of a distinct subpopulation with an advantage that has yet to be selected for, such as hypoxia tolerance or treatment resistance. A logarithmic dependence on the size of the mutated population is also observed. A significant probability of emergence is observed for subpopulations with any growth advantage that comprise even 0.1% of the proliferative cells in a tumor. The impact of even two clonal populations within a tumor is shown to be sufficient such that a prognosis based on the assumption of a monoclonal tumor can be markedly inaccurate.

We have developed a novel and versatile three-dimensional cellular automaton model of brain tumor growth. We show that macroscopic tumor behavior can be realistically modeled using microscopic parameters. Using only four parameters, this model simulates Gompertzian growth for a tumor growing over nearly three orders of magnitude in radius. It also predicts the composition and dynamics of the tumor at selected time points in agreement with medical literature. We also demonstrate the flexibility of the model by showing the emergence, and eventual dominance, of a second tumor clone with a different genotype. The model incorporates several important and novel features, both in the rules governing the model and in the underlying structure of the model. Among these are a new definition of how to model proliferative and non-proliferative cells, an isotropic lattice, and an adaptive grid lattice.

Recent progress in three different areas involving the modeling of the physical properties of composites is reviewed. These include: (i) theoretical approaches to microstructure/property relations; (ii) X-ray microtomography, an imaging technique that enables one to obtain high-resolution three-dimensional microstructural phase information of a composite sample in a non-intrusive manner; and (ii) topology optimization, a promising numerical technique that enables one to design composites with tailored material properties. Current limitations and future research needs are described.

The apparent stiffness tensors of two-dimensional elastic composite samples smaller thanthe representative volume element (RVE) are studied as a function of system size. Numericalexperiments are used to investigate how the apparent properties of the composite converge withincreasing scale factor n, defined to be the ratio between the linear size of the composite and thelinear size of the unit cell. Under affine (Dirichlet-type) or homogeneous stress (Neumann-type) boundary conditions, the apparent elastic moduli overestimate orunderestimate, respectively, the effective elastic moduli of the infinitely periodic system. Theresults show that the difference between the Dirichlet, Neumann and the effective stiffnesstensors depends strongly on the phase stiffness contrast ratio. Dirichlet boundary conditionsprovide a more accurate estimate of the effective elastic properties of stiff matrix composites,whereas Neumann boundary conditions provide a more accurate estimate for compliant matrixstructures. It is shown that the apparent bulk and shear moduli may lie outside of theHashin–Shtrikman bounds. However, these bounds provide good upper and lower estimates forthe apparent bulk and shear moduli of structures with a scale factor n ⩾ 2. A similar approach isused to study hierarchical composites containing two distinct structural levels with a finiteseparation of length scales. It is shown, numerically, that the error associated with replacing thesmallest-scale regions by an equivalent homogeneous medium is very small, even when the ratiobetween the length scales is as low as three.

Accurate approximate relations for the effective elastic moduli of two- and three-dimensional isotropic dispersions are obtained by truncating, after third-order terms, an exact series expansion for the effective stiffness tensor of d-dimensional two-phase composites (obtained in the first paper) that perturbs about certain optimal dispersions. Our third-order approximations of the effective bulk modulus Ke and shear modulus Ge are compared to benchmark data, rigorous bounds and popular self-consistent approximations for a variety of macroscopically isotropic dispersions in both two and three dimensions, for a wide range of phase moduli and volume fractions. Generally, for the cases considered, the third-order approximations are in very good agreement with benchmark data, always lie within rigorous bounds, and are superior to popular self-consistent approximations.

Recent progress that we have made on the problem of determining the effective properties of random heterogeneous media from knowledge of the morphology is reviewed and extended. A variety of different effective properties are considered, including the elastic moduli and electrical conductivity of composites, time scales associated with diffusion and reaction among traps, and fluid permeability of porous media. We also remark on the importance of microstructure fluctuations in influencing the failure characteristics of composites. The preponderance of work thus far has focused on heterogeneous media that are statistically homogeneous. We propose a model for statistically inhomogeneous two-phase random media (including functionally graded materials) consisting of inhomogeneous fully penetrable spheres that permits one to represent and evaluate certain n-point correlation functions that statistically characterize the microstructure for this model. Unlike the case of statistically homogeneous media, the microstructure functions depend upon the absolute positions of their arguments. Finally, we describe a novel procedure to reconstruct heterogeneous media from limited knowledge about the microstructure and discuss applications of this technique.

The general goal of this paper is to develop a new approach for bounding the effective moduli (e.g. elastic moduli, conductivity, or thermal expansion coefficients) of composite materials. Specifically, this paper aims to : (1) formulate a procedure that combines the translation method and the Beran procedure into one powerful method that has the advantages of both approaches ; and (2) apply the new method to study effective properties of viscoelastic composites. The new method enables one to get the most restrictive three-point geometrical-parameter bounds. In particular, we obtain new three-point bounds on the complex bulk modulus of an isotropic viscoelastic composite. The new bounds are given by the outermost of several circular arcs in the complex bulk modulus plane. They take into account three-point statistical information and thus are much more restrictive than previously known two-point bounds.

We utilize two different approaches, homogenization theory and discrete network analyses, to study the mechanical and transport properties of two-dimensional cellular solids (honeycombs) consisting of either hexagonal, triangular, square or Voronoi cells. We exploit results from homogenization theory for porous solids (in the low-density limit) to establish rigorous bounds on the effective thermal conductivity of honeycombs in terms of the elastic moduli and vice versa. It is shown that for hexagonal, triangular or square honeycombs, the cross-property bound relating the bulk modulus to the thermal conductivity turns out to be an exact and optimal result. The same is true for the cross-property bound linking the shear or Young's modulus of the triangular honeycomb to its conductivity. For low-density honeycombs, we observe that all of the elastic moduli do not depend on the Poisson's ratio of the solid phase. The elastic-viscoelastic correspondence principle enables us to conclude that all of the viscoelastic moduli of honeycombs in the low-density limit are proportional to the complex Young's modulus of the solid phase. Such structures have real Poisson's ratios and the loss tangent is the same for any load.

Sharp bounds on the effective thermal expansion coefficients of isotropic multiphase composites and isotropic polycrystals are obtained by using classical variational principles and the translation method. Our bounds are appreciably narrower than the known Schapery-Rosen-Hashin bounds. Conditions are formulated that guarantee a one-to-one correspondence between the bulk modulus and thermal expansion coefficient of a polycrystal. All of our results can be readily applied to the poroelasticity problem. Generalizations of the results to treat anisotropic composites comprised of anisotropic phases are discussed.

We consider an optimal design of composite hydrophones consisting of parallel piezoelectric PZT rods that are embedded in a porous polymer matrix. Given the material properties of the polymer and PZT ceramic, we have optimally designed the piezocomposite to maximize the hydrostatic coupling factor, hydrophone figure of merit, or electromechanical coupling factor, using the methods of homogenization theory. The optimal composite is obtained by using a two-step procedure : (i) first we find the ideal structure of the matrix material by weakening the polymer by an optimal arrangement of pores, and (ii) then we embed the PZT rods in this matrix. The design parameters are the shape, volume fraction, and spatial arrangement of the piezoceramic rods, and the structure of the matrix material. It turns out that the optimal matrix is highly anisotropic and is characterized by negative Poisson’s ratios in certain directions. The optimal composites possess performance characteristics that are significantly higher than those of a piezocomposite with an isotropic polymer matrix. The results can be viewed as theoretical upper bounds on the hydrophone performance. 0 1997 Elsevier Science Ltd. All rights reserved.

To what extent can the structure of a disordered heterogeneous material be reconstructed using limited but essentially exact structural information about the original system? We formulate a methodology based on simulated annealing to reconstruct both equilibrium and non-equilibrium dispersions of particles based only on correlation functions which statistically characterize the system. To test this method, we reconstruct dispersions from the radial distribution function (RDF) associated with the original system. Other statistical correlation functions are evaluated to compare how well the reconstructed system matches the original system. We show that for low-density systems or high-density systems with little particle aggregation, our reconstruction from the RDF reproduces the system fairly well. However, for dense systems with extensive clustering, RDF information is somewhat inadequate in being able to reconstruct the original system. We also show that one can produce a system with an RDF that is similar to the reference system, but with appreciably different structure. Finally, system-size effects are analyzed analytically.

For isotropic two-phase composites, we derive phase-interchange inequalities for the bulk and shear moduli in two and three space dimensions. We find optimal microstructures that realize part of the bull, and shear moduli bounds in two dimensions. Geometrical-parameter bounds and the translation method are used to prove the results. The phase-interchange relations are applied to composites with cavities or a perfectly rigid phase, composites near the percolation threshold, incompressible composites, composites with equal bulk or shear phase moduli, and symmetric composites.

We find bounds on the effective elastic moduli of cracked materials in terms of the effective conductivity of such media. These represent the first non-trivial bounds on the effective properties of cracked media which are independent of the shapes and spatial distribution of the cracks. Different approximations for the elastic moduli of cracked media are tested against our bounds. The microgeometries of cracks that satisfy the bounds exactly are identified.

We study bounds on the effective conductivity and elastic moduli of two-phase isotropic composites that depend on geometrical parameters that take into account up to three-point statistical information concerning the composite microstructure. We summarize existing bounds, apply a special fractional linear transformation to simplify their functional forms, and describe two approaches to improve such bounds. These approaches allow us to get new and improved geometrical-parameter bounds on the elastic moduli of two-dimensional composites. Applications of the bounds for effective-medium geometries as well as random arrays of aligned fibers in a matrix are discussed.

We will review recent progress in our understanding of classical transport in porous media. Theoretical concepts will be illustrated with two distinct kinds of calculations. The first involve the grain consolidation model and are based on a particular multisize packing of spherical grains. The computational methods developed here are sufficiently accurate that we propose to combine them with direct measurements of the pore and grain geometry based on X-ray microtomography. Our preliminary results indicate that this approach may well play an important part in future studies of transport in porous media.

Recent progress that we have made on the problem of determining the effective properties of random heterogeneous media from the morphology is reviewed. A variety of different effective properties are considered, including the electrical conductivity and elastic moduli of composites, time scales associated with diffusion and reaction among traps, and fluid permeability of porous media. Studying seemingly disparate properties from a general viewpoint is shown to be fruitful.

One of the fundamental quantities which statistically characterizes a random system of interacting particles is the nearest-neighbor distribution function. We present computersimulation results for two different types of nearest-neighbor distribution functions for random distributions of identical impenetrable (hard) spheres. We also report, for such systems, computer-simulation data for closely related quantities such as the associated cumulative distributions. From this information, we calculate the "mean nearest-neighbor distance" between particles. Our computer-simulation results are compared to the various sets of theoretical expressions derived recently by Torquato, Lu and Rubinstein. One of these sets of expressions is shown to be in excellent agreement with the simulation data.

Computer simulation results are reported for the two-point matrix probability function S/sub 2/ of two-phase random media composed of disks distributed with an arbitrary degree of impenetrability lambda. The novel technique employed to sample S/sub 2/(r) (which gives the probability of finding the endpoints of a line segment of length r in the matrix) is very accurate and has a fast execution time. Results for the limiting cases lambda = 0 (fully penetrable disks) and lambda = 1 (hard disks), respectively, compare very favorably with theoretical predictions made by Torquato and Beasley and by Torquato and Lado. Results are also reported for several values of lambda that lie between these two extremes; cases which heretofore have not been examined. copyright 1988 Academic Press, Inc.

We evaluate third-order bounds on the effective transverse bulk and shear moduli of transversely isotropic fiber-reinforced materials for a distribution of fully penetrable cylinders in a matrix. The third-order bounds not only incorporate the simplest of statistical quantities, the fiber volume fraction φ2, but also involve microstructural parameters which depend upon the threepoint matrix probability function of the model. The third-order bounds, for the fully penetrable-cylinder model and for a wide range of conditions, significantly improve upon second-order bounds on the effective transverse elastic moduli, due to Hill and to Hashin, which incorporate φ2 only. In particular, when the fiber phase is as much as two orders of magnitude more rigid than the matrix phase, the Silnutzer bounds, for the model considered here, reduce the second-order bound widths by over 50% for 0 ≤ φ2 ≤ 0.5.

Third-order and fourth-order bounds on the effective transverse thermal conductivity k e of fiber-reinforced materials (statistically isotropic in the transverse plane) due to Silnutzer and to Milton, respectively, are evaluated for a distribution of fully penetrable cylinders in a matrix. Both of these bounds involve a multidimensional integral which depends upon the three-point matrix probability function of the model. For nontrivial values of the fiber volume fraction φ 2 (i.e. 0 φ 2 φ 2 ⩽ 0.3 to give quantitatively useful estimates of k e even when the conductivity of the fiber phase k 2 is as much as two orders of magnitude greater than the matrix conductivity k 1 . For 0.8 ⩽ φ 2 ⩽ 1, the Milton bounds are sufficiently restrictive to provide reasonable approximations of k e for an insulating fiber phase such that 0.01 ⩽ α ⩽ 1, where α = k 2 k 1 . The results obtained in this study for k e translate immediately into equivalent results for the axial shear modulus of fiber-reinforced materials.

We evaluate the third-order Beran-Molyneux bounds on the effective bulk modulus K e and the third-order McCoy bounds on the effective shear modulus μ e of a model of a two-phase composite in which one of the phases consists of spherical inclusions (or voids), with bulk and shear moduli, K 2 and μ 2 , respectively, and volume fraction φ 2 , dispersed randomly throughout a matrix phase, with bulk and shear moduli, K 1 and μ 1 , respectively, and volume fraction φ 1 . We tabulate the two fundamental microstructural parameters I 1 and L 1 required to evaluate the bounds, which depend upon the three-point matrix probability function of the model, for the aforementioned fully-penetrablesphere model. We compare the third-order bounds on K e and μ e to the second-order bounds due to Hashin and Shtrikman and to Walpole. We find that the third-order bounds for our model are always more restrictive than the corresponding second-order bounds. When the moduli of the phases differ by an order of magnitude, the third-order bounds are sharp enough to provide quantitatively useful estimates of K e and μ e for all φ 2 . The third-order bounds are very restrictive at low φ 2 values (e.g., φ 2 = 0.1 ) where they remain useful for cases in which the moduli of the phases differ by two orders of magnitude. Experimental values of μ e measured by Corson for a tungstenlead composite are found to lie within the McCoy bounds for our model, with the lower bound giving a good estimate of the data.