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Polycrystalline metals, porous rocks, colloidal suspensions, epitaxial thin films, gels, foams, granular aggregates, sea ice, shape-memory metals, magnetic materials, and electro-rheological fluids are all examples of materials where an understanding of the mathematics on the different length scales is a key to interpreting their physical behavior. In their analysis of these media, scientists coming from a number of disciplines have encountered similar mathematical problems, yet it is rare for researchers in the various fields to meet. The 1995-1996 program at the Institute for Mathematics and its Applications was devoted to Mathematical Methods in Material Science, and was attended by materials scientists, physicists, geologists, chemists engineers, and mathematicians. The present volume contains chapters which have emerged from four of the workshops held during the year, focusing on the following areas: Disordered Materials; Interfaces and Thin Films; Mechanical Response of Materials from Angstroms to Meters; and Phase Transformation, Composite Materials and Microstructure. The scales treated in these workshops ranged from the atomic to the microstructural to the macroscopic, the microstructures from ordered to random, and the treatments from "purely" theoretical to the highly applied. Taken together, these works form a compelling and broad account of many aspects of the science of multiscale materials, and will hopefully inspire research across the self-imposed barriers of twentieth century science.



Scaling Limit for the Incipient Spanning Clusters

Scaling limits of critical percolation models show major differences between low and high dimensional models. The article discusses the formulation of the continuum limit for the former case. A mathematical framework is proposed for the direct description of the limiting continuum theory. The resulting structure is expected to exhibit strict conformal invariance, and facilitate the mathematical discussion of questions related to universality of critical behavior, conformal invariance, and some relations with a number of field theories.
Michael Aizenman

Bounded and Unbounded Level Lines in Two-Dimensional Random Fields

Every two-dimensional incompressible flow follows the level lines of some scalar function ψ on ∝2; transport properties of the flow depend on the qualitative structure e.g. boundedness—of these level lines. We discuss some recent results on the boundedness of level lines when ψ is a stationary random field. Under mild hypotheses there is only one possible alternative to bounded level lines: the “treelike” random fields, which, for some interval of values of a, have a unique unbounded level line at each level a, with this line “winding through every region of the plane.” If the random field has the FKG property then only bounded level lines are possible. For stationary C 2 Gaussian random fields with covariance function decaying to 0 at infinity, the treelike property is the only alternative to bounded level lines provided the density of the absolutely continuous part of the spectral measure decays at infinity “slower than exponentially,” and only bounded level lines are possible if the covariance function is nonnegative.
Kenneth S. Alexander

Transversely Isotropic Poroelasticity Arising from Thin Isotropic Layers

Percolation phenomena play central roles in the field of poroelasticity, where two distinct sets of percolating continua intertwine. A connected solid frame forms the basis of the elastic behavior of a poroelastic medium in the presence of external confining forces, while connected pores permit a percolating fluid (if present) to influence the mechanical response of the system from within The present paper discusses isotropic and anisotropic poroelastic media and establishes general formulas for the behavior of transversely isotropic poroelasticity arising from laminations of isotropic components. The Backus averaging method is shown to provide elementary means of constructing general formulas. The results for confined fluids are then compared with the more general Gassmann formulas that must be satisfied by any anisotropic poroelastic medium and found to be in complete agreement.
James G. Berryman

Bounds on the Effective Elastic Properties of Martensitic Polycrystals

We draw attention to the problem of estimation of elastic energies in martensitic polycrystals. In particular we introduce a tensorial parameter η=ηijkl which contains information about the microgeometry and disorder of the polycrystalline structure. Under the assumption of isotropic elasticity and mild hypothesis on the statistics of the polycrystal, this parameter allows for explicit calculation of rigorous and stringent upper bounds on the effective energy. For circular grains in two dimensions η gives the elastic energy resulting from transformation of a single circular inclusion in an elastic matrix and the bounds coincide with those derived recently by Bruno, Reitich and Leo. Consideration of such particular cases shows that our bounds can yield substantial improvements over those obtained under Taylor’s constant strain hypothesis. For arbitrary microgeometries the statistical parameter η can be calculated by means of two-point correlations functions.
Oscar P. Bruno, Fernando Reitich

Statistical Models for Fracture

Recent developments in statistical physics studying fracture phenomena are reviewed. A quantity of experimental interest is the breaking characteristics of the system (force vs. displacement): we discuss its universal scaling behaviour. Moreover, the distribution of local strain has multifractal scaling properties just before the system breaks fully apart. All the presented results appear to be universal with respect to different distributions of quenched disorder in the thresholds and other models for fracture. Universality is also found in the analysis of the scaling properties of the fracture surface roughness, where the existence of two universality classes, one for brittle the other for ductile fracture, is suggested. We present a network model for the cracking of the surface of a material subjected to an imposed strain along one boundary, or to a surface shrinkage induced by drying or cooling. Finally, cracks can be grown on a lattice by deterministic rules and their patterns can be fractal due only to the interplay of anisotropy and memory.
Lucilla De Arcangelis

Anomalous Diffusion in Random Flows

The simplest model of turbulent transport is the random motion of Brownian particles passively convected by random, incompressible velocity fields.
Albert Fannjiang

Calculating the Mechanical Properties of Materials from Interatomic Forces

Determining the mechanical properties of materials from microscopic models of their electronic structure is difficult, first because of the difference between microscopic and macroscopic length scales, and second because the linear relation between stress and strain fails in the plastic limit. The purpose of this paper is to describe how these difficulties may be avoided by use of Green’s functions and the black body theorem. Although the relation between stress and strain can be non-linear, the density of atomic positions and momenta in phase space evolves linearly, and consequently can be expressed using Green’s functions whose singular frequency dependence describes the behavior of the systems on macroscopic time scales. The black body theorem says that for quantities which obey wave equations, the local density of modes is insensitive to distant parts of system. The Green’s functions describing the evolution of local disturbances of the atomic position and momentum distributions obey just such a wave equation and are hence insensitive to the structure of the system on large length scales.
Roger Haydock

Granular Media: Some New Results

Granular materials consist of a large collection of grains, like sand or powder. They can present very intriguing effects. When shaken, sheared or poured they show segregation, convection and spontaneous fluctuations in densities and stresses. I will discuss the modelling of a granular medium on a computer by simulating a packing of elastic spheres via Molecular Dynamics which includes issipation of energy and shear friction at collisions. On a vibrating plate the formation of convection cells due to walls or amplitude modulations can be observed. Segregation of larger particles in deep beds is found to be always accompanied by these convection cells. The simulations show the existence of spontaneous density patterns in granular material flowing through pipes or hoppers and that these density fluctuations follow a 1/f α spectrum. In a dense packing non-linear acoustic phenomena, like the pressure dependence of the sound velocity are studied. Finally the plastic shear bands occuring in large scale deformations of compactified granular media are investigated using an explicit Lagrangian technique.
H. J. Herrmann

Elastic Freedom in Cellular Solids and Composite Materials

The question of how much freedom is to be incorporated in a continuum theory must ultimately be decided by experiment. There are several theories which describe behavior of materials. An early uniconstant theory was proposed based on atomic interaction theory; it was abandoned since it predicted a Poisson’s ratio of 1/4 for all materials. The elasticity theory currently accepted as classical allows Poisson’s ratios in isotropic materials in the range -1 to 1/2. Common materials exhibit a Poisson’s ratio from 1/4 to nearly 1/2. We have prepared materials with a Poisson’s ratio as small as -0.8. Deformation mechanisms in these materials include relative rotation of micro-elements, and non-affine micro-deformation. The relation between properties and structure is exploited to prepare viscoelastic composites with high stiffness combined with high damping. Generalized continuum theories exist with more freedom than classical theory. For example, in Cosserat elasticity there are characteristic lengths as additional engineering elastic constants. Recent experimental work discloses a variety of cellular and fibrous materials to exhibit such freedom, and the characteristic lengths have been measured. In hierarchical solids structural elements themselves have structure. Several examples of natural structural hierarchy are considered, with consequences related to optimality of material properties.
Roderic Lakes

Weakly Nonlinear Conductivity and Flicker Noise Near Percolation

This short review describes developments over the last decade in the study of a special type of transport, namely, weakly nonlinear conductivity and the closely related conductance fluctuation noise, in percolation systems. These phenomena are much more sensitive than linear transport properties to the special microgeome-try of the infinite percolation cluster. As a result they exhibit a much richer critical behavior in the vicinity of a percolation threshold. In particular, the noise always as-sumes a maximum at the threshold and the weakly nonlinear conductivity may exhibit either monotonic or non-monotonic critical behavior, depending on the properties of the components. In the case of non-monotonic behavior, the nonlinear conductivity at the threshold may be larger than that of both components, which leads to an enhancement of nonlinear effects in such systems. Either the good conductor or the poor conductor may dominate the critical behavior both above and below the threshold. These prop-erties were studied by numerical simulations on random resistor networks, an effective medium type approximation and a scaling theory.
Ohad Levy

Fine Properties of Solutions to Conductivity Equations With Applications to Composites

The simplest possible approach in bounding effective moduli enjoying a variational definition, consists in plugging into the variational principle an “admissible test field”. If only the volume fraction of the phases is known, the traditional approach is to use a field which is constant throughout the media. In the context of conductivity, this very simple choice leads to the so called Wiener (or harmonic and arithmetic mean) bounds. On the other hand rank one laminates provide us with composites which saturate these elementary bounds.
If one considers “sum of energies” and restricts attention to isotropic composites, Wiener bounds are no longer attainable. This is because the constant test field no longer matches the actual solution for any microgeometry.
We explain how to use recent results from the theory of quasiconformal mappings, to make a more flexible and in general more appropriate choice of test field when a sum of energies is considered. Our method only requires information about the volume fraction and, if used in combination with the translation method, is at least as successful as any other known method.
The method, as developed so far, works only in two dimensional conductivity. However it applies to non linear problems as well. In the non linear context, when applied to a polycrystalline material, the method delivers a new lower bound when the energy is subquadratic. The only other known method to do so has been recently developed by D. Talbot and J. R. Willis. A brief review of recently established very fine properties of solutions to the conductivity equations in the linear context will be given.
V. Nesi

Composite Sensors and Actuators

Composite materials have found a number of structural applications but their use in the electronics industry has been relatively limited. As the advantages and disadvantages of electroceramic composites are better understood, we can expect this picture to change.
Robert E. Newnham

Bounding the Effective Yield Behavior of Mixtures

The yield surface of a mixture of elastic/plastic materials is examined through mathematical techniques. A variational principle is derived from the local yield behavior (which may be anisotropic) and is used to generate upper bounds on the effective yield surface of the mixture. These bounds effectively reduce estimates on the nonlinear problem of yield to estimates on the elastic properties of mixtures oflinearly elastic“reference materials.” Upper bounds obtained from the variational principle are shown to be tighter than previous bounds, provided one uses a linear bound which is tighter than the arithmetic-mean bound. Finally, explicit bounds are derived for the yield surface of anisotropic mixtures of two von Mises type materials when one phase takes the form of ellipsoidal inclusions embedded in a matrix of the second phase, and difficulties arising in applications to polycrystals are discussed.
Tamara Olson

Upper Bounds on Electrorheological Properties

Electrorheological (ER) fluids are a class of materials whose rheological properties transform from liquid-like to solid-like upon the application of an external electric field. We consider the simplest model of ER fluids: a collection of identical dielectric solid spheres dispersed in a liquid. By transforming the problem to one of effective dielectric constant optimization, it is shown that the ground state of the ER fluid and its various electrical and rheological properties may be calculated from first principles through the Bergman-Milton representation. In particular, we obtain the upper bounds on the dielectric constant, the shear modulus, and the static yield stress of the ER fluid in its high field (solid) state.
Ping Sheng, Hongru Ma

On Spatiotemporal Patterns in Composite Reactive Media

Motivated by experimental results of spatiotemporal pattern formation during heterogeneous chemical reactions on composite catalyst surfaces, we present a numerical study of such phenomena on model one-dimensional reactive media. Typical composite geometries employed in the simulations are in the form of alternating catalyst stripes with different reactivity (media with spatially varying kinetic constants). By varying the geometry and nature of the composite, and using the system size as bifurcation parameter, we explore a wealth of dynamic patterns, ranging from nonuniform steady states and effective travelling pulses to spatiotemporal chaos. We use simulation as well as numerical continuation, bifurcation and stability analysis techniques in an attempt to characterize and classify the nature and symmetry of these solutions and their transitions.
S. Shvartsman, A. K. Bangia, M. Bär, I. G. Kevrekidis

Equilibrium Shapes of Islands in Epitaxially Strained Solid Films

We calculate the equilibrium morphology of a strained layer, for the case where it wets the substrate (Stranski-Krastonow growth). Assuming isotropic surface energy and equal elastic constants in the film and substrate, we are able to calculate two-dimensional equilibrium shapes as a function of the island size and spacing. We present asymptotic results for the equilibrium shape of a thin island where the island height is much smaller than the island width. We also present numerical results of the full equations to describe the island shape for separated islands, allowing us to characterize the features of the island morphology as a function of island volume.
Brian J. Spencer, J. Tersoff

Numerical Simulation of the Effective Elastic Properties of a Class of Cell Materials

This study deals with the effective properties of a special class of two-phase linear composite materials called cell materials. This class is generated by covering the whole space by identical cells and by randomly choosing the properties in each individual cell. First, bounds and estimates for two-dimensional isotropic elastic systems are recalled. Second, the results of numerical simulations for specific microstructures similar to cell materials are presented. The ensemble average of the numerical simulations is well approximated by the prediction of the classical self-consistent scheme.
Pierre Suquet, Hervé Moulinec


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