Mathematics of Multiscale Materials

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.

Inhalt
Scaling limit for the incipient spanning clusters.- Bounded and unbounded level lines in two-dimensional random fields.- Transversely isotropic poroelasticity arising from thin isotropic layers.- Bounds on the effective elastic properties of martensitic polycrystals.- Statistical models for fracture.- Normal and anomalous diffusions in random flows.- Calculating the mechanical properties of materials from interatomic forces.- Granular media: some new results.- Elastic freedom in cellular solids and composite materials.- Weakly nonlinear conductivity and flicker noise near percolation.- Fine properties of solutions to conductivity equations with applications to composites.- Composite sensors and actuators.- Bounding the effective yield behavior of mixtures.- Upper bounds on electrorheological properties.- On spatiotemporal patterns in composite reactive media.- Equilibrium shapes of islands in epitaxially strained solid films.- Numerical simulation of the effective elastic properties of a class of cell materials.