In this section attention is focused on problems in analysis and modeling in which the relevant length scales are on the order of microns or more, that is, much larger than the distances between neighboring atoms. When modeling the behavior of materials at such large length scales, one generally does not need to keep track of the positions of the individual atoms. Rather, it suffices to deal with local average properties—e.g., density, temperature, strain, and magnetization—and to describe the behavior of these quantities by continuum equations in which it is assumed that all variations are extremely slow when viewed on atomic length scales. Thus one uses diffusion equations to describe the transport of heat, composition, or chemical reagents; hydrodynamic equations to describe the motion of fluids; and elasticity theory to relate strains in solids to applied stresses. Of course, many of the ingredients of such models—the transport coefficients, for example—ultimately are determined by fundamental, atomistic principles. But the basic point of view is macroscopic in the sense that it pertains to length scales that are much larger than atomic, and classical in the sense that it makes no explicit use of quantum mechanics.

The advent of the computer has brought about an important change in the perspective from which scientists view continuum analysis. Because continuum models in principle derive from atomistic theories, they often have been viewed as less fundamental, less of a venture into uncharted territory, less apt to produce surprises. It now seems that just the opposite may be true, at least as regards many of the questions that are most relevant to materials research. Now that we actually can explore the consequences of the continuum models—whose ingredients have been known and trusted for decades or longer—we are discovering a wealth of unexpected phenomena and challenging mathematical problems.

In order to describe the implications of some of these recent developments, two broad classes of continuum problems that are part of the traditional core of materials research—microstructural solidification patterns in alloys and fracture mechanics—are discussed below. These are by no means the only areas of materials research where analysis and modeling at continuum length scales are appropriate. Note, for example, the wide variety of materials processing problems in which hydrodynamics is important, or the yet more complicated problems in which fluid motion is coupled to diffusion and chemical reactions. Some of these more complex modeling problems will be referred to toward the end of this section in the discussion of integrated approaches to materials technology.

Microstructures in Alloys

When a molten alloy is solidified by quenching, its chemical constituents tend to segregate. This happens even in situations where equilibrium ther-

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