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Mathematical Research in Materials Science: Opportunities and Perspectives
interactions among defects. Such interactions are responsible for defect aggregation. In modeling internal fields, questions arise regarding what is the statistical nature of defect distribution, what (if any) correlations there are between defects, and what length scales are required to take these correlations into consideration. Defects appear to have many intermediate configurations that may be near local minima of the energy. In response to internal and external fields, defect configurations evolve in time in a complex manner. For example, the way that dislocations arrange and rearrange into network, cell, and subgrain boundaries and eventually into grain boundaries during deformation is not yet understood. The evolution of defects is highly dynamic in nature and presents significant challenges to materials scientists and mathematical scientists. Currently, it is simply not known whether there exists a unique minimum or multiple local minima for groups of dislocations.
The modeling of the evolution of defects is important in materials science because, in many cases, defects directly control microstructural evolution, and knowledge of microstructural evolution is a basis for designing thermomechanical treatments in industry. Defects determine the mechanical constitutive equations in nearly all cases: plastic deformation, fatigue, and creep. Constitutive equations are usually formulated at the continuum length scale and omit the details at the mesoscopic scale. Many (but not all) deformation and fracture problems of interest to materials scientists and structural engineers are well defined. Applied mathematicians, known as applied mechanicians, have for years had a tremendous impact in this domain in analyzing deformation and fracture problems (elasticity, plasticity, fracture mechanics, and micromechanics). Standard mathematical methods, such as partial differential equations, singularity analysis, finite-element methods, and so on have been intensely utilized and further developed in this area. Applied mathematicians have contributed to creating materials with improved mechanical performance that provide greater margins of safety in structural engineering applications.
In many materials, especially brittle materials, eventual failure is attributed mostly to inherent flaws in the materials. Often, the most severe flaws control the probability of failure of the material. This situation is not unique to mechanical failure; it is also encountered in other types of failure such as dielectric breakdown and failure in superconducting networks. Treatment of such problems demands statistical methods. In recent years, however, materials scientists have also resorted to discrete modeling to treat such problems at a scale traditionally appropriate for continuum mechanics.
In the following sections, additional problem areas are presented, including modeling methodology, mechanics of defects and interfaces, plasticity and fracture, large local fields and instability in random systems, dynamic fracture, liquid crystals, and a few topics concerning the structure of interfaces on the atomistic level and the macroscopic properties that result. They provide a glimpse into some of the current activities in the areas of defects, deformations, and interfaces and point out some issues that may interest mathematical scientists. These problem areas are representative but not exhaustive, and the reader is encouraged to refer to the large current literature of applied mechanics (for example, appearing in the Journal of Applied Mechanics) and materials science (for example, appearing in the Journal of Materials Research, the Journal of the American Ceramics Society , and so on) for further exploration of this field.