Valery Smyshlyaev has deployed his distinctive combination of powerful analytical technique and deep physical insight in two distinct areas of applied mathematics. One of these is the theory of diffraction. The other concerns the mechanics of composite materials, and related problems of materials science. The two areas are disjoint, except that both make use of asymptotic methods.

The geometrical theory of diffraction combines the high-frequency asymptotic approximation of ray theory, which deals with the propagation of waves, with the solution of certain canonical problems, of reflection or diffraction of waves from obstacles. The most elementary canonical problem is that of reflection from a plane boundary. However, diffraction from corners has also to be understood. Until a few years ago, the canonical problem of diffraction from a conical point remained outstanding. It was solved by Smyshlyaev. He reduced the problem to one of solving a one-dimensional integral equation around a cross-section of the cone. This integral equation is then treated numerically. The technique was first developed in the (scalar) context of acoustics; it has recently been extended to electromagnetic waves, with important applications relating to radar and to telecommunications.

Composite media typically display very strong heterogeneity on a very small scale (the microscale), while appearing smooth at a larger scale (the mesoscale). Smyshlyaev has made key contributions to the mathematical theory of “homogenization”, which concerns the derivation of the smooth mesoscale properties from the detail of the properties of the constituent materials, and their local geometry, at the microscale. Smyshlyaev developed practically-useful approximations, of “self-consistent” type, for problems in which the macroscopic variation of the stress and strain fields is not sufficiently slow for the “homogenization limit” to provide an adequate description. This occurs, for example, during the propagation of ultrasonic waves through a composite. He also obtained results for composites containing spheroidal inclusions, either aligned or randomly oriented, including the important limiting case of a body weakened by cracks. Smyshlyaev considered material nonlinearity, extending the “self-consistent” methodology to the dynamics of a body containing cracks. Subsequent work treated static problems for composites, mostly displaying nonlinear material response, obtaining strict bounds on material response, linear as well as nonlinear. He has recently combined the “translation method” with another established variational structure (due to Talbot and Willis) to obtain the best bounds known at present for the yield stress of a model rigid-plastic polycrystal. He has also made a study of meso-scale non-locality of response, when the constituent materials are conventionally elastic.

Some of his most profound work applies some of the methodology of homogenization to finding bounds on the response of a material such as a shape-memory alloy,that changes phase in response to applied macroscopic deformation, so as to minimize its energy. The first innovation here was to formulate the problem as one of optimizing the “H-measure” associated with the microstructure. This simultaneously introduced flexibility and generality (any anisotropic elastic response could be treated), and directly confronted the essential physical quantity involved. The case of a material with three energy wells was treated in detail. When the optimal H-measure is attainable, the exact energy is found; when it is not, the construction yields a lower bound for the energy (an upper bound could be obtained by optimizing over a subset of H-measures, known to be attainable). Related methodology was applied to obtain bounds on the response of phase-transforming polycrystals. In other on-going work, Valery is using techniques of asymptotic analysis, together with ergodic theory, to address problems concerning the kinetics of such phase transformations, studying “gradient flows” when the energy landscape contains fine-scale undulations (“wiggly energies”).

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