Instabilities in Materials Science

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1714287Grabovsky The investigator and his collaborators study several important phenomena where the main underlying feature is instability, or extreme sensitivity to measurement errors, imperfections of shape, or load. Three projects, unified by the general idea of instability, are pursued. One of them deals with understanding instabilities in nonlinear elasticity associated with martensitic phase transitions, whereby sharp phase boundaries are observed experimentally. Such phase transitions are responsible for shape memory effects in alloys, and for a giant magnetostrictive effect exhibited by some materials, to name just a few examples. Another project applies new tools to analyze the extreme sensitivity to imperfections of the buckling stress of axially compressed circular cylindrical shells -- an essential structural component in a vast variety of structures, from grain silos to airplanes and space ships. The third project focuses on quantitative understanding of a well-known confluence of rigidity and flexibility -- a seemingly contradictory combination of properties -- of Herglotz functions that are ubiquitous in applications from materials science to nuclear physics. For example, the investigator's research quantifies the uncertainty due to experimental errors in the estimates of complex electromagnetic permittivity, which describes how a material interacts with electromagnetic waves of various frequencies from radio waves through visible light to gamma rays. This project constitutes Ph.D. research of a graduate student working under the investigator's guidance. The most salient feature in martensitic shape transformations, which in particular is responsible for shape memory effect, is the presence of phase boundaries -- sharp interfaces separating different martensitic variants. Stability of such phase boundaries can be completely characterized in terms of the concept of quasiconvexity. With the understanding that characterization of quasiconvexity is often regarded as unachievable, the investigator and his collaborators focus on eminently computable algebraic conditions of stability of phase boundaries and assess their "strength" through specific examples that show how far off algebraic conditions are from true answers, which could be computed either analytically or numerically. Another project deals with buckling of cylindrical shells -- one of the extensively studied, yet poorly understood problems in mechanics, where the classical formula cannot be used to predict buckling stress observed in experiments due to initial imperfections of load and shape. This project creates a mathematically rigorous theory of the buckling of slender bodies that is capable of explaining why initial imperfections of shape and load have strong influence on the buckling stress in some structures, while having negligible effect in others. The third project studies the somewhat paradoxical behavior of Herglotz functions, that are at the same time extremely rigid and surprisingly flexible. Its understanding contributes to several areas of mathematics and physics and answers many important questions. The problems of data validation, and extrapolation are addressed. This project constitutes Ph.D. research of the investigator's graduate student.