The investigator studies problems related to composite materials, martensitic phase transformations, and morphological stability in materials. First, he considers exact relations and links for effective tensors of fiber-reinforced elastic composites, by applying the general theory developed by him and his collaborators. The general theory provides a strategy for finding every relation and link. However, the actual execution of that strategy is far from trivial. The case of fiber-reinforced elastic composites is both important for applications and incredibly challenging technically, because the microstructure is two-dimensional while the properties of the composite are represented by three-dimensional fully anisotropic elastic tensors. This topic builds on the successful solution of a similar, simpler problem in the context of Hall-effect conductivity. A second problem concerns morphological stability of phase boundaries in materials capable of undergoing martensitic phase transformations. The investigator studies the possibility of a configuration in which the only mode of instability is the global motion of the phase boundary. Mathematically this corresponds to the failure of the generalized second variation to stay positive. The difficulty is that all other stability criteria are required to be satisfied. The resolution of this issue opens the way to a complete understanding of all instabilities of configurations with smooth phase boundaries. In the context of periodic composites, morphological instabilities may occur locally within a period cell. Their interaction with the effective behavior of a composite is investigated, following prior work for structural and material instabilities.
Heterogeneous media, or media with internal structure, are of great importance in applications. Composite materials, which have by now become ubiquitous, are one example. Another example is shape memory alloys and other smart materials. These two groups of materials are very different, yet have some important common features. A well-known connection between the two theories can be used to shed light on the less well-understood theory of martensitic phase transformations responsible for much of the shape memory behavior. One aim of this project is to understand elastic properties of fiber-reinforced elastic composites. Such composites may combine two cheap but imperfect materials to produce a new material that is either rare or not found in nature at all. For example, modern skis use composites to create a material that bends easily but is incredibly stiff with respect to torsion. However, there are limitations to what can be achieved by composites. The investigator maps out those instances where the benefits of creating composites are dramatically reduced due to the presence of "exact relations," i.e. relations between material moduli that cannot be altered by making composites, no matter how hard one tries. The present work focuses on a very widely used class of fiber-reinforced composites. Another aim of the project is to understand instabilities in models of non-linear elasticity and shape memory alloys. One widely known type of elastic instability is buckling. Martensitic phase transitions responsible for shape memory behavior are another kind of instability. These examples show how important instabilities are in applications. The investigator classifies all possible instabilities in a systematic way, with the secondary goal of solving one of the long-standing mathematical problems: finding "correct" sufficient conditions for stability.
" dealt with two interconnected themes: heterogeneous media with microstructure and instability. Instability in materials capable of martensitic phase transformations, for instance, leads to the creation of a microstructure. The buckling instability of axially compressed cylindrical shells also leads to the emergence of multiscale deformations that can be regarded, at least mathematically, as microstructures.This project is a part of a larger effort to shed more light on the fundamental questions of formation of microstructure via instabilities and its effect on the overal properties of materials. The latter question comes to the forefront in the theory of composite materials. More specifically, the PI investigated the fundamental properties of homogenization as applied to fiber-reinforced elastic composites, buckling instability in axially compressed circular cylindrical shells, and stability of phase boundaries in nonlinearly elastic materials. The goal of the first part of the project was to achieve a deeper understanding of the effective properties of fiber-reinforced composites, which are commonly used in industry. For example, injecting carbon fibers into epoxy is a widely used method for creating strong and lightweight materials. Steel-reinforced concrete is another example. The project focused on finding all exact relations between a composite's constituent materials and its effective behavior that hold regardless of the microstructure. In this way exact relations express fundamental properties of homogenization - the theory underlying all mathematical models of composite materials. Building on the decisive advances made by the PI's Ph.D. student Meredith Hegg, the PI was able to obtain the complete list of exact relations in the contex of fiber-reinforced elastic composites. It consists of 23 beautiful formulas, with deep implications about the anisotropy of composites, which can be mined by practitioners to improve the design and manufacture of composites. For instance, the independence of exact relations of the particular details of the microstructure can be exploited to test the effects of the violations of such commonly made assumptions as perfect adhesion and absence of prestress. This project had a substantial educational impact. In addition to training a new Ph.D., the PI involved two undergraduate students in research, one of whom has decided to pursue a Ph.D. in mathematics. Also, two graduate students from other disciplines contributed to the project with computational work, engaging in cross-disciplinary research. The second theme addressed in the project was instability. One commonly encountered type of instability is buckling. In his prior work, the PI and his collaborator Lev Truskinovsky developed a general theory of buckling of slender structures. In this project, this theory was further developed and applied to the buckling of cylindrical shells, yielding the first rigorous derivation of the classical formula for buckling load. The new method of buckling analysis also revealed that random perturbations of stress in the body would lead to a different scaling of the buckling load as a function of shell thickness. It turns out that the classical formula predicts a buckling load 4 to 5 times larger than observed in experiments. The reason for this discrepancy is the buckling load's high sensitivity to imperfections. This suggests that the PI's method, if applied to an imperfect cylindrical shell, may give a quantitative characterization of the effect of imperfections on the actual buckling load. The key outcome of this research was the observation that the scaling of the buckling load, corresponding to a randomly perturbed stress field, agrees with the one observed in experiments. This suggests a possible avenue of future research and lends credibility to the PI's novel approach to understanding imperfection sensitivity. In the course of this project, the PI trained a Postdoctoral Associate, Davit Harutyunyan, who is now seeking an academic position at a research university. Mathematical analysis of stability is especially challenging for materials undergoing martensitic phase transformations, such as shape memory alloys. In the process of a phase transformation, a new phase (martensite) nucleates in the matrix of the original phase (austenite) and continues to grow under the influence of stress or temperature changes. The surface separating the two phases is called a phase boundary, and as a result of this project, its stability is now better understood. The project's idea was to distinguish the stability of each individual phase from the stability of the phase boundary itself. The outcome of the project was the rather surprising realization that when each individual phase is stable, the only local instability associated with the phase boundary is "material interchange", causing roughening of the phase boundary. This improved understanding of local stability of phase boundaries should be incorporated into the analysis of the global structure, which is one of the the directions of the PI's future research project.
