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# Brief Introduction to Mathematics of Materials Science

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Advances in technology are often due to fundamental advances in our understanding of the materials of which things are made. Obvious examples include the transistor and superconductors. Promising new technologies have been spawned by the recent discovery of Photonic Band Gap Crystals. Less spectacular but equally important developments in materials science are providing steady improvements in infrastructure, aerospace, and micro-electronics. The study of materials science has evolved to the point at which scientists from physics, engineering and applied mathematics are working on problems of common interest.

The need to describe material behavior resulting from physical interactions across several length scales is forcing scientists to develop new mathematics. One of the ongoing goals of the newly formed research group in materials science at LSU is the development of the mathematical underpinnings that facilitate the rational design of materials for applications.

Prof. Yaniv Almog studies the Ginzburg-Landau model of superconductivity. Superconductors are metals that below a certain critical temperature lose their electrical resistivity. Strong applied magnetic fields destroy superconductivity and revert the material to the normal state even if the temperature is lower than the critical one. If the applied magnetic field is then decreased superconductivity nucleates once again. In the course of nucleation one observes, both analytically and experimentally, first a thin superconducting boundary layer and then a bifurcation of a periodic solution away from the boundary.

The tools necessary in the study of these problems are mostly analytic: direct methods of the calculus of variations, semi-classical analysis, linear bifurcation analysis etc. There is also need for formal asymptotic and numerical calculations. Other physical models analogous in some sense to the Ginzburg-Landau model that exhibit a similar pattern of behavior include liquid crystals, Bose-Einstein condensates and more.

Prof. Yuri Antipov's research involves integral and functional equations of fracture mechanics, atomic diffusion in thin films and electromagnetic and acoustic scattering. It was only recently discovered that cracks can propagate faster than the speed of sound waves in a solid. Prior to this being observed in the laboratory, it was believed that such rapid propagation was impossible. Now it is being exploited in contexts as diverse as the failure of certain advanced engineering structures, and explanation of the damage patterns caused in certain earthquakes. Dr. Antipov develops the mathematical solutions using the Wiener-Hopf technique for describing this type of propagation when the medium is viscoelastic. Atomic diffusion from a material surface into grain boundaries is an important mechanism for stress relaxation and deformation in thin metal films that are now widely used in micro-electronics. There is still a general lack of rigorous mathematical modeling of this important class of materials phenomena. He analyzes mathematical models of grain boundary diffusion and shielding of a material wedge by dislocations. These models are described by singular integro-differential equations which are solved exactly by the machinery of the Riemann-Hilbert problem on an unclosed contour, or numerically by the Bubnov-Galerkin method.

Analytical devices are also essential for the geometrical theory of diffraction which deals with high frequency asymptotics. This theory needs solving canonical problems of acoustic and electromagnetic scattering with effective boundary conditions describing different types of the surfaces of scatters (e.g. anisotropic impedance plates and perforated sandwich panels). Dr Antipov works out a new analytical method for a class of vector functional Wiener-Hopf equations and vector-difference equations arising in scattering theory. The technique uses the theory of the Riemann-Hilbert boundary-value problem, Riemann surfaces of algebraic functions and Jacobi's inversion problem.

Prof. Blaise Bourdin's research interests are in the mathematics of material science and topology optimization. One
of the main concepts in these fields is the
*energy functional*. Choosing the correct energy functional in an optimizing process is crucial, for example in the
theory of
*fracture mechanics* and
*edge recognition*. Efficient
*numerical methods* and
*parallel processing* are important in the practical application of the theory. This aspect of the research is supported
by LSU's new Beowulf Cluster parallel process computer. The large cluster is in the Computer Center on campus, and the
department has a substantial smaller version which is compatible for testing software in preparation for running on the
big machine.

Prof. Robert Lipton investigates the effect of microstructure on the solutions of Partial Differential Equations. The aim is to develop mathematical tools for the understanding of physical processes that occur across several length scales. Accurate modeling of multi-scale phenomena is necessary for the development of computational tools for the design of high performance composites.

Recent work of Lipton seeks to reveal the interaction between applied forces and microstructure and its effect on failure initiation in composites. A second research direction characterizes the macroscopic behavior of random media with bulk and surface transport processes occurring at the scale of the microstructure. Dr. Lipton represented the American Mathematical Society at the 11th annual exhibition of the Coalition for National Science Funding held June 21, 2005 on Capitol Hill. He presented an exhibit on Mathematics for Advanced Composites Technology.

Prof. Daniel Sage investigates the G-closure problem for composite materials. A composite is a material that, while appearing homogeneous at a macroscopic level, is actually a mixture of two or more materials with the mixing occurring on a scale much larger than the atomic scale. A central problem in the theory of composites is the study of how physical properties of composites such as conductivity and elasticity depend on the properties of their constituents. In general, these properties depend strongly on the microstructure. A natural question thus arises. For fixed materials, what is the set of all possible values of a given physical property obtained as one varies the microstructure of the composite? This set is called a G-closure; it will be a subset of an appropriate tensor space.

The general description of G-closure sets is difficult and seems intractable with current techniques. A more accessible problem is suggested by the fact that generically, the G- closure will have nonempty interior in the given tensor space. Intuitively, this means that small changes in microstructure typically allow for small changes in the effective tensor of the composite in any direction in the tensor space. This, however, does not always occur; in exceptional cases, the G-closure set degenerates to lie in a submanifold, which is called an exact relation. Finding an exact relation can be viewed as proving an ``impossibility theorem'' in engineering; namely, they describe situations where an engineer can know {\em a priori} that a composite with desired properties cannot be manufactured from a given set of constituents. Dr. Sage, in recent joint work with Grabovsky and Milton, has shown how the classification of exact relations can be reduced to purely algebraic problems involving group representation theory and has developed new algebraic techniques to solve these algebraic problems. More generally, Dr. Sage is interested in applications of algebra and representation theory to material science.

One of
Prof. Stephen Shipman's interests is in photonic crystals. In electronic devices such as antennas, wave guides, filters,
and lasers, materials that have
*periodic* electromagnetic properties are of fundamental importance. Such structures are called
*photonic crystals*, and their distinguishing characteristic is that they prohibit the passage of electromagnetic
frequencies in certain spectral regions called
*band-gaps*. The other important feature is that specific
*eigenvalue* frequencies resonate with the structure, resulting in special physical phenomena such as surface waves
and plasmons and anomalous response to applied fields. This interesting behavior occurs in the regime in which the wavelength
scale is at the same order of magnitude as the periodic structure. For example, microwaves will interact in an interesting
way with a screen containing an array of holes on the order of a centimeter apart. The underlying mathematical theory
of photonic crystals is the
*Floquet-Bloch* theory of
*pseudo-periodic waves*.

Electromagnetic resonance in a random crystal |

*Resonant phenomena* in photonic crystals are facilitated by
*defects* in their structure, such as cavities, passageways, or surface irregularities. For the important class of
structures that consists of
*infinite slabs* (or sheets) or infinite pillars of a photonic crystal material, defects have a profound effect on
the way electromagnetic waves are scattered, transmitted, and reflected by the structure.
*Dispersion relations* for the traveling
*modes* and
*quasi-modes*, or leaky modes, play a central role in the theory. We are thus presented with a rich source of possibilities
for designing crystals that have desired properties, such as enhancement of transmission or reflection of specific frequencies
or the guiding of specific frequencies through a channel. A variety of theoretical and numerical techniques are used
in the study of photonic crystals, typically in combination. They include Green's functions, boundary-integral methods,
energy methods, spectral theory, asymptotic methods, and finite element and finite difference numerical schemes.

You are invited to visit the Applied Analysis Seminar.