Variational Principles and Partial Differential Equations for Some Model Problems in Polycrystal PlasticityTuesday, September 30, 2008 - 15:15 , Location: Skiles 255 , Marian Bocea , North Dakota State University, Fargo , Organizer:
The yield set of a polycrystal may be characterized using variational principles associated to suitable supremal functionals. I will describe some model problems for which these can be obtained via Gamma-convergence of a class of "power-law" functionals acting on fields satisfying appropriate differential constraints, and I will indicate some PDEs which play a role in the analysis of these problems.
Tuesday, September 23, 2008 - 15:15 , Location: Skiles 255 , Michael Shearer , Department of Mathematics, North Carolina State University , Organizer:
Granular materials are important in a wide variety of contexts, such as avalanches and industrial processing of powders and grains. In this talk, I discuss some of the issues in understanding how granular materials flow, and especially how they tend to segregate by size. The segregation process, known scientifically as kinetic sieving, and more colorfully as The Brazil Nut Effect, involves the tendency of small particles to fall into spaces created by large particles. The small particles then force the large particles upwards, as in a shaken can of mixed nuts, in which the large Brazil nuts tend to end up near the lid. I'll describe ongoing physics experiments, mathematical modeling of kinetic sieving, and the results of analysis of the models (which are nonlinear partial differential equations). Movies of simulations and exact solutions illustrate the role of shock waves after layers of small and large particles have formed.
Tuesday, September 9, 2008 - 15:15 , Location: Skiles 255 , Marta Lewicka , School of Mathematics, University of Minnesota , Organizer:
A longstanding problem in the mathematical theory of elasticity is to predict theories of lower-dimensional objects (such as rods, plates or shells), subject to mechanical deformations, starting from the 3d nonlinear theory. For plates, a recent effort (in particular work by Friesecke, James and Muller) has lead to rigorous justification of a hierarchy of such theories (membrane, Kirchhoff, von Karman). For shells, despite extensive use of their ad-hoc generalizations present in the engineering applications, much less is known from the mathematical point of view. In this talk, I will discuss the limiting behaviour (using the notion of Gamma-limit) of the 3d nonlinear elasticity for thin shells around an arbitrary smooth 2d mid-surface S. We prove that the minimizers of the 3d elastic energy converge, after suitable rescaling, to minimizers of a hierarchy of shell models. The limiting functionals (which for plates yield respectively the von Karman, linear, or linearized Kirchhoff theories) are intrinsically linked with the geometry of S. They are defined on the space of infinitesimal isometries of S (which replaces the 'out-of-plane-displacements' of plates), and the space of finite strains (which replaces strains of the `in-plane-displacements'), thus clarifying the effects of rigidity of S on the derived theories. The different limiting theories correspond to different magnitudes of the applied forces, in terms of the shell thickness. This is joint work with M. G. Mora and R. Pakzad.
Tuesday, September 2, 2008 - 15:00 , Location: Skiles 255 , Zhiwu Lin , School of Mathematics, Georgia Tech , Organizer:
Shear flow instability is a classical problem in hydrodynamics. In particular, it is important for understanding the transition from laminar to turbulent flow. First, I will describe some results on shear flow instability in the setting of inviscid flows in a rigid wall. Then the effects of a free surface (or water waves) and viscosity will be discussed.
Tuesday, August 26, 2008 - 15:00 , Location: Skiles 255 , Eldad Haber , Mathematics & Computer Science, Emory University , Organizer:
Optimization problems with PDE constraints are commonly solved in different areas of science and engineering. In this talk we give an introduction to this field. In particular we discuss discretization techniques and effective linear and nonlinear solvers. Examples are given from inverse problems in electromagnetics.
Tuesday, August 19, 2008 - 15:00 , Location: Skiles 255 , Giovanni Leoni , Carnegie Mellon University, Pittsburgh , Organizer:
In this talk we will discuss some recent existence and regularity results for equilibrium configurations of epitaxially strained crystalline films.