December 28, 2011 Wednesday 15:30
Mikhail Katanaev, Steklov Mathematical Institute, Russian Academy of Sciences
Geometric theory of defects

Abstract:Defects in elastic media define almost all physical properties of solids, and therefore theory of defects is of great importance for applications. One attempt to construct a fundamental theory of defects - dislocations and disclinations - in elastic media is based on Riemann-Cartan geometry where metric and torsion are considered as independent physical variables. The torsion two-form is interpreted as the surface density of Burgers vector for dislocations, and the curvature two-form yields the surface density of Frank vector for disclinations. The elasticity equations enter the theory through the gauge conditions on the triad. In a similar way, the Lorentz gauge for the SO(3)-connection yields the SO(3)-principle chiral field theory as the gauge condition. It means that the geometric theory of defects in the absence of dislocations and disclinations reduces to the elasticity theory for the displacement vector field and to the SO(3)-principal chiral field theory for the spin structure. This geometric approach not only reproduces some simple results obtained within the ordinary elasticity theory but also permits one to consider continuous distribution of defects. At the moment the Riemann-Cartan geometry provides a link between solid state physicists and general relativists: the former obtain a mighty mathematical background for solving the problems while the latter obtain a new insight in the physical foundation of gravity theory.