ACM/CDS COLLOQUIUM: Geometrization of Continuum and Discrete Mechanics

Dr. Arash Yavari
School of Civil Engineering and Environmental Engineering
Georgia Institute of Technology

In this seminar we present some applications of modern differential geometry
in solid mechanics. Traditionally, mechanics has been formulated in Euclidean space, mainly for convenience and simplicity as in the early days of modern mechanics, engineering scientists were interested in the simplest possible models. Working with the simplest possible models for the practical problems of the time laid the foundations of continuum mechanics. However, experience in physics shows that, in general, configuration space of a physical system is not globally Euclidean; instead it can only be locally described by a Euclidean space. In other words, physical theories should be formulated on manifolds.

We first motivate the recent geometric studies in continuum and discrete mechanics by some important applications of geometric ideas, e.g. systematic discritizations of elasticity and elastic cloaking. We review some previous attempts in geometrization of different solid mechanics problems, e.g. continua with distributed defects, etc. We then discuss several applications of geometric techniques in solid mechanics, namely, covariant formulation of elasticity, understanding the nature of balance laws in continua with microstrcture, material evolutions and their role in continuum mechanics, covariant formulation of particle mechanics, kinematics of defect mechanics, and geometric discretization of elasticity.

Refreshments in 204 Firestone at 3:45pm.

http://www.acm.caltech.edu/colloq.shtml