Cartan media: geometric continuum mechanics in homogeneous spaces

Abstract

We present a geometric formulation of the mechanics of a field that takes values in a homogeneous space \mathbb{X} on which a Lie group G acts transitively. This generalises the mechanics of Cosserat media where \mathbb{X} is the frame bundle of Euclidean space and G is the special Euclidean group. Kinematics is described by a map from a space-time manifold to the homogeneous space. This map is characterised locally by generalised strains (representing spatial deformations) and generalised velocities (representing temporal motions). These are, respectively, the spatial and temporal components of the Maurer-Cartan one-form in the Lie algebra of G. Cartan's equation of structure provides the fundamental kinematic relationship between generalised strains and velocities. Dynamics is derived from a Lagrange-d'Alembert principle in which generalised stresses and momenta, taking values in the dual Lie algebra of G, are paired, respectively, with generalised strains and velocities. For conservative systems, the dynamics can be expressed completely through a generalised Euler-Poincare action principle. The geometric formulation leads to accurate and efficient structure-preserving integrators for numerical simulations. We provide an unified description of the mechanics of Cosserat solids, surfaces and rods using our formulation. We further show that, with suitable choices of \mathbb{X} and G, a variety of systems in soft condensed matter physics and beyond can be understood as instances of a class of materials we provisionally call Cartan media.