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Title
Abstract
Introduction
Local Charts on the Material Base Space
The Resulting Expression is
Summary and Conclusion
References
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Mathematics
Geometry

Cartan media: geometric continuum mechanics in homogeneous spaces

Profile image of Lukas KikuchiLukas Kikuchi

2023, arXiv (Cornell University)

https://doi.org/10.48550/ARXIV.2310.01388
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32 pages

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Abstract

We present a geometric formulation of the mechanics of a field that takes values in a homogeneous space X on which a Lie group G acts transitively. This generalises the mechanics of Cosserat media where 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 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.

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Raffaele Barretta

International Journal of Non-Linear Mechanics, 2013

The need for a proper geometric approach to constitutive theory in non-linear continuum mechanics (NLCM) is witnessed by lasting debates about basic questions concerning time-invariance, integrability, conservativeness and frame invariance. Our aim is to bring geometry to play a central role in theoretical and computational issues of NLCM. This demand is imposed by the present state of art, dominated by a mainly algebraic approach which, being a modified heritage of the linearized theory, is inadequate to manage concepts and methods in a non-linear framework. A proper definition of spatial and material fields and the statement of the ensuing covariance paradigm, provide a firm foundation to the theory of constitutive behavior in NLCM. The notion of constitutive frame invariance (CFI) is introduced as geometric correction to the formulation of material frame indifference (MFI). Standard models of constitutive behavior are critically discussed and compared with the ones consistent with the new approach. The outcome is a physically testable theory which eventually results in new effective computation tools for structural engineers.

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