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Advances in Mathematical Physics
Volume 2015, Article ID 828475, 11 pages
http://dx.doi.org/10.1155/2015/828475
Review Article

On Finsler Geometry and Applications in Mechanics: Review and New Perspectives

1Impact Physics, US ARL, Aberdeen, MD 21005-5066, USA
2A. James Clark School of Engineering (Adjunct Faculty), University of Maryland, College Park, MD 20742, USA

Received 21 November 2014; Accepted 18 January 2015

Academic Editor: Mahouton N. Hounkonnou

Copyright © 2015 J. D. Clayton. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In Finsler geometry, each point of a base manifold can be endowed with coordinates describing its position as well as a set of one or more vectors describing directions, for example. The associated metric tensor may generally depend on direction as well as position, and a number of connections emerge associated with various covariant derivatives involving affine and nonlinear coefficients. Finsler geometry encompasses Riemannian, Euclidean, and Minkowskian geometries as special cases, and thus it affords great generality for describing a number of phenomena in physics. Here, descriptions of finite deformation of continuous media are of primary focus. After a review of necessary mathematical definitions and derivations, prior work involving application of Finsler geometry in continuum mechanics of solids is reviewed. A new theoretical description of continua with microstructure is then outlined, merging concepts from Finsler geometry and phase field theories of materials science.