Preliminaries to Studying Non-Euclidean Geometry

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Introduction

Continuum mechanics applies the tools of differential geometry to study stress and strain fields within a material by treating that material as an idealized continuum. In most cases, we assume that we assume Euclidean geometry for the continuum, but there are several situations when the assumptions of Euclidean geometry no longer apply.

One example is using the tools of continuum mechanics at astronomical length-scale where the effects of gravity, i.e. the curvature of space, become non-negligible, such as when studying neutron stars. [1] [2]

Another example is when applying continuum mechanics at the mesoscale to study material defects such as dislocations, voids and disclinations. [3] [4] [5] Normally, the continuum idealization would have meant that we cannot model explicitly discontinuities in the material such as due to dislocations and disclinations, but as the aforementioned references show, this problem can be overcome by applying non-Euclidean geometry to the discontinuous material. Here is an intuitive way to see how non-Euclidean geometry helps: Consider a wall map of the Earth and notice that any such map would have a one or more discontinuities, e.g. the line where east-west hemispheres appear separated on the projection. A globe, however, does not manifest such discontinuity, because the globe's geometry is no longer Euclidean (flat) but spherical.

There are other potential future applications for why it maybe worth pursuing the generalization of continuum mechanics to non-Euclidean geometry. For example. non-Euclidean computational models could be more accurate as they will be able to naturally handle discontinuities and incompatibilities.

Nevertheless, understanding how the geometry of space should reflect in the equations of continuum mechanics is a challenging exercise, which is why it is not a part of the typical continuum mechanics curriculum. Non-Euclidean geometry is often veiled in highly abstracted mathematical language (tangent bundles, vector fields, Lie groups and Lie algebra, differential forms, ... ). While some of these abstractedness is necessary to ensure that we are not overly limiting or overly extending the applicability of the math, it is often more helpful to first grasp the basic concepts intuitively and then grok the generalization.

Here we will focus on an attempt to intuitively present the most basic abstractions in a way that would hopefully lend itself to better intuitive and not just syntactic understanding. The following sections attempt to introduce and motivate some of the key vocabulary of differential geometry. The reader is encourage to lookup the references for further reading.

Preliminary concepts: Metric (like a measuring stick), Co-variant and Contra-variant vectors (different representations of a vector once we have have a metric defined), Geometric connections: (how can we compare vectors at different points in space),

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References

1. Gerard A Maugin, Magnetized deformable media in general relativity, Annales de l'I.H.P., section A, tome 15, #4(1971), p.275-302
2. G.A.Maugin, On the covariant equaitons of the relativistic electrodynamics of continua. I. General equations, Universite de Paris VI, Laboratoire de Mecanique Theorique associe au C.N.R.S., tour 66, 75230 Paris, Cedex 05, France
3. Kazuo Kondo, ON THE ANALYTICAL AND PHYSICAL FOUNDATIONS OF THE THEORY OF DISLOCATIONS AND YIELDING BY THE DIFFERENTIAL GEOMETRY OF CONTINUA, International Journal of Engineering Science, Vol. 2, pp. 219-251, Pergamon Press 1964
4. John D. Clayton, Douglas J. Bammann, and David L. McDowell, A Geometric Framework for the Kinematics of Crystals With Defects, Philosophical Magazine, vol 85, nos. 33-35, pp. 3983-4010, February 2005
5. J.D. Clayton, D.L. McDowell, D.J. Bammann, Modeling dislocations and disclinations with finite micropolar elastoplasticity, International Journal of Plasticity, 22 (2006) 210-256