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

Advances in Mathematical Physics ◽

10.1155/2015/828475 ◽

2015 ◽

Vol 2015 ◽

pp. 1-11 ◽

Cited By ~ 6

Author(s):

J. D. Clayton

Keyword(s):

Materials Science ◽

Finsler Geometry ◽

Theoretical Description ◽

Mechanics Of Solids ◽

Metric Tensor ◽

Covariant Derivatives ◽

Special Cases ◽

Great Generality ◽

Primary Focus ◽

Mathematical Definitions

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.

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Geodesies of an affine connection and electromagnetism with radiation reaction

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700046505 ◽

1971 ◽

Vol 4 (2) ◽

pp. 225-240 ◽

Cited By ~ 1

Author(s):

R.R. Burman

Keyword(s):

Electromagnetic Field ◽

Affine Connection ◽

Radiation Reaction ◽

External Electromagnetic Field ◽

Geodesic Equation ◽

Conformally Flat ◽

Metric Tensor ◽

Special Cases ◽

Test Charge ◽

Point Test

This paper deals with the motion of a point test charge in an external electromagnetic field with the effect of electromagnetic radiation reaction included. The equation of motion applicable in a general Riemannian space-time is written as the geodesic equation of an affine connection. The connection is the sum of the Christoffel connection and a tensor which depends on, among other things, the external electromagnetic field, the charge and mass of the particle and the Ricci tensor. The affinity is not unique; a choice is made so that the covariant derivative of the metric tensor with respect to the connection vanishes. The special cases of conformally flat spaces and the space of general relativity are discussed.

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Generalized pseudo-Finsler geometry applied to the nonlinear mechanics of torsion of crystalline solids

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s021988781850113x ◽

2018 ◽

Vol 15 (07) ◽

pp. 1850113 ◽

Cited By ~ 2

Author(s):

J. D. Clayton

Keyword(s):

Boron Carbide ◽

Internal State ◽

Broad Class ◽

Torsional Rigidity ◽

Nonlinear Partial Differential Equations ◽

Finsler Geometry ◽

Coupled System ◽

Cylindrical Sample ◽

Deformable Solid ◽

Metric Tensor

A continuum theory of the mechanical behavior of solid materials is presented wherein fundamental geometric quantities such as the metric tensor and connection coefficients can depend on one or more director vectors, also called internal state vectors. This theory, referred to as generalized pseudo-Finsler geometric continuum mechanics, enables depiction of a very broad class of physical phenomena in deformable solid bodies. The general nonlinear theory is reported first, primarily summarizing prior work by the author. Next, a new application of the theory to torsional deformation of solids is presented, whereby a cylindrical sample of material may be simultaneously subjected to twisting, extension or compression along its axis, as well as possible radial confinement or contraction. The internal state variable, when normalized by a regularization length, is identified with an order parameter associated with inelastic deformation that may include slippage, fracture, and/or structural collapse. Evolution of the internal state follows a generalized Ginzburg–Landau type of kinetic equation. For axially homogeneous fields, a coupled system of nonlinear partial differential equations is obtained that can be integrated numerically. Results are first documented for generic solids representative of many crystals that exist in nature. Then solutions corresponding to realistic properties of crystals of boron carbide ceramic and ice are reported. Results for boron carbide predict a dominant effect of shearing over compression on the structural transformation process, in agreement with observations from atomistic simulations. Results for ice demonstrate periods of steady plastic flow under constant applied average shear stress as well as torsional rigidity varying with sample size. Existence of both phenomena agrees with experimental observations.

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ON TOLMAN'S MASS-ENERGY RELATION AND A NEW TOLMAN-TYPE RELATION

International Journal of Modern Physics A ◽

10.1142/s0217751x89000133 ◽

1989 ◽

Vol 04 (02) ◽

pp. 327-334

Author(s):

B. M. BARKER ◽

R. F. O'CONNELL

Keyword(s):

Gravitational Field ◽

Coordinate System ◽

Energy Relation ◽

Mass Energy ◽

Schwarzschild Solution ◽

Metric Tensor ◽

Harmonic Coordinates ◽

Isotropic Coordinates ◽

Special Cases ◽

Type Relation

Tolman derived the mass-energy relation [Formula: see text] using a particular choice of coordinates, viz. the Schwarzschild solution for the metric tensor in isotropic coordinates for a body of mass m at rest at the origin. Here we show that this relation retains the same form for the case of a very general coordinate system. The latter includes the Schwarzschild and harmonic coordinates as special cases. In addition, we give a new Tolman-type relation [Formula: see text]. The quantities [Formula: see text] and [Formula: see text] are the energy-momentum densities for matter and the gravitational field, respectively.

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On a projectively invariant pseudo-distance in Finsler geometry

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887815500437 ◽

2015 ◽

Vol 12 (04) ◽

pp. 1550043 ◽

Cited By ~ 1

Author(s):

Behroz Bidabad ◽

Maryam Sepasi

Keyword(s):

Nonlinear Analysis ◽

Ricci Tensor ◽

Finsler Geometry ◽

Finsler Metrics ◽

Analysis Method ◽

Covariant Derivatives

Here, a nonlinear analysis method is applied rather than classical one to study projective changes of Finsler metrics. More intuitively, a projectively invariant pseudo-distance is introduced and characterized with respect to the Ricci tensor and its covariant derivatives.

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On the applicability of bloch’s theorem to the total wave function in the dynamical theory of electron diffraction

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100131875 ◽

1992 ◽

Vol 50 (2) ◽

pp. 1448-1449

Author(s):

H. S. Kim ◽

S. S. Sheinin

Keyword(s):

Electron Microscope ◽

Electron Diffraction ◽

Materials Science ◽

High Energy ◽

Dynamical Theory ◽

Crystal Defects ◽

Special Cases ◽

Energy Approximation ◽

Diffraction Patterns ◽

Bloch’S Theorem

The dynamical theory of electron diffraction is widely used in materials science problems such as determining the contrast in electron microscope images of crystal defects, calculations of structure images and calculations of diffracted beam intensities in electron diffraction patterns. In carrying out these calculations, the high energy approximation is normally made and it is usually assumed that the crystal is in a symmetrical Laue orientation. In practice, however, a specimen in the electron microscope will generally be oriented so that the non-symmetrical Laue case is obtained. Even in those special cases where the symmetrical Laue case is obtained for the zero order Laue zone reflections, non-symmetrical Laue effects may occur if reflections in higher order zones are important. It has been shown in the literature that the Bloch functions are not orthogonal in more general forms of the dynamical theory in which the high energy approximation is not made and the nonsymmetrical Laue case is considered,.

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Curved Space

The Physical World ◽

10.1093/oso/9780198795933.003.0006 ◽

2017 ◽

Author(s):

Nicholas Manton ◽

Nicholas Mee

Keyword(s):

Arbitrary Dimension ◽

Spherical Geometry ◽

Flat Space ◽

Geodesic Equation ◽

Metric Tensor ◽

Curved Space ◽

Covariant Derivatives ◽

Christoffel Symbols ◽

Geometry Configuration ◽

Fisheye Lenses

This chapter develops the mathematical technology required to understand general relativity by taking the reader from the traditional flat space geometry of Euclid to the geometry of Riemann that describes general curved spaces of arbitrary dimension. The chapter begins with a comparison of Euclidean geometry and spherical geometry. The concept of the geodesic is introduced. The discovery of hyperbolic geometry is discussed. Gaussian curvature is defined. Tensors are introduced. The metric tensor is defined and simple examples are given. This leads to the use of covariant derivatives, expressed in terms of Christoffel symbols, the Riemann curvature tensor and all machinery of Riemannian geometry, with each step illustrated by simple examples. The geodesic equation and the equation of geodesic deviation are derived. The final section considers some applications of curved geometry: configuration space, mirages and fisheye lenses.

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Large deformation possible in every isotropic elastic membrane

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences ◽

10.1098/rsta.1977.0143 ◽

1977 ◽

Vol 287 (1341) ◽

pp. 145-187 ◽

Cited By ~ 27

Keyword(s):

Strain Energy ◽

Closed Surface ◽

Elastic Membrane ◽

Energy Response ◽

Normal Surface ◽

Metric Tensor ◽

Alternative Description ◽

Special Cases ◽

Static Solutions ◽

Elastic Membranes

This paper is concerned with static solutions of finitely deformed elastic membranes regarded as thin shells. It deals with deformations that can be maintained, in the absence of body force, in every isotropic elastic membrane by the application of edge loads and/or uniform normal surface loads on the major surfaces of the thin shell-like body. The solutions, which are valid for both compressible and incompressible materials, are obtained with the use of a strain energy response function which depends on the metric tensor of the membrane in its deformed configuration. The main results are summarized by several theorems and their corollaries in accordance with three mutually exclusive cases for which the initial undeformed surface of the membrane (which may be a sector of a complete or closed surface) is, respectively, developable, spherical and a surface of variable Gaussian curvature satisfying certain differential criteria. The corresponding deformed surfaces are, respectively, a plane or a right circular cylinder, a sphere and a surface of constant mean curvature. These results are exhaustive in that they represent all finite deformation solutions possible in every isotropic elastic material characterized by the strain energy response mentioned above. Also discussed are some special cases of the general results and several families of solutions in terms of an alternative description which should be useful in application and which permit easy interpretations.

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Analytical models for fishery reference points

Canadian Journal of Fisheries and Aquatic Sciences ◽

10.1139/f97-212 ◽

1998 ◽

Vol 55 (2) ◽

pp. 515-528 ◽

Cited By ~ 14

Author(s):

Jon T Schnute ◽

Laura J Richards

Keyword(s):

Fishery Management ◽

Reference Points ◽

Analytical Models ◽

Model Parameters ◽

Harvest Management ◽

Structured Population Models ◽

General Utility ◽

Special Cases ◽

Age Structured ◽

Mathematical Definitions

Fishery reference points are widely applied in formulating harvest management policies. We supply precise mathematical definitions for several reference points in common use. We then derive analytical expressions for these quantities from age-structured population models. In particular, we explain how the maximum sustainable harvest rate and catch (h*, C*), two quantities of management importance, can replace the classical recruitment parameters ( alpha , beta ) in the Beverton-Holt and Ricker recruitment curves. We also demonstrate dependencies of various reference points on subsets of model parameters. Although our analysis is restricted to special cases, our models still have general utility. For example, simple calculations from analytical formulas enable checks on the output from more complex models and guide the choice of reference points for fishery management.

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