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Continuum kinematics with incompatible-compatible decomposition.

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Continuum mechanics, stresses, currents and electrodynamics.

Reuven Segev1

  • 1Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel rsegev@bgu.ac.il.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|March 23, 2016
PubMed
Summary
This summary is machine-generated.

This study introduces a new weak theory of forces and stresses using de Rham currents and vector bundles. It formulates pre-metric, p-form electrodynamics within the Eulerian approach to continuum mechanics.

Keywords:
Maxwell's equationscontinuum mechanicsp-form electrodynamicspre-metric electrodynamicsstressvector-valued de Rham currents

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Area of Science:

  • Continuum Mechanics
  • Electrodynamics
  • Differential Geometry

Background:

  • The Eulerian approach in continuum mechanics defines fields on space-time manifolds, not body manifolds.
  • Generalized velocities can be represented by sections of vector bundles, independent of material point motion.

Purpose of the Study:

  • To formulate a weak theory of forces and stresses using de Rham currents and generalized sections of vector bundles.
  • To present a weak formulation of pre-metric, p-form electrodynamics as an application of the developed theory.

Main Methods:

  • Utilizing de Rham currents and generalized sections of vector bundles.
  • Representing generalized velocities by differential forms as generalized potential fields.
  • Employing the flat topology of forms for continuity conditions.

Main Results:

  • A weak theory of forces and stresses is formulated using vector-valued currents.
  • A weak formulation of pre-metric, p-form electrodynamics is presented.
  • The continuity of the force functional with respect to the flat topology of forms is shown to replace prior assumptions.

Conclusions:

  • The Eulerian approach can be extended to formulate advanced theories in continuum mechanics and electrodynamics.
  • The use of differential forms and currents provides a powerful framework for generalized physical theories.
  • The developed theory offers a novel perspective on pre-metric electrodynamics and its underlying assumptions.