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Foundations of Geometric Continuum Mechanics

Geometry and Duality in Continuum Mechanics

  • Book
  • © 2023

Overview

Authors:
  1. Reuven Segev

    1. Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel

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  • Provides a general framework for the geometric foundations of continuum mechanics
  • Highlights the relationship between the physical aspects of continuum mechanics and the underlying mathematical notions
  • Appeals to an audience of mathematicians, physicists, and engineers interested in the foundational problems of the area

Part of the book series: Advances in Mechanics and Mathematics (AMMA, volume 49)

Part of the book sub series: Advances in Continuum Mechanics (ACM)

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Table of contents (24 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Introduction

    • Reuven Segev
    Pages 1-6

  3. Prelude: Finite-Dimensional Systems

    • Reuven Segev
    Pages 7-14

  4. Part I

    1. Front Matter

      Pages 17-17
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    2. Simplices in Affine Spaces and Their Boundaries

      • Reuven Segev
      Pages 17-28

    3. Uniform Fluxes in Affine Spaces

      • Reuven Segev
      Pages 29-40

    4. From Uniform Fluxes to Exterior Algebra

      • Reuven Segev
      Pages 41-69

  5. Part II

    1. Front Matter

      Pages 73-73
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    2. Smooth Analysis on Manifolds: A Short Review

      • Reuven Segev
      Pages 73-114

    3. Interlude: Smooth Distributions of Defects

      • Reuven Segev
      Pages 115-122

    4. Smooth Fluxes

      • Reuven Segev
      Pages 123-140

    5. Frames, Body Points, and Spacetime Structure

      • Reuven Segev
      Pages 141-147

    6. Stresses

      • Reuven Segev
      Pages 149-174

    7. Smooth Electromagnetism on Manifolds

      • Reuven Segev
      Pages 175-185

    8. The Elasticity Problem

      • Reuven Segev
      Pages 187-229

    9. Symmetry and Dynamics

      • Reuven Segev
      Pages 231-246
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Keywords

About this book

This monograph presents the geometric foundations of continuum mechanics. An emphasis is placed on increasing the generality and elegance of the theory by scrutinizing the relationship between the physical aspects and the mathematical notions used in its formulation. The theory of uniform fluxes in affine spaces is covered first, followed by the smooth theory on differentiable manifolds, and ends with the non-smooth global theory. Because continuum mechanics provides the theoretical foundations for disciplines like fluid dynamics and stress analysis, the author’s extension of the theory will enable researchers to better describe the mechanics of modern materials and biological tissues. The global approach to continuum mechanics also enables the formulation and solutions of  practical optimization problems.


Foundations of Geometric Continuum Mechanics will be an invaluable resource for researchers in the area, particularly mathematicians, physicists, and engineers interested in the foundational notions of continuum mechanics.

Authors and Affiliations

  • Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel

    Reuven Segev

About the author

Reuven Segev is the H. Greenhill Chair Professor of Theoretical and Applied Mechanics at the Department of Mechanical Engineering at Ben-Gurion University, Beer-Sheva, Israel. His research focuses on the applications of geometrical and analytical methods in mechanics in general and the mechanics of continuous media, in particular. He also serves as the President of the Israeli Society for Theoretical and Applied Mechanics, and is on the editorial board of the Journal of Geometric Mechanics.

Bibliographic Information

  • Book Title: Foundations of Geometric Continuum Mechanics

  • Book Subtitle: Geometry and Duality in Continuum Mechanics

  • Authors: Reuven Segev

  • Series Title: Advances in Mechanics and Mathematics

  • DOI: https://doi.org/10.1007/978-3-031-35655-1

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

  • Hardcover ISBN: 978-3-031-35654-4Published: 01 November 2023

  • Softcover ISBN: 978-3-031-35657-5Due: 12 December 2023

  • eBook ISBN: 978-3-031-35655-1Published: 31 October 2023

  • Series ISSN: 1571-8689

  • Series E-ISSN: 1876-9896

  • Edition Number: 1

  • Number of Pages: XVI, 411

  • Number of Illustrations: 100 b/w illustrations, 17 illustrations in colour

  • Topics: Differential Geometry, Classical and Continuum Physics

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