Foundations of Geometric Continuum Mechanics : Geometry and Duality in Continuum Mechanics / by Reuven Segev.

Format:

Book

Author/Creator:

Segev, Reuven.

Series:

Advances in Continuum Mechanics, 2524-4647 ; 49

Language:

English

Subjects (All):

Geometry, Differential.

Continuum mechanics.

Differential Geometry.

Continuum Mechanics.

Local Subjects:

Differential Geometry.

Continuum Mechanics.

Physical Description:

1 online resource (410 pages)

Edition:

1st ed. 2023.

Place of Publication:

Cham : Springer International Publishing : Imprint: Birkhäuser, 2023.

Summary:

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.

Contents:

1. Introduction

2. Prelude: Finite Dimensional Systems

Part I Algebraic Theory: Uniform Fluxes

3. Simplices in Affine Spaces and Their Boundaries

4. Uniform Fluxes in Affine Spaces

5. From Uniform Fluxes to Exterior Algebra

Part II: Smooth Theory

6. Smooth Analysis on Manifolds: A Short Review

7. Interlude: Smooth Distributions of Defects

8. Smooth Fluxes

9. Frames, Body Points, and Spacetime Structure

10. Stresses

11. Smooth Electromagnetism on Manifolds

12. The Elasticity Problem

13. Symmetry and Dynamics

Part III Non-Smooth, Global Theories

14. Banachable Space of Sections of Vector Bundles over Compact Manifolds

15. Manifolds of Sections and Embeddings

16. The General Framework for Global Analytic Stress Theory

17. Dual Spaces Corresponding to Spaces of Differentiable Sections of a Vector Bundle: Localization of Sections and Functionals

18. de Rham Currents

19. Interlude: Singular Distributions of Defects in Bodies

20. Vector-Valued Currents

21. The Representation of Forces by Stresses and Hyperstresses

22. Simple Forces and Stresses

23. Whitney's Geometric Integration Theory and Non-Smooth Bodies

24. Optimal Fields and Load Capacity of Bodies

Index.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Segev, Reuven Foundations of Geometric Continuum Mechanics

ISBN:

3-031-35655-1

OCLC:

1407318129