Foundations of Geometric Continuum Mechanics

Geometry and Duality in Continuum Mechanics

Reuven Segev (Autor)

Buch | Softcover

XVI, 411 Seiten

2024
Springer International Publishing (Verlag)
978-3-031-35657-5 (ISBN)

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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.

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.

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.

Erscheinungsdatum	02.11.2024
Reihe/Serie	Advances in Continuum Mechanics
Reihe/Serie	Advances in Mechanics and Mathematics
Zusatzinfo	XVI, 411 p. 117 illus., 17 illus. in color.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Themenwelt	Naturwissenschaften ► Physik / Astronomie ► Mechanik
Schlagworte	Banach spaces • classical field theories • Configuration space • Continuum Mechanics • Differential Geometry • Global Analysis • Manifolds and bundles • Non-smooth manifolds • Smooth analysis • Smooth fluxes • Smooth manifolds • Smooth theory • Uniform fluxes • Vector Bundles
ISBN-10	3-031-35657-8 / 3031356578
ISBN-13	978-3-031-35657-5 / 9783031356575
Zustand	Neuware