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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.

Geometry, Differential. --- Continuum mechanics. --- Differential Geometry. --- Continuum Mechanics. --- Mecànica dels medis continus --- Matemàtica

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This contributed volume explores the applications of various topics in modern differential geometry to the foundations of continuum mechanics. In particular, the contributors use notions from areas such as global analysis, algebraic topology, and geometric measure theory. Chapter authors are experts in their respective areas, and provide important insights from the most recent research. Organized into two parts, the book first covers kinematics, forces, and stress theory, and then addresses defects, uniformity, and homogeneity. Specific topics covered include: Global stress and hyper-stress theories Applications of de Rham currents to singular dislocations Manifolds of mappings for continuum mechanics Kinematics of defects in solid crystals Geometric Continuum Mechanics will appeal to graduate students and researchers in the fields of mechanics, physics, and engineering who seek a more rigorous mathematical understanding of the area. Mathematicians interested in applications of analysis and geometry will also find the topics covered here of interest.

Differential geometry. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Mechanics. --- Mechanics, Applied. --- Continuum physics. --- Mathematical physics. --- Differential Geometry. --- Global Analysis and Analysis on Manifolds. --- Theoretical and Applied Mechanics. --- Classical and Continuum Physics. --- Mathematical Applications in the Physical Sciences. --- Physical mathematics --- Physics --- Classical field theory --- Continuum physics --- Continuum mechanics --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Dynamics --- Quantum theory --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Differential geometry --- Mathematics --- Geometry, Differential. --- Global analysis (Mathematics) --- Manifolds (Mathematics) --- Field theory (Physics)

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