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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.
List of contents
Part I: Kinematics, Forces, and Stress Theory.- Manifolds of Mappings for Continuum Mechanics.- Notes on Global Stress and Hyper-Stress Theories.- Applications of Algebraic Topology in Elasticity.- De Donder Construction for Higher Jets.- Part II: Defects, Uniformity, and Homogeneity.- Regular and Singular Dislocations.- Homogenization of Edge-Dislocations as a Weak Limit of de-Rham Currents.- A Kinematics of Defects in Solid Crystals.- Limits of Distributed Dislocations in Geometric and Constitutive Paradigms.- On the Homogeneity of Non-Uniform Material Bodies.
