Geometrical foundations of continuum mechanics

Viac o knihe

This book explores the foundational aspects of geometrically nonlinear kinematics in generalized continuum mechanics through the lens of differential geometry. It begins with a motivation from geometrically linear first- and second-order crystal plasticity, followed by a review of essential differential geometry concepts, including connection, parallel transport, torsion, curvature, and metrics, applicable to both holonomic and anholonomic coordinate transformations. The kinematics of geometrically nonlinear continuum mechanics are examined, applying differential geometry principles to first- and second-order models. Key topics include compatibility, integrability conditions for distortions and double-distortions, and the introduction of dislocation, disclination, and point-defect density tensors. The text provides a thorough discussion of (multiplicative) first- and second-order elasto-plasticity, culminating in a detailed analysis of various dislocation and defect density tensors, which may aid in modeling geometrically necessary defects and hardening in crystalline materials. The final section summarizes integrability findings, differentiating between conditions for distortion and double-distortion integrability and the more complex requirements for strain and double-strain integrability. This work is aimed at readers interested in continuum modeling of solids, requiring a solid understanding of tensor calculus and continuu