Differential Geometry and Kinematics of Continua

This book provides definitions and mathematical derivations of fundamental relationships of tensor analysis encountered in nonlinear continuum mechanics and continuum physics, with a focus on finite deformation kinematics and classical differential geometry. Of particular interest are anholonomic aspects arising from a multiplicative decomposition of the deformation gradient into two terms, neither of which in isolation necessarily obeys the integrability conditions satisfied by the gradient of a smooth vector field. The concise format emphasizes clarity and ease of reference, and detailed step-by-step derivations of most analytical results are provided.

Sample Chapter(s)
Chapter 1: Introduction (117 KB)

• Introduction
• Geometric Fundamentals
• Kinematics of Integrable Deformation
• Geometry of Anholonomic Deformation
• Kinematics of Anholonomic Deformation
• List of Symbols
• Bibliography
• Index

• Presentation of mathematical operations and examples in anholonomic space associated with a multiplicative decomposition (e.g., of the gradient of motion) is more general and comprehensive than any given elsewhere and contains original ideas and new results
• Line-by-line derivations are frequent and exhaustive, to facilitate practice and enable verification of final results
• General analysis is given in generic curvilinear coordinates; particular sections deal with applications and examples in Cartesian, cylindrical, spherical, and convected coordinates. Indicial and direct notations of tensor calculus enable connections with historic and modern literature, respectively