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. Contents: IntroductionGeometric FundamentalsKinematics of Integrable DeformationGeometry of Anholonomic DeformationKinematics of Anholonomic DeformationList of SymbolsBibliographyIndex Readership: Researchers in mathematical physics and engineering mechanics. Key Features: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 resultsLine-by-line derivations are frequent and exhaustive, to facilitate practice and enable verification of final resultsGeneral 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, respectivelyKeywords:Differential Geometry;Tensor Analysis;Continuum Mechanics;Kinematics;Deformation;Anholonomic Coordinates

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 offers a concise introduction to some basic notions of modern differential geometry and their applications to solid mechanics and physics. Concepts such as manifolds, groups, fibre bundles and groupoids are first introduced within a purely topological framework. They are shown to be relevant to the description of space-time, configuration spaces of mechanical systems, symmetries in general, microstructure and local and distant symmetries of the constitutive response of continuous media. Once these ideas have been grasped at the topological level, the differential structure needed for the description of physical fields is introduced in terms of differentiable manifolds and principal frame bundles. These mathematical concepts are then illustrated with examples from continuum kinematics, Lagrangian and Hamiltonian mechanics, Cauchy fluxes and dislocation theory. This book will be useful for researchers and graduate students in science and engineering.

This book illustrates the deep roots of the geometrically nonlinear kinematics of generalized continuum mechanics in differential geometry. Besides applications to first- order elasticity and elasto-plasticity an appreciation thereof is particularly illuminating for generalized models of continuum mechanics such as second-order (gradient-type) elasticity and elasto-plasticity. After a motivation that arises from considering geometrically linear first- and second- order crystal plasticity in Part I several concepts from differential geometry, relevant for what follows, such as connection, parallel transport, torsion, curvature, and metric for holonomic and anholonomic coordinate transformations are reiterated in Part II. Then, in Part III, the kinematics of geometrically nonlinear continuum mechanics are considered. There various concepts of differential geometry, in particular aspects related to compatibility, are generically applied to the kinematics of first- and second- order geometrically nonlinear continuum mechanics. Together with the discussion on the integrability conditions for the distortions and double-distortions, the concepts of dislocation, disclination and point-defect density tensors are introduced. For concreteness, after touching on nonlinear fir st- and second-order elasticity, a detailed discussion of the kinematics of (multiplicative) first- and second-order elasto-plasticity is given. The discussion naturally culminates in a comprehensive set of different types of dislocation, disclination and point-defect density tensors. It is argued, that these can potentially be used to model densities of geometrically necessary defects and the accompanying hardening in crystalline materials. Eventually Part IV summarizes the above findings on integrability whereby distinction is made between the straightforward conditions for the distortion and the double-distortion being integrable and the more involved conditions for the strain (metric) and the double-strain (connection) being integrable. The book addresses readers with an interest in continuum modelling of solids from engineering and the sciences alike, whereby a sound knowledge of tensor calculus and continuum mechanics is required as a prerequisite.

This book describes thermoelastic and inelastic deformation processes in crystalline solids undergoing loading by shock compression. Constitutive models with a basis in geometrically nonlinear continuum mechanics supply these descriptions. Large deformations such as finite strains and rotations, are addressed. The book covers dominant mechanisms of nonlinear thermoelasticity, dislocation plasticity, deformation twinning, fracture, flow, and other structure changes. Rigorous derivations of theoretical results are provided, with approximately 1300 numbered equations and an extensive bibliography of over 500 historical and modern references spanning from the 1920s to the present day. Case studies contain property data, as well as analytical, and numerical solutions to shock compression problems for different materials. Such materials are metals, ceramics, and minerals, single crystalline and polycrystalline. The intended audience of this book is practicing scientists (physicists, engineers, materials scientists, and applied mathematicians) involved in advanced research on shock compression of solid materials.

On the roots of continuum mechanics in differential geometry -- a review.- Cosserat media.- Cosserat-type shells.- Cosserat-type rods.- Micromorphic media.- Electromagnetism and generalized continua.- Computational methods for generalized continua. The need of generalized continua models is coming from practice. Complex material behavior sometimes cannot be presented by the classical Cauchy continua. At present the attention of the scientists in this field is focused on the most recent research items • new models, • application of well-known models to new problems, • micro-macro aspects, • computational effort, and • possibilities to identify the constitutive equations The new research directions are discussed in this volume - from the point of view of modeling and simulation, identification, and numerical methods.

The second of three parts comprising Volume 54, the proceedings of the Summer Research Institute on Differential Geometry, held at the University of California, Los Angeles, July 1990 (ISBN for the set is 0-8218-1493-1). Among the subjects of Part 2 are gauge theory, symplectic geometry, complex ge

With a pioneering methodology, the book covers the fundamental aspects of kinematic analysis and synthesis of linkage, and provides a theoretical foundation for engineers and researchers in mechanisms design. • The first book to propose a complete curvature theory for planar, spherical and spatial motion • Treatment of the synthesis of linkages with a novel approach • Well-structured format with chapters introducing clearly distinguishable concepts following in a logical sequence dealing with planar, spherical and spatial motion • Presents a pioneering methodology by a recognized expert in the field and brought up to date with the latest research and findings • Fundamental theory and application examples are supplied fully illustrated throughout