Differential Geometry And Continuum Mechanics
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Geometric Continuum Mechanics
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Author : Reuven Segev
language : en
Publisher: Springer Nature
Release Date : 2020-05-13
Geometric Continuum Mechanics written by Reuven Segev and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-05-13 with Mathematics categories.
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 And Continuum Mechanics
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Author : Gui-Qiang G. Chen
language : en
Publisher:
Release Date : 2015
Differential Geometry And Continuum Mechanics written by Gui-Qiang G. Chen and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015 with categories.
This book examines the exciting interface between differential geometry and continuum mechanics, now recognised as being of increasing technological significance. Topics discussed include isometric embeddings in differential geometry and the relation with microstructure in nonlinear elasticity, the use of manifolds in the description of microstructure in continuum mechanics, experimental measurement of microstructure, defects, dislocations, surface energies, and nematic liquid crystals. Compensated compactness in partial differential equations is also treated. The volume is intended for specialists and non-specialists in pure and applied geometry, continuum mechanics, theoretical physics, materials and engineering sciences, and partial differential equations. It will also be of interest to postdoctoral scientists and advanced postgraduate research students. These proceedings include revised written versions of the majority of papers presented by leading experts at the ICMS Edinburgh Workshop on Differential Geometry and Continuum Mechanics held in June 2013. All papers have been peer reviewed.
Geometry Of Incompatible Deformations
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Author :
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2019-03-04
Geometry Of Incompatible Deformations written by and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-03-04 with Science categories.
Differential Geometry
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Author : Marcelo Epstein
language : en
Publisher: Springer
Release Date : 2014-07-02
Differential Geometry written by Marcelo Epstein and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-07-02 with Mathematics categories.
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.
The Geometrical Language Of Continuum Mechanics
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Author : Marcelo Epstein
language : en
Publisher: Cambridge University Press
Release Date : 2010-07-26
The Geometrical Language Of Continuum Mechanics written by Marcelo Epstein and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-07-26 with Science categories.
Epstein presents the fundamental concepts of modern differential geometry within the framework of continuum mechanics. Divided into three parts of roughly equal length, the book opens with a motivational chapter to impress upon the reader that differential geometry is indeed the natural language of continuum mechanics or, better still, that the latter is a prime example of the application and materialisation of the former. In the second part, the fundamental notions of differential geometry are presented with rigor using a writing style that is as informal as possible. Differentiable manifolds, tangent bundles, exterior derivatives, Lie derivatives, and Lie groups are illustrated in terms of their mechanical interpretations. The third part includes the theory of fiber bundles, G-structures, and groupoids, which are applicable to bodies with internal structure and to the description of material inhomogeneity. The abstract notions of differential geometry are thus illuminated by practical and intuitively meaningful engineering applications.
Differential Geometry Applied To Continuum Mechanics
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Author : Daniel Aubram
language : en
Publisher:
Release Date : 2009
Differential Geometry Applied To Continuum Mechanics written by Daniel Aubram and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with categories.
Geometrical Foundations Of Continuum Mechanics
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Author : Paul Steinmann
language : en
Publisher: Springer
Release Date : 2015-03-25
Geometrical Foundations Of Continuum Mechanics written by Paul Steinmann and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-03-25 with Science categories.
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.
Differential Geometry And Kinematics Of Continua
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Author : John D Clayton
language : en
Publisher: World Scientific
Release Date : 2014-07-31
Differential Geometry And Kinematics Of Continua written by John D Clayton and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-07-31 with Mathematics categories.
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.
Material Geometry Groupoids In Continuum Mechanics
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Author : Manuel De Leon
language : en
Publisher: World Scientific
Release Date : 2021-04-23
Material Geometry Groupoids In Continuum Mechanics written by Manuel De Leon and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-04-23 with Mathematics categories.
This monograph is the first in which the theory of groupoids and algebroids is applied to the study of the properties of uniformity and homogeneity of continuous media. It is a further step in the application of differential geometry to the mechanics of continua, initiated years ago with the introduction of the theory of G-structures, in which the group G denotes the group of material symmetries, to study smoothly uniform materials.The new approach presented in this book goes much further by being much more general. It is not a generalization per se, but rather a natural way of considering the algebraic-geometric structure induced by the so-called material isomorphisms. This approach has allowed us to encompass non-uniform materials and discover new properties of uniformity and homogeneity that certain material bodies can possess, thus opening a new area in the discipline.
Categories In Continuum Physics
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Author : F. William Lawvere
language : en
Publisher: Springer
Release Date : 2006-11-14
Categories In Continuum Physics written by F. William Lawvere and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-14 with Mathematics categories.