Material Geometry Groupoids In Continuum Mechanics

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Material Geometry: Groupoids In Continuum Mechanics

Manuel De Leon,Marcelo Epstein,Victor Manuel Jimenez
Material Geometry Groupoids In Continuum Mechanics Book PDF

Author : Manuel De Leon,Marcelo Epstein,Victor Manuel Jimenez
Publisher : World Scientific
Page : 226 pages
File Size : 50,9 Mb
Release : 2021-04-23
Category : Mathematics
ISBN : 9789811232565

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Material Geometry: Groupoids In Continuum Mechanics by Manuel De Leon,Marcelo Epstein,Victor Manuel Jimenez Pdf

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.

Geometric Continuum Mechanics

Reuven Segev,Marcelo Epstein
Geometric Continuum Mechanics Book PDF

Author : Reuven Segev,Marcelo Epstein
Publisher : Springer Nature
Page : 416 pages
File Size : 52,6 Mb
Release : 2020-05-13
Category : Mathematics
ISBN : 9783030426835

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Geometric Continuum Mechanics by Reuven Segev,Marcelo Epstein Pdf

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.

The Geometrical Language of Continuum Mechanics

Marcelo Epstein
The Geometrical Language of Continuum Mechanics Book PDF

Author : Marcelo Epstein
Publisher : Cambridge University Press
Page : 325 pages
File Size : 50,9 Mb
Release : 2010-07-26
Category : Science
ISBN : 9781139490467

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The Geometrical Language of Continuum Mechanics by Marcelo Epstein Pdf

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

Marcelo Epstein
Differential Geometry Book PDF

Author : Marcelo Epstein
Publisher : Springer
Page : 139 pages
File Size : 51,5 Mb
Release : 2014-07-02
Category : Mathematics
ISBN : 9783319069203

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Differential Geometry by Marcelo Epstein Pdf

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.

Continuum Mechanics and Theory of Materials

Peter Haupt
Continuum Mechanics and Theory of Materials Book PDF

Author : Peter Haupt
Publisher : Springer Science & Business Media
Page : 666 pages
File Size : 53,7 Mb
Release : 2013-03-14
Category : Technology & Engineering
ISBN : 9783662047750

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Continuum Mechanics and Theory of Materials by Peter Haupt Pdf

The new edition includes additional analytical methods in the classical theory of viscoelasticity. This leads to a new theory of finite linear viscoelasticity of incompressible isotropic materials. Anisotropic viscoplasticity is completely reformulated and extended to a general constitutive theory that covers crystal plasticity as a special case.

Material Inhomogeneities and their Evolution

Marcelo Epstein,Marek Elzanowski
Material Inhomogeneities and their Evolution Book PDF

Author : Marcelo Epstein,Marek Elzanowski
Publisher : Springer Science & Business Media
Page : 261 pages
File Size : 53,5 Mb
Release : 2007-08-03
Category : Science
ISBN : 9783540723738

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Material Inhomogeneities and their Evolution by Marcelo Epstein,Marek Elzanowski Pdf

With its origins in the theories of continuous distributions of dislocations and ofmetalplasticity,inhomogeneitytheoryisarichandvibrant?eldofresearch. The recognition of the important role played by con?gurational or material forces in phenomena such as growth and remodelling is perhaps its greatest present-day impetus. While some excellent comprehensive works approa- ing the subject from di?erent angles have been published, the objective of this monograph is to present a point of view that emphasizes the di?erenti- geometric aspects of inhomogeneity theory. In so doing, we follow the general lines of thought that we have propounded in many publications and presen- tions over the last two decades. Although based on these sources, this book is a stand-alone entity and contains some new results and perspectives. At the same time, it does not intend to present either a historical account of the - velopment of the subject or a comprehensive picture of the various schools of thought that can be encountered by perusing scholarly journals and attending specialized symposia. The book is divided into three parts, the ?rst of which is entirely devoted to the formulation of the theory in the absence of evolution. In other words, time is conspicuously absent from Part I. It opens with the geometric ch- acterization of material inhomogeneity within the context of simple bodies in Chapter 1, followed by extensions to second-grade and Cosserat media in Chapters 2 and 3.

Geometrical Foundations of Continuum Mechanics

Paul Steinmann
Geometrical Foundations of Continuum Mechanics Book PDF

Author : Paul Steinmann
Publisher : Springer
Page : 517 pages
File Size : 49,6 Mb
Release : 2015-04-07
Category : Technology & Engineering
ISBN : 3662464594

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Geometrical Foundations of Continuum Mechanics by Paul Steinmann Pdf

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 first- 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 Continuum Mechanics

Gui-Qiang G. Chen,Michael Grinfeld,R. J. Knops
Differential Geometry and Continuum Mechanics Book PDF

Author : Gui-Qiang G. Chen,Michael Grinfeld,R. J. Knops
Publisher : Unknown
Page : 128 pages
File Size : 43,5 Mb
Release : 2015
Category : Electronic
ISBN : 3319185748

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Differential Geometry and Continuum Mechanics by Gui-Qiang G. Chen,Michael Grinfeld,R. J. Knops Pdf

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.

Material Inhomogeneities in Elasticity

G.A. Maugin
Material Inhomogeneities in Elasticity Book PDF

Author : G.A. Maugin
Publisher : CRC Press
Page : 292 pages
File Size : 48,9 Mb
Release : 2020-09-11
Category : Mathematics
ISBN : 9781000110012

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Material Inhomogeneities in Elasticity by G.A. Maugin Pdf

Self contained, this book presents a thorough introduction to the complementary notions of physical forces and material (or configurational) forces. All the required elements of continuum mechanics, deformation theory and differential geometry are also covered. This book will be a great help to many, whilst revealing to others a rather new facet of continuum mechanics in general, and elasticity in particular. An organized exposition of continuum mechanics on the material manifold is given which allows for the consideration of material inhomogeneities in their most appropriate framework. In such a frame the nonlinear elasticity of anisotropic inhomogenous materials appears to be a true field theory. Extensions to the cases of electroelasticity and magnetelasticity are then straightforward. In addition, this original approach provides systematic computational means for the evaluation of characteristic parameters which are useful in various branches of applied mechanics and mathematical physics. This is the case for path-independent integrals and energy-release rates in brittle fracture, the influence of electromagnetic fields on fracture criteria (such as in ceramics), the notion of momentum of electromagnetic fields in matter in optics, and the perturbation of solitons propagating in elastic dispersive systems.

Advanced Methods of Continuum Mechanics for Materials and Structures

Konstantin Naumenko,Marcus Aßmus
Advanced Methods of Continuum Mechanics for Materials and Structures Book PDF

Author : Konstantin Naumenko,Marcus Aßmus
Publisher : Springer
Page : 558 pages
File Size : 47,8 Mb
Release : 2016-05-12
Category : Science
ISBN : 9789811009594

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Advanced Methods of Continuum Mechanics for Materials and Structures by Konstantin Naumenko,Marcus Aßmus Pdf

This volume presents a collection of contributions on advanced approaches of continuum mechanics, which were written to celebrate the 60th birthday of Prof. Holm Altenbach. The contributions are on topics related to the theoretical foundations for the analysis of rods, shells and three-dimensional solids, formulation of constitutive models for advanced materials, as well as development of new approaches to the modeling of damage and fractures.

Mechanics of Generalized Continua

Gérard A. Maugin,Andrei V. Metrikine
Mechanics of Generalized Continua Book PDF

Author : Gérard A. Maugin,Andrei V. Metrikine
Publisher : Springer Science & Business Media
Page : 337 pages
File Size : 44,7 Mb
Release : 2010-03-24
Category : Mathematics
ISBN : 9781441956958

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Mechanics of Generalized Continua by Gérard A. Maugin,Andrei V. Metrikine Pdf

In their 1909 publication Théorie des corps déformables, Eugène and François Cosserat made a historic contribution to materials science by establishing the fundamental principles of the mechanics of generalized continua. The chapters collected in this volume showcase the many areas of continuum mechanics that grew out of the foundational work of the Cosserat brothers. The included contributions provide a detailed survey of the most recent theoretical developments in the field of generalized continuum mechanics and can serve as a useful reference for graduate students and researchers in mechanical engineering, materials science, applied physics and applied mathematics.

Continuum Mechanics of Single-Substance Bodies

A. Cemal Eringen
Continuum Mechanics of Single Substance Bodies Book PDF

Author : A. Cemal Eringen
Publisher : Academic Press
Page : 632 pages
File Size : 55,7 Mb
Release : 2016-11-08
Category : Science
ISBN : 9781483276670

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Continuum Mechanics of Single-Substance Bodies by A. Cemal Eringen Pdf

Continuum Physics, Volume II: Continuum Mechanics of Single-Substance Bodies discusses the continuum mechanics of bodies constituted by a single substance, providing a thorough and precise presentation of exact theories that have evolved during the past years. This book consists of three parts—basic principles, constitutive equations for simple materials, and methods of solution. Part I of this publication is devoted to a discussion of basic principles irrespective of material geometry and constitution that are valid for all kinds of substances, including composites. The geometrical notions, kinematics, balance laws, and thermodynamics of continua are also deliberated. Part II focuses on materials consisting of a single substance, followed by a general theory of constitutive equations and special types of bodies. The thermoelastic solids, thermoviscous fluids, and memory-dependent materials are likewise considered. Part III is devoted to a discussion of a variety of nonlinear and linear problems, as well as nonlinear deformations of elastic solids, viscometric fluids, singular surfaces and waves, and complex function technique. This volume is a good source for researchers and students conducting work on the continuum mechanics of single-substance bodies.

Mathematics Applied to Continuum Mechanics

Lee A. Segel
Mathematics Applied to Continuum Mechanics Book PDF

Author : Lee A. Segel
Publisher : SIAM
Page : 613 pages
File Size : 42,7 Mb
Release : 1977-01-01
Category : Science
ISBN : 0898719089

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Mathematics Applied to Continuum Mechanics by Lee A. Segel Pdf

This book focuses on the fundamental ideas of continuum mechanics by analyzing models of fluid flow and solid deformation and examining problems in elasticity, water waves, and extremum principles. Mathematics Applied to Continuum Mechanics gives an excellent overview of the subject, with an emphasis on clarity, explanation, and motivation. Audience: upper-level undergraduate and graduate students in the fields of applied mathematics, science, and engineering.

Advances in Continuum Mechanics and Thermodynamics of Material Behavior

Donald E. Carlson,Yi-Chao Chen
Advances in Continuum Mechanics and Thermodynamics of Material Behavior Book PDF

Author : Donald E. Carlson,Yi-Chao Chen
Publisher : Springer Science & Business Media
Page : 431 pages
File Size : 51,9 Mb
Release : 2012-12-06
Category : Science
ISBN : 9789401007283

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Advances in Continuum Mechanics and Thermodynamics of Material Behavior by Donald E. Carlson,Yi-Chao Chen Pdf

The papers included in this volume were presented at the Symposium on Advances in the Continuum Mechanics and Thermodynamics of Material Behavior, held as part of the 1999 Joint ASME Applied Mechanics and Materials Summer Conference at Virginia Tech on June 27-30, 1999. The Symposium was held in honor of Professor Roger L. Fosdick on his 60th birthday. The papers are written by prominent researchers in the fields of mechanics, thermodynamics, materials modeling, and applied mathematics. They address open questions and present the latest development in these and related areas. This volume is a valuable reference for researchers and graduate students in universities and research laboratories.

Continuum Mechanics

I-Shih Liu
Continuum Mechanics Book PDF

Author : I-Shih Liu
Publisher : Springer Science & Business Media
Page : 316 pages
File Size : 51,5 Mb
Release : 2002-05-28
Category : Science
ISBN : 3540430199

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Continuum Mechanics by I-Shih Liu Pdf

This concise textbook develops step by step the fundamental principles of continuum mechanics. Emphasis is on mathematical clarity, and an extended appendix provides the required background knowledge in linear algebra and tensor calculus. After introducing the basic notions about general kinematics, balance equations, material objectivity and constitutive functions, the book turns to the presentation of rational thermodynamics by stressing the role of Lagrange multipliers in deriving constitutive funcitions from the underlying entropy principle. A brief lecture on extended thermodynamics closes the book. Many examples and exercises round off the material presendted in the chapters. The book addresses primarily advanced undergraduate students in theoretical physics, applied mathematics and materials sciences.