Geometrical Foundations of Continuum Mechanics (eBook)

An Application to First- and Second-Order Elasticity and Elasto-Plasticity

Paul Steinmann (Autor)

eBook Download: PDF

2015 | 2015
XXIV, 517 Seiten
Springer Berlin (Verlag)
978-3-662-46460-1 (ISBN)

Infos

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.

 

 

Inhalt

Preface 7
Acknowledgements 9
Contents 10
Part I: Prologue 24
Motivation: Linear Crystal Plasticity 26
1.1 Introduction 26
1.2 First-Order Continuum 29
1.3 Second-Order Continuum 41
Part II:Differential Geometry 53
Preliminaries 55
2.1 History of Differential Geometry 55
2.2 Necessity of Differential Geometry 62
2.3 Classification of Differential Geometry 64
Geometry on Connected Manifolds 67
3.1 Manifolds 67
3.2 Connection 72
3.3 Torsion 83
3.4 Curvature 90
Geometry on Metric Manifolds 141
4.1 Metric 142
4.2 Metric Connection 144
4.3 Curvature Based on a Metric Connection 158
4.4 Riemann Geometry 167
4.5 Non-Metric Connection 173
4.6 Curvature Based on a Non-Metric Connection 179
Representations in Four-, Three-, Two-Space 190
5.1 Representation in Four-Space 190
5.2 Representation in Three-Space 197
5.3 Representation in Two-Space 207
Part III:Nonlinear Continuum Mechanics 220
Continuum Kinematics 222
6.1 Coordinates in Euclidean Space 223
6.2 Position and Distortions 231
6.3 Embedded General Metric Manifold 241
6.4 Integrability of Distortion and Double-Distortion 250
Elasticity 303
7.1 First-Order Continuum 304
7.2 Second-Order Continuum 311
Elasto-Plasticity 380
8.1 First-Order Continuum 380
8.2 Second-Order Continuum 409
Part IV:Epilogue 509
Integrability and Non-Integrability in a Nutshell 511
9.1 First-Order Continuum 511
9.2 Second-Order Continuum 513
References 518
Index 528

Produkt-Details

• Erscheint lt. Verlag: 25.3.2015
• Reihe/Serie: Lecture Notes in Applied Mathematics and Mechanics
• Reihe/Serie: Lecture Notes in Applied Mathematics and Mechanics
• Zusatzinfo: XXIV, 517 p. 59 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: Applied mathematics • Applied Mechanics • Differential Geometry • Geometrical Foundations of Continuum Mechanics • nonlinear continuum mechanics
• ISBN-10: 3-662-46460-8 / 3662464608
• ISBN-13: 978-3-662-46460-1 / 9783662464601