Variational Principles of Continuum Mechanics (eBook)

II. Applications

Victor Berdichevsky (Autor)

eBook Download: PDF

2009 | 2010
X, 430 Seiten
Springer Berlin (Verlag)
978-3-540-88469-9 (ISBN)

Infos

The book reviews the two features of the variational approach: its use as a universal tool to describe physical phenomena and as a source for qualitative and quantitative methods of studying particular problems.

Berdichevsky's work differs from other books on the subject in focusing mostly on the physical origin of variational principles as well as establishing their interrelations. For example, the Gibbs principles appear as a consequence of the Einstein formula for thermodynamic fluctuations rather than as the first principles of the theory of thermodynamic equilibrium. Mathematical issues are considered as long as they shed light on the physical outcomes and/or provide a useful technique for the direct study of variational problems. In addition, a thorough account of variational principles discovered in various branches of continuum mechanics is given.

This book, the second volume, describes how the variational approach can be applied to constructing models of continuum media, such as the theory of elastic plates; shells and beams; shallow water theory; heterogeneous mixtures; granular materials; and turbulence. It goes on to apply the variational approach to asymptotical analysis of problems with small parameters, such as the derivation of the theory of elastic plates, shells and beams from three-dimensional elasticity theory; and the basics of homogenization theory. A theory of stochastic variational problems is considered in detail too, along with applications to the homogenization of continua with random microstructures.

Inhalt

Contents - II. Applications 5
Contents - I. Fundamentals 7
Part III Some Applications of Variational Methods to Development of Continuum Mechanics Models 11
14 Theory of Elastic Plates and Shells 12
Preliminaries from Geometry of Surfaces 13
Classical Shell Theory: Phenomenological Approach 21
Plates 43
Derivation of Classical Shell Theory from Three-Dimensional Elasticity 49
Short Wave Extrapolation 63
Refined Shell Theories 65
Theory of Anisotropic Heterogeneous Shells 88
Laminated Plates 102
Sandwich Plates 111
Nonlinear Theory of Hard-Skin Plates and Shells 124
15 Elastic Beams 138
Phenomenological Approach 138
Variational Problem for Energy Density 148
Asymptotic Analysis of the Energy Functional of Three-Dimensional Elasticity 164
16 Some Stochastic Variational Problems 174
Stochastic Variational Problems 174
Stochastic Quadratic Functionals 179
Extreme Values of Energy 184
Probability Distribution of Energy: Gaussian Excitation 191
Probability Distribution of Energy: Small Excitations 194
Probability Distribution of Energy: Large Excitations 212
Probability Distribution of Linear Functionals of Minimizers 220
Variational Principle for Probability Densities 224
17 Homogenization 239
The Problem of Homogenization 239
Homogenization of Periodic Structures 240
Some Non-asymptotic Features of Homogenization Problem 255
Homogenization of Random Structures 262
Homogenization in One-Dimensional Problems 271
A One-Dimensional Nonlinear Homogenization Problem: Spring Theory 278
Two-Dimensional Structures 284
Two-Dimensional Incompressible Elastic Composites 297
Some Three-Dimensional Homogenization Problems 305
Estimates of Effective Characteristics of Random Cell Structures in Terms of that for Periodic Structures 314
18 Homogenization of Random Structures: a Closer View 320
More on Kozlov's Cell Problem 320
Variational Principle for Probability Densities 339
Equations for Probability Densities 343
Approximations of Probability Densities 348
The Choice of Probabilistic Measure 352
Entropy of Microstructure 355
Temperature of Microstructure 360
Entropy of an Elastic Bar 364
19 Some Other Applications 381
Shallow Water Theory 381
Models of Heterogeneous Mixtures 386
A Granular Material Model 396
A Turbulence Model 398
Bibliographic Comments 407
Bibliography 411
Index 424
Notation 430

Produkt-Details

• Erscheint lt. Verlag: 18.9.2009
• Reihe/Serie: Interaction of Mechanics and Mathematics
• Zusatzinfo: X, 430 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: Continuum Mechanics • Development • Elastic Body • fluid- and aerodynamics • Fluids • Mechanics • microstructure • Variational Principles
• ISBN-10: 3-540-88469-6 / 3540884696
• ISBN-13: 978-3-540-88469-9 / 9783540884699