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© 2009

Variational Principles of Continuum Mechanics

II. Applications

Book

Part of the Interaction of Mechanics and Mathematics book series (IMM)

Table of contents

  1. Front Matter
    Pages 1-9
  2. Some Applications of Variational Methods to Development of Continuum Mechanics Models

    1. Front Matter
      Pages 587-587
    2. Victor L. Berdichevsky
      Pages 589-714
    3. Victor L. Berdichevsky
      Pages 715-750
    4. Victor L. Berdichevsky
      Pages 751-815
    5. Victor L. Berdichevsky
      Pages 817-897
    6. Victor L. Berdichevsky
      Pages 899-959
    7. Victor L. Berdichevsky
      Pages 961-986
  3. Back Matter
    Pages 1-27

About this book

Introduction

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.

Keywords

Elastic Body Fluids Variational Principles continuum mechanics development mechanics microstructure

Authors and affiliations

  1. 1.Mechanical Engineering Dept.Wayne State UniversityDetroitU.S.A.

Bibliographic information

Reviews

From the reviews:

“This new book goes far beyond anything currently available concerning variational principles in continuum mechanics. … We have at hand a monument that all students and professionals in applied mathematics, physics and engineering will praise and naturally keep handy on their bookshelf. Teacher will find in the book a wealth of pedagogical material for many one- semester courses. They and their students will appreciate the clarity simplicity and ingenuity of many arguments offered without pedantry and sacrifice of rigour.” (Gerard A. Maugin, Mathematical Reviews, Issue 2011 a)