Variational and Potential Methods in the Theory of Bending of Plates with Transverse Shear Deformation

Variational and Potential Methods in the Theory of Bending of Plates with Transverse Shear Deformation

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Description

Elastic plates form a class of very important mechanical structures that appear in a wide range of practical applications, from building bodies to microchip production. As the sophistication of industrial designs has increased, so has the demand for greater accuracy in analysis. This in turn has led modelers away from Kirchoff's classical theory for thin plates and toward increasingly refined models that yield not only the deflection of the middle section, but also account for transverse shear deformation. The improved performance of these models is achieved, however, at the expense of a much more complicated system of governing equations and boundary conditions.

In this Monograph, the authors conduct a rigorous mathematical study of a number of boundary value problems for the system of partial differential equations that describe the equilibrium bending of an elastic plate with transverse shear deformation. Specifically, the authors explore the existence, uniqueness, and continuous dependence of the solution on the data. In each case, they give the variational formulation of the problems and discuss their solvability in Sobolev spaces. They then seek the solution in the form of plate potentials and reduce the problems to integral equations on the contour of the domain.

This treatment covers an extensive range of problems and presents the variational method and the boundary integral equation method applied side-by-side. Readers will find that this feature of the book, along with its clear exposition, will lead to a firm and useful understanding of both the model and the methods.
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Product details

  • Hardback | 248 pages
  • 162.6 x 243.8 x 19.8mm | 497.65g
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 2003.
  • 1584881550
  • 9781584881551

Table of contents

Introduction
FORMULATION OF THE PROBLEMS
The Equilibrium Equations for Plates
The Boundary Value Problems
The Plate Potentials and their Properties
Boundary Integral Equations
VARIATIONAL FORMULATION OF THE DIRICHLET AND NEUMANN PROBLEMS
Function Spaces
Solvability of the Interior Problems
Weighted Sobolev Spaces
Solvability of the Exterior Problems
BOUNDARY INTEGRAL EQUATIONS FOR THE DIRICHLET AND NEUMANN PROBLEMS
The Area Potential and its Properties
The Poincare-Steklov Operators
Further Properties of the Plate Potentials
Solvability of the Boundary Equations
TRANSMISSION BOUNDARY VALUE PROBLEMS
Formulation and Solvability of the Problems
Infinite Plate with a Finite Inclusion
Multiply Connected Finite Plate
Finite Plate with an Inclusion
PLATE WEAKENED BY A CRACK
Formulation and Solvability of the Problems
The Poincare-Steklov Operator
The Single Layer and Double Layer Potentials
Infinite Plate with a Crack
Finite Plate with a Crack
BOUNDARY VALUE PROBLEMS WITH OTHER TYPES OF BOUNDARY CONDITIONS
Mixed Boundary Conditions
Boundary Equations for Mixed Conditions
Combined Boundary Conditions
Elastic Boundary Conditions
PLATE ON A GENERALIZED ELASTIC FOUNDATION
Formulation and Solvability of the Problems
A Fundamental Matrix of Solutions
Properties of the Boundary Operators
Solvability of the Boundary Equations
APPENDIX: An Elementary Introduction to Sobolev Spaces
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Review quote

"It is amazing that the authors have managed to cover so many fundamental boundary-value problems and present the variational method and the boundary integral equation method applied side-by-side in a single volumeThis feature of the book will certainly strengthen understanding of both the model and the methods. The writing style is very clear, the book is self-contained and easy to read, and it should be extremely valuable to researchers interested in applied analysis and mathematical models in elasticity." -Proceedings of the Edinburgh Mathematical Society (2002, vol. 45) "This book will be useful for mathematicians, theoretical engineers, and all interested in mathematical modeling in elasticity." -European Mathematical Society Newsletter, No. 40 (June 2001)
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