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

Product details

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

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)show more

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 Spacesshow more