Variational Methods in Theoretical Mechanics

Variational Methods in Theoretical Mechanics by J.T. Oden, J.N. Reddy.

This is a textbook written for use in a graduate-level course for students of mechanics and engineering science. It is designed to cover the essential features of modern variational methods and to demonstrate how a number of basic mathematical concepts can be used to produce a unified theory of vari...

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Bibliographic Details
Main Authors: Oden, J.T (Author), Reddy, J.N (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1976.
Edition:1st ed. 1976.
Series:Universitext,
Springer eBook Collection.
Subjects:
Mechanics.
Mechanics, Applied.
Mathematics.
Mathematical physics.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • Tale of Contents
  • 1. Introduction
  • 1.1 The Role of Variational Theory in Mechanics
  • 1.2 Some Historical Comments
  • 1.3 Plan of Study
  • 2. Mathematical Foundations of Classical Variational Theory
  • 2.1 Introduction
  • 2.2 Nonlinear Operators
  • 2.3 Differentiation of Operators
  • 2.4 Mean Value Theorems
  • 2.5 Taylor Formulas
  • 2.6 Gradients of Functionals
  • 2.7 Minimization of Functionals
  • 2.8 Convex Functionals
  • 2.9 Potential Operators and the Inverse Problem
  • 2.10 Sobolev Spaces
  • 3. Mechanics of Continua — A Brief Review
  • 3.1 Introduction
  • 3.2 Kinematics
  • 3.3 Stress and the Mechanical Laws of Balance
  • 3.4 Thermodynamic Principles
  • 3.5 Constitutive Theory
  • 3.6 Jump Conditions for Discontinuous Fields
  • 4. Complementary and Dual Variational Principles in Mechanics
  • 4.1 Introduction
  • 4.2 Boundary Conditions and Green’s Formulas
  • 4.3 Examples from Mechanics and Physics
  • 4.4 The Fourteen Fundamental Complementary-Dual Principles
  • 4.5 Some Complementary-Dual Variational Principles of Mechanics and Physics
  • 4.6 Legendre Transformations
  • 4.7 Generalized Hamiltonian Theory
  • 4.8 Upper and Lower Bounds and Existence Theory
  • 4.9 Lagrange Multipliers
  • 5. Variational Principles in Continuum Mechanics
  • 5.1 Introduction
  • 5.2 Some Preliminary Properties and Lemmas
  • 5.3 General Variational Principles for Linear Theory of Dynamic Viscoelasticity
  • 5.4 Gurtin’s Variational Principles for the Linear Theory of Dynamic Viscoelasticity
  • 5.5 Variational Principles for Linear Coupled Dynamic Thermoviscoelasticity
  • 5.6 Variational Principles in Linear Elastodynamics
  • 5.7 Variational Principles for Linear Piezoelectric Elastodynamic Problems
  • 5.8 Variational Principles for Hyperelastic Materials
  • 5.9 Variational Principles in the Flow Theory of Plasticity
  • 5.10 Variational Principles for a Large Displacement Theory of Elastoplasticity
  • 5.11 Variational Principles in Heat Conduction
  • 5.12 Biot’s Quasi-Variational Principle in Heat Transfer
  • 5.13 Some Variational Principles in Fluid Mechanics and Magnetohydrodynamics
  • 5.14 Variational Principles for Discontinuous Fields
  • 6. Variational Boundary-Value Problems, Monotone Operators, and Variational Inequalities
  • 6.1 Direct Variational Methods
  • 6.2 Linear Elliptic Variational Boundary-Value Problems
  • 6.3 The Lax-Milgram-Babuska Theorem
  • 6.4 Existence Theory in Linear Incompressible Elasticity
  • 6.5 Monotone Operators
  • 6.6 Existence Theory in Nonlinear Elasticity
  • 6.7 Variational Inequalities
  • 6.8 Applications in Mechanics
  • 7. Variational Methods of Approximation
  • 7.1 Introduction
  • 7.2 Several Variational Methods of Approximation
  • 7.3 Finite-Element Approximations
  • 7.4 Finite-Element Interpolation Theory
  • 7.5 Existence and Uniqueness of Galerkin Approximations
  • 7.6 Convergence and Accuracy of Finite-Element Galerkin Approximations
  • References.

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