Variational Methods in Theoretical Mechanics

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.- The Principle of Conservation of Mass.- The Principle of Balance of Linear Momentum.- The Principle of Balance of Angular Momentum.- 3.4 Thermodynamic Principles.- The Principle of Conservation of Energy.- The Clausius-Duhem Inequality.- 3.5 Constitutive Theory.- Rules of Constitutive Theory.- Special Forms of Constitutive Equations.- 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.- Linear (Coupled) Thermoelasticity.- 5.6 Variational Principles in Linear Elastodynamics.- 5.7 Variational Principles for Linear Piezoelectric Elastodynamic Problems.- 5.8 Variational Principles for Hyperelastic Materials.- Finite Elasticity.- Quasi-Static Problems.- 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.- Non-Newtonian Fluids.- Perfect Fluids.- An Alternate Principle for Invicid Flow.- Magnetohydrodynamics.- 5.14 Variational Principles for Discontinuous Fields.- Hybrid Variational Principles.- 6. Variational Boundary-Value Problems, Monotone Operators, and Variational Inequalities.- 6.1 Direct Variational Methods.- 6.2 Linear Elliptic Variational Boundary-Value Problems.- Regularity.- 6.3 The Lax-Milgram-Babuska Theorem.- 6.4 Existence Theory in Linear Incompressible Elasticity.- 6.5 Monotone Operators.- 6.6 Variational Inequalities.- 6.7 Applications in Mechanics.- 7. Variational Methods of Approximation.- 7.1 Introduction.- 7.2 Several Variational Methods of Approximation.- Galerkin's Method.- The Rayleigh-Ritz Method.- Semidiscrete Galerkin Methods.- Methods of Weighted Residuals.- Least Square Approximations.- Collocation Methods.- Functional Imbeddings.- 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.