Mechanics of Solids and Shells: Theories and Approximations

Introduction Purpose and Scope. Mechanical Concepts and Mathematical Representations. Index Notation. Systems. Summation Convention. Position of Indices. Vector Notation. Kronecker Delta. Permutation Symbol. Symmetrical and Antisymmetrical Systems. Abbreviation for Partial Derivatives. Terminology. Specific Notations. Vectors, Tensors, and Curvilinear Coordinates Introduction. Curvilinear Coordinates, Base Vectors, and Metric Tensor. Products of Base Vectors. Components of Vectors. Surface and Volume Elements. Derivatives of Vectors. Tensors and Invariance. Associated Tensors. Covariant Derivative. Transformation from Cartesian to Curvilinear Coordinates. Integral Transformations. Deformation Concept of a Continuous Medium. Geometry of the Deformed Medium. Dilation of Volume and Surface. Vectors and Tensors Associated with the Deformed System. Nature of Motion in Small Regions. Strain. Transformation of Strain Components. Principal Strains. Maximum Shear Strain. Determination of Principal Strains and Principal Directions. Determination of Extremal Shear Strain. Engineering Strain Tensor. Strain Invariants and Volumetric Strain. Decomposition of Motion into Rotation and Deformation. Physical Components of the Engineering Strain. Strain-Displacement Relations. Compatibility of Strain Components. Rates and Increments of Strain and Rotation. Eulerian Strain Rate. Strain Deviator. Approximation of Small Strain. Approximations of Small Strain and Moderate Rotation. Approximations of Small Strain and Small Rotation. Stress Stress Vector. Couple Stress. Actions upon an Infinitesimal Element. Equations of Motion. Tensorial and Invariant Forms of Stress and Internal Work. Transformation of Stress-Physical Basis. Properties of a Stressed State. Hydrostatic Stress. Stress Deviator. Alternative Forms of the Equations of Motion. Significance of Small Strain. Approximation of Moderate Rotations. Approximations of Small Strains and Mall Rotations Linear Theory. Example: Buckling of a Beam. Behavior of Materials Introduction. General Considerations. Thermodynamic Principles. Excessive Entropy. Heat Flow. Entropy, Entropy Flux, and Entropy Production. Work of Internal Forces. Alternative Forms of the First and Second Laws. Saint-Venant's Principle. Observations of Simple Tests. Elasticity. Inelasticity. Linearly Elastic Material. Monotropic, Orthotropic, Transversely Isotropic , and Isotropic Hookean Material. Heat Conduction. Heat Conduction in the Hookean Material. Coefficients of Isotropic Elasticity. Alternative Forms of the Energy Potentials. Hookean Behavior in Plane-Stress and Plane-Strain. Justification of Saint-Venant's Principle. Yield Condition. Yield Condition for Isotropic Materials. Tresca Yield Condition. von Mises Yield Criterion. Plastic Behavior. Incremental Stress-Strain Relations. Geometrical Interpretation of the Flow Condition. Thermodynamic Interpretation. Tangent Modulus of Elasto-plastic Deformations. The Equations of Saint-Venant, Levy, Prandtl, and Reuss. Hencky Stress-Strain Relations. Plasticity without a Yield Condition Endochronic Theory. An Endochronic Form of Ideal Plasticity. Viscous Behavior. Newtonian Fluid. Linear Viscoelasticity. Isotropic Linear Viscoelasticity. Principles of Work and Energy Introduction. Historical Remarks. Terminology. Work, Kinetic Energy, and Fourier's Inequality. The Principle of Virtual Work. Conservative Forces and Potential Energy. Principle of Stationary Potential Energy. Complementary Energy. Principle of Minimum Potential Energy. Structural Stability. Stability at the Critical Load. Equilibrium States near the Critical Load. Effect of Small Imperfections upon the Buckling Load. Principle of Virtual Work Applied to a Continuous Body. Principle of Stationary Potential Applied to a Continuous Body. Generalization of the Principle of Stationary Potential. General Functional and Complementary Parts. Principle of Stationary Complementary Potential. Extremal Properties of the Complementary Potentials. Functionals and Stationary Theorem of Hellinger-Reissner. Functionals and Stationary Criteria for the Continuous Body Summary. Generalization of Castigliano's Theorem on Displacement. Variational Formulations of Inelasticity. Linear Theories of Isotropic Elasticity and Viscoelasticity Introduction. Uses and Limitations of the Linear Theories. Kinematic Equations of a Linear Theory. Linear Equations of Motion. Linear Elasticity. The Boundary-Value Problems of Linear Elasticity. Kinematic Formulation. Solutions via Displacements. Formulation in Terms of Stresses. Plane Strain and Plane Stress. Airy Stress Function. Stress Concentration at a Circular Hole in a Plate. General Solution by Complex Variables. Simple Bending of a Slender Rod. Torsion of a Cylindrical Bar. Linear Viscoelasticity. Kinematic Formulation. Quasistatic Problems and Separation of Variables. Quasistatic Problems in Terms of Displacements. Quasistatic Problems in Terms of Stresses. Laplace Transforms and Correspondence with Elastic Problems. Differential Geometry of a Surface Introduction. Base Vectors and Metric Tensors of the Surface. Products of the Base Vectors. Derivatives of the Base Vectors. Metric Tensor of the Three-Dimensional Space. Fundamental Forms. Curvature and Torsion. Volume and Area Differentials. Vectors, Derivatives, and Covariant Derivatives. Surface Tensors. Green's Theorem for a Surface. Equations of Gauss and Codazzi. Theory of Shells Introduction. Historical Perspective. The Essence of Shell Theory. Scope of the Current Treatment. Kinematics. Strains and Stresses. Equilibrium. Complementary Potentials. Physical Interpretations. Theory of Membranes. Approximations of Small Strain. The Meaning of Thin. Theory of Hookean Shells with Trnsverse Shear Strain. Theories under the Kirchhoff-Love Constraint Kinematics. Stresses and Strains. Equilibrium. Compatibility Equations, Stress Functions, and the Static-Geometric Analoty. Constitutive Equations of the Hookean Shell. Constitutive Equations of the Thin Hookean Shell. Intrinsic Kirchhoff-Love Theories. Plasticity of the Kirchhoff-Love Shell. Strain-Displacement Equations. Approximation of Small Strains and Moderate Rotations. Theory of Shallow Shells. Refinements-Limitations-References. Concepts of Approximation Introduction. Alternative Means of Approximation. Brief Retrospection. Concept of Finite Differences. Stationarity of Functionals Solutions and Forms of Approximation. Nodal Approximations via the Stationarity of a Functional. Higher-Order Approximations with Continuous Derivatives. Approximation by Finite Elements Physical and Mathematical Implications. Approximation via the Potential Convergence. Valid Approximations, Excessive Stiffness, and Some Cures. Approximation via the Modified Potential Convergence and Efficiency. Nonconforming Elements Approximations with Discontinuous Displcements. Finite Elements of Shells Basic Features. Supplementary Remarks on Elemental Approsimations. Approximation of Nonlinear Paths. References