Course No: (1 st version: for graduate students) Course Name: Continuum Mechanics Offered by: Chyanbin Hwu

Transcription

1 Course No: (1 st version: for graduate students) Course Name: Continuum Mechanics Offered by: Chyanbin Hwu Contents: 1. Introduction 1.1 Basic Concepts of Continuum Mechanics 1.2 The Need for Tensor Analysis 2. Matrices, Vectors, and Tensors 2.1 Index Notation 2.2 Matrices Basic Operation of Matrices Square Marix Eigenvalues and Eigenvectors Cayley-Hamilton Theorem Non-symmetric Matrix 2.3 Vector Analysis Gradient, Divergence, and Curl Coordinate Rotations Orthogonal Curvilinear Coordinates Integral Theorems 2.4 Tensor Analysis Cartesian Tensors Curvilinear Coordinates Tensors of Order One Tensors of Order Two The Metric Tensor Tensor Equations Differentiation of a Tensor Christoffel Symbols and Riemann-Christoffel Tensors 3. Kinematics 3.1 Deformation 3.2 Polar Decomposition 3.3 Geometrical Interpretation of Polar Decomposition Eigenvalues and Eigenvectors of U, V, C and B Determinant of R, U and F Strain Ellipsoid and Cauchy Ellipsoid Examples of calculation of U, C, B, R and F 3.4 Strains Lagrangian Strains Eluerian Strains Small Strains 3.5 Strain Compatibility Equations

2 3.6 Description of Motions 3.7 Relative Deformation 3.8 Stretching and Spin 4. Fundamental Laws of Mechanics 4.1 Conservation of Mass Material Form of the Continuity Conditions Some Identities 4.2 The Balance of Momentum 4.3 Cauchy s Laws of Motion Cauchy s Fundamental Theorem Cauchy s First Law of Motion Cauchy s Second Law of Motion 4.4 The Piola-Kirchhoff Stress Tensors The First Piola-Kirchhoff Stress Tensor The Second Piola-Kirchhoff Stress Tensor 4.5 Equilibrium Equations in Orthogonal Curvilinear Coordinates 4.6 Thermodynamics The First Law of thermodynamics The Second Law of Thermodynamics 5. Constitutive Equations 5.1 Theories of General Constitutive Equations Fundamental Postulates of a Purely Mechanical Theory Reduced Constitutive Equations Constitutive Equations for Materials with Internal Constraints 5.2 Material Symmetry, Isotropic Materials 5.3 Hyperelastic Materials Frame Indifference Isotropic Hyperelastic Materials 5.4 Viscoelasticity (incomplete) 5.5 Hypoelasticity (incomplete) Appendix A: Change of Frame Appendix B: Tensor Functions and Tensor Functional B.1 Isotropic Tensor Functions B.2 Representation Theorem for Isotropic Tensor Functions Course No: MÉ32B, IPSA (2 nd version: for undergraduate students) Course Name: Solid Continuum Mechanics for Elastic Bodies Offered by: Chyanbin Hwu Introduction 1.1 Basic Concepts of Continuum Mechanics 1.2 Elementary Topics

3 1.3 The Need for Tensor Analysis 2. Vectors and Tensors 2.1 Indicial Notation 2.2 Matrices (omit) Basic Operation of Matrices Square Marix Eigenvalues and Eigenvectors Cayley-Hamilton Theorem Non-symmetric Matrix 2.3 Vector Analysis Gradient, Divergence, and Curl Coordinate Rotations Orthogonal Curvilinear Coordinates (omit) Integral Theorems 2.4 Tensor Analysis Cartesian Tensors Curvilinear Coordinates (omit) Tensors of Order One (omit) Tensors of Order Two (omit) The Metric Tensor (omit) Tensor Equations (omit) Differentiation of a Tensor (omit) Christoffel Symbols and Riemann-Christoffel Tensors (omit) Appendix 3. Stress 3.1 The Balance of Momentum 3.2 Cauchy s Laws of Motion Cauchy s Fundamental Theorem Cauchy s First Law of Motion Cauchy s Second Law of Motion 3.3 Equilibrium Equations in Orthogonal Curvilinear Coordinates 3.4 The Piola-Kirchhoff Stress Tensors (omit) 3.5 State of Stress in Cartesian Coordinates Principal stresses and Maximum Shear Stresses Classical Failure Criteria Appendix 3A: Fundamental Laws of Mechanics (omit) 3A.1 Conservation of Mass (omit) 3A.1.1 Material Form of the Continuity Conditions 3A.1.2 Some Identities 3A.2 Thermodynamics (omit) 3A.2.1 The First Law of thermodynamics 3A.2.2 The Second Law of Thermodynamics 4. Deformation

4 4.1 Deformation Tensor 4.2 Polar Decomposition Geometrical Interpretation of Polar Decomposition Eigenvalues and Eigenvectors of U, V, C and B (omit) Determinant of R, U and F (omit) Examples of calculation of U, C, B, R and F (omit) 4.3 Strain Ellipsoid and Cauchy Ellipsoid 4.4 Strains Lagrangian Strains Eluerian Strains Small Strains Strain Compatibility Equations 4.5 State of Strain in Cartesian Coordinates Principal strains and Maximum Shear Strains Strain Measurements 4.6 Description of Motions (omit) 4.7 Relative Deformation (omit) 4.8 Stretching and Spin (omit) 5. Constitutive Equations 5.1 General Description of Solids 1D Elastic Materials Plastic Materials Viscoelastic Materials 5.2 Stress-Strain Relations for Isotropic Elastic Solids Relations for Pure Tension and Pure Shear Relations for Plane Stress and Plane Strain Relations for General Three-Dimensional Problems 5.3 Constitutive Equations for Anisotropic Elastic Solids 3D Generalized Hooke s Law Material Symmetry Engineering Constants 5.4 Constitutive Equations for Elastic Solids 2D Isotropic Materials Anisotropic Materials Monoclinic Materials Orthotropic Materials 5.5 Constitutive Equations for Composite Laminates Specially Orthotropic Lamina Generally Orthotropic Lamina Classical Lamination Theory 5.6 Extension of Generalized Hooke s Law Effect of Temperature Piezoelectric Materials Viscoelastic Materials Appendix 5A: Theories of General Constitutive Equations (omit)

5 Appendix 5B: Derivation for Sec Standard linear viscoelastic solids (omit) 6. Some Simple Problems in Elasticity 6.1 Theory of Elasticity for Anisotropic Elastic Solids State of Stress (see Section 3.5) Deformation (see Section 4.5) Constitutive Laws (see Section 5.3) Boundary conditions 6.2 Analysis of One-Dimensional Structures Axially Loaded Members Torsion of Bars Beams Columns 6.3 Analysis of Two-Dimensional Structures Plane Problems Inplane Loading Plate Bending Problems Transverse Loading Two-Dimensional Anisotropic Elasticity References: 1. C. Truesdell, The Elements of Continuum Mechanics, Springer-Verlag, Newy York, I.S. Sokolnikoff, Tensor Analysis Theory and Applications to Geometry and Mechanics of Continua, John Wiley & Sons, Inc., L.E. Malvern, Introduction to the Mechanics of a Continuous Medium, Prentice-Hall, Englewood Cliffs, M.N.L. Narasimhan, Principles of Continuum Mechanics, John Wiley & Sons, Inc., New York, Y.C. Fung, A First Course in Continuum Mechanics, Prentice-Hall, Englewood Cliffs, C. Hwu, Anisotropic Elastic Plates, Springer, New York, T.J. Chung, Continuum Mechanics, Prentice-Hall, Englewood Cliffs, C. Truesdell and W. Noll, The Non-linear Field Theories of Mechanics, in Encyclopedia of Physics, Ed. by S. Flugge, Vol.III/3, Springer-Verlag, Berlin, C. Truesdell and R.A. Toupin, The Classical Field Theories, Encyclopedia of Physics, Ed. by S. Flugge, Vol.III/1, pp , Springer-Verlag, Berlin, W. Jaunzemis, Continuum Mechanics, The Macmillan Co., N.Y., A.C. Eringen, Mechanics of Continua, 2nd ed., W. Flugge, Tensor Analysis and Continuum Mechanics, Springer-Verlag, N.Y., 1972.