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Contents

I CONTINUUM MECHANICS 0-7

1 MATHEMATICAL PRELIMINARIES; Part I Vectors and Tensors 1-1

1.1.1 Operations 1-2

1.1.2 Coordinate Transformation 1-4

1.1.2.1 jGeneral Tensors 1-4

1.1.2.1.1 jContravariant Transformation 1-5

1.1.2.1.2 Covariant Transformation 1-6

1.1.2.2 Cartesian Coordinate System 1-6

1.2.1 Indicial Notation 1-8

1.2.2 Tensor Operations 1-10

1.2.2.2 Multiplication by a Scalar 1-10

1.2.2.3 Contraction 1-10

1.2.2.4 Products 1-11

1.2.2.4.1 Outer Product 1-11

1.2.2.4.2 Inner Product 1-11

1.2.2.4.3 Scalar Product 1-11

1.2.2.4.4 Tensor Product 1-11

1.2.2.5 Product of Two Second-Order Tensors 1-13

1.2.4 Rotation of Axes 1-13

1.2.6 Inverse Tensor 1-14

1.2.7 Principal Values and Directions of Symmetric Second Order Tensors 1-14

1.2.8 Powers of Second Order Tensors; Hamilton-Cayley Equations 1-15

2 KINETICS 2-1

2.1 Force, Traction and Stress Vectors 2-1

2.2 Traction on an Arbitrary Plane; Cauchy's Stress Tensor 2-3

E 2-1 Stress Vectors 2-4

2.3 Symmetry of Stress Tensor 2-5

2.3.1 Cauchy's Reciprocal Theorem 2-5

2.4 Principal Stresses 2-6

2.4.1 Invariants 2-8

2.4.2 Spherical and Deviatoric Stress Tensors 2-8

2.5 Stress Transformation 2-9

E 2-2 Principal Stresses 2-9

E 2-3 Stress Transformation 2-10

2.5.1 Plane Stress 2-10

2.5.2 Mohr's Circle for Plane Stress Conditions 2-10

E 2-4 Mohr's Circle in Plane Stress 2-12

2.5.3 jMohr's Stress Representation Plane 2-14

2.6 Simplified Theories; Stress Resultants 2-14

2.6.2 Plates 2-17

3 MATHEMATICAL PRELIMINARIES; Part II VECTOR DIFFERENTIATION 3-1

3.1 Introduction 3-1

3.2 Derivative WRT to a Scalar 3-1

3.3 Divergence 3-4

E 3-2 Divergence 3-6

3.3.2 Second-Order Tensor 3-6

3.4 Gradient 3-6

E 3-3 Gradient of a Scalar 3 8

E 3-4 Stress Vector normal to the Tangent of a Cylinder 3-8

E 3-5 Gradient of a Vector Field 3-10

3.4.3 Mathematica Solution 3-11

E 3-6 Curl of a vector 3-11

3.6 Some useful Relations 3-13

4 KINEMATIC 4-1

4.1 Elementary Definition of Strain 4-1

4.1.1 Small and Finite Strains in 1D 4-1

4.1.2 Small Strains in 2D 4-2

4.2 Strain Tensor 4-2

4.2.1 Position and Displacement Vectors; (x, X) 4-4

E 4-1 Displacement Vectors in Material and Spatial Forms 4-4

4.2.1.1 Lagrangian and Eulerian Descriptions; x(X,t),X(x,t) 4-7

E 4-2 Lagrangian and Eulerian Descriptions 4-8

4.2.2 Gradients 4-8

4.2.2.1.1 | Change of Area Due to Deformation 4-9

4.2.2.1.2 | Change of Volume Due to Deformation 4-10

E 4-3 Change of Volume and Area 4-10

4.2.2.3 Examples 4-11

E 4-4 Material Deformation and Displacement Gradients 4-11

4.2.3 Deformation Tensors 4-12

4.2.3.1 Cauchy's Deformation Tensor; (dX)2 4-12

4.2.3.2 Green's Deformation Tensor; (dx)2 4-14

E 4-5 Green's Deformation Tensor 4-14

4.2.4.1 Finite Strain Tensors 4-15

4.2.4.1.1 Lagrangian/Green's Strain Tensor 4-15

E 4-6 Lagrangian Tensor 4-16

4.2.4.1.2 Eulerian/Almansi's Tensor 4-16

4.2.4.2 Infinitesimal Strain Tensors; Small Deformation Theory 4-17

4.2.4.2.1 Lagrangian Infinitesimal Strain Tensor 4-17

4.2.4.2.2 Eulerian Infinitesimal Strain Tensor 4-18

4.2.4.3 Examples 4-18

E 4-7 Lagrangian and Eulerian Linear Strain Tensors 4-18

4.2.5 jPhysical Interpretation of the Strain Tensor 4-19

4.2.5.1 Small Strain 4-19

4.2.5.2 Finite Strain; Stretch Ratio 4-21

4.3 Strain Decomposition 4-22

4.3.1 jLinear Strain and Rotation Tensors 4-22

4.3.1.1 Small Strains 4-22

4.3.1.1.1 Lagrangian Formulation 4-22

4.3.1.1.2 Eulerian Formulation 4-24

4.3.1.2 Examples 4-25

E 4-8 Relative Displacement along a specified direction 4-25

E 4-9 Linear strain tensor, linear rotation tensor, rotation vector 4-25

4.3.2 Finite Strain; Polar Decomposition 4-26

E 4-10 Polar Decomposition I 4-26

E 4-11 Polar Decomposition II 4-27

E 4-12 Polar Decomposition III 4-28

4.4 Summary and Discussion 4-30

4.5 Compatibility Equation 4-30

E 4-13 Strain Compatibility 4-32

4.6 Lagrangian Stresses; Piola Kirchoff Stress Tensors 4-32

4.6.2 Second 4-33

E 4-14 Piola-Kirchoff Stress Tensors 4-34

4.7 Hydrostatic and Deviatoric Strain 4-34

4.8 Principal Strains, Strain Invariants, Mohr Circle 4-36

E 4-15 Strain Invariants & Principal Strains 4-36

E 4-16 Mohr's Circle 4-38

4.9 Initial or Thermal Strains 4-39

4.10 j Experimental Measurement of Strain 4-39

4.10.1 Wheatstone Bridge Circuits 4-40

4.10.2 Quarter Bridge Circuits 4-41

5 MATHEMATICAL PRELIMINARIES; Part III VECTOR INTEGRALS 5-1

5.1 Integral of a Vector 5-1

5.2 Line Integral 5-1

5.3 Integration by Parts 5-2

5.4 Gauss; Divergence Theorem 5-2

5.5 Stoke's Theorem 5-2

5.6 Green; Gradient Theorem 5-2

E 5-1 Physical Interpretation of the Divergence Theorem 5-2

6 FUNDAMENTAL LAWS of CONTINUUM MECHANICS 6-1

6.1 Introduction 6-1

6.1.1 Conservation Laws 6-1

6.2 Conservation of Mass; Continuity Equation 6-3

6.2.1 Spatial Form 6-3

6.2.2 Material Form 6-4

6.3 Linear Momentum Principle; Equation of Motion 6-4

6.3.1 Momentum Principle 6-4

E 6-1 Equilibrium Equation 6-5

6.3.2 Moment of Momentum Principle 6-6

6.3.2.1 Symmetry of the Stress Tensor 6-6

6.4 Conservation of Energy; First Principle of Thermodynamics 6-7

6.4.1 Spatial Gradient of the Velocity 6-7

6.4.2 First Principle 6-7

6.5 Equation of State; Second Principle of Thermodynamics 6-9

6.5.1 Entropy 6-9

6.5.1.1 Statistical Mechanics 6-10

6.5.1.2 Classical Thermodynamics 6-10

6.5.2 Clausius-Duhem Inequality 6-11

6.6 Balance of Equations and Unknowns 6-11

6.7 | Elements of Heat Transfer 6-12

6.7.1 Simple 2D Derivation 6-13

6.7.2 jGeneralized Derivation 6-14

7 CONSTITUTIVE EQUATIONS; Part I LINEAR 7-1

7.1 | Thermodynamic Approach 7-1

7.1.1 State Variables 7-1

7.1.2 Gibbs Relation 7-2

7.1.3 Thermal Equation of State 7-3

7.1.4 Thermodynamic Potentials 7-3

7.1.5 Elastic Potential or Strain Energy Function 7-4

7.2 Experimental Observations 7-5

7.2.2 Bulk Modulus 7-5

7.3 Stress-Strain Relations in Generalized Elasticity 7-6

7.3.1 Anisotropic 7-6

7.3.2 Monotropic Material 7-7

7.3.3 Orthotropic Material 7-7

7.3.4 Transversely Isotropic Material 7-8

7.3.5 Isotropic Material 7-9

7.3.5.1 Engineering Constants 7-10

7.3.5.1.1 Isotropic Case 7-10

7.3.5.1.1.2 Bulk's Modulus; Volumetric and Deviatoric Strains 7-11

7.3.5.1.1.3 Restriction Imposed on the Isotropic Elastic Moduli . . . 7-12

7.3.5.1.2 Transversly Isotropic Case 7-12

7.3.5.2 Special 2D Cases 7-13

7.3.5.2.1 Plane Strain 7-13

7.3.5.2.2 Axisymmetry 7-13

7.4 Linear Thermoelasticity 7-14

7.5 Fourrier Law 7-15

7.6 Updated Balance of Equations and Unknowns 7-16

8 INTERMEZZO 8-1

II ELASTICITY/SOLID MECHANICS 8-3

9 BOUNDARY VALUE PROBLEMS in ELASTICITY 9-1

9.1 Preliminary Considerations 9-1

9.2 Boundary Conditions 9-1

9.3 Boundary Value Problem Formulation 9-3

9.4 Compacted Forms 9-3

9.4.1 Navier-Cauchy Equations 9-3

9.4.2 Beltrami-Mitchell Equations 9-5

9.4.3 Ellipticity of Elasticity Problems 9-5

9.5 Strain Energy and Extenal Work 9-5

9.6 Uniqueness of the Elastostatic Stress and Strain Field 9-5

9.7 Saint Venant's Principle 9-6

9.8 Cylindrical Coordinates 9-6

9.8.1 Strains 9-7

9.8.2 Equilibrium 9-8

9.8.3 Stress-Strain Relations 9-9

9.8.3.1 Plane Strain 9-10

10 SOME ELASTICITY PROBLEMS 10-1

10.1 Semi-Inverse Method 10-1

10.1.1 Example: Torsion of a Circular Cylinder 10-1

10.2 Airy Stress Functions 10-3

10.2.1 Cartesian Coordinates; Plane Strain 10-3

10.2.1.1 Example: Cantilever Beam 10-5

10.2.2 Polar Coordinates 10-7

10.2.2.1 Plane Strain Formulation 10-7

10.2.2.2 Axially Symmetric Case 10-7

10.2.2.3 Example: Thick-Walled Cylinder 10-8

10.2.2.4 Example: Hollow Sphere 10-10

10.2.2.5 Example: Stress Concentration due to a Circular Hole in a Plate 10-10

11 THEORETICAL STRENGTH OF PERFECT CRYSTALS 11-1

11.1 Introduction 11-1

11.2 Theoretical Strength 11-3

11.2.1 Ideal Strength in Terms of Physical Parameters 11-3

11.2.2 Ideal Strength in Terms of Engineering Parameter 11-5

11.3 Size Effect; Griffith Theory 11-6

12 BEAM THEORY 12-1

12.1 Introduction 12-1

12.2 Statics 12-1

12.2.1 Equilibrium 12-1

12.2.2 Reactions 12-3

12.2.3 Equations of Conditions 12-3

12.2.4 Static Determinacy 12-4

12.2.5 Geometric Instability 12-4

12.2.6 Examples 12-5

E 12-1 Simply Supported Beam 12-5

12.3 Shear & Moment Diagrams 12-6

12.3.1 Design Sign Conventions 12-6

12.3.2 Load, Shear, Moment Relations 12-6

12.3.3 Examples 12-8

E 12-2 Simple Shear and Moment Diagram 12-8

12.4 Beam Theory 12-10

12.4.1 Basic Kinematic Assumption; Curvature 12-10

12.4.2 Stress-Strain Relations 12-11

12.4.3 Internal Equilibrium; Section Properties 12-11

12.4.4 Beam Formula 12-12

12.4.5 Limitations of the Beam Theory 12-13

12.4.6 Example 12-13

E 12-3 Design Example 12-13

13 VARIATIONAL METHODS 13-1

13.1 Preliminary Definitions 13-1

13.1.1 Internal Strain Energy 13-2

13.1.2 External Work 13-3

13.1.3 Virtual Work 13-4

13.1.3.1 Internal Virtual Work 13-4

13.1.3.2 External Virtual Work 6W 13-5

13.1.4 Complementary Virtual Work 13-5

13.1.5 Potential Energy 13-6

13.2 Principle of Virtual Work and Complementary Virtual Work 13-6

13.2.1 Principle of Virtual Work 13-6

E 13-1 Tapered Cantiliver Beam, Virtual Displacement 13-7

13.2.2 Principle of Complementary Virtual Work 13-9

E 13-2 Tapered Cantilivered Beam; Virtual Force 13-10

13.3 Potential Energy 13-11

13.3.1 Derivation 13-11

13.3.2 Rayleigh-Ritz Method 13-13

E 13-3 Uniformly Loaded Simply Supported Beam; Polynomial Approximation 13-14

13.4 Summary 13-16

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