## Draft

List of Figures

1.1 Direction Cosines (to be corrected) 1-2

1.3 Cross Product of Two Vectors 1-3

1.4 Cross Product of Two Vectors 1-4

1.6 Arbitrary 3D Vector Transformation 1-7

1.7 Rotation of Orthonormal Coordinate System 1-8

2.1 Stress Components on an Infinitesimal Element 2-2

2.2 Stresses as Tensor Components 2-2

2.3 Cauchy's Tetrahedron 2-3

2.4 Cauchy's Reciprocal Theorem 2-6

2.5 Principal Stresses 2-7

2.6 Mohr Circle for Plane Stress 2-11

2.7 Plane Stress Mohr's Circle; Numerical Example 2-13

2.8 Unit Sphere in Physical Body around O 2-14

2.9 Mohr Circle for Stress in 3D 2-15

2.10 Differential Shell Element, Stresses 2-15

2.11 Differential Shell Element, Forces 2-16

2.12 Differential Shell Element, Vectors of Stress Couples 2-16

2.13 Stresses and Resulting Forces in a Plate 2-17

3.1 Examples of a Scalar and Vector Fields 3-2

3.2 Differentiation of position vector p 3-2

3.3 Curvature of a Curve 3-3

3.4 Mathematica Solution for the Tangent to a Curve in 3D 3-4

3.5 Vector Field Crossing a Solid Region 3-4

3.6 Flux Through Area dA 3-5

3.7 Infinitesimal Element for the Evaluation of the Divergence 3-5

3.8 Mathematica Solution for the Divergence of a Vector 3-7

3.9 Radial Stress vector in a Cylinder 3-8

3.10 Gradient of a Vector 3-10

3.11 Mathematica Solution for the Gradients of a Scalar and of a Vector 3-11

3.12 Mathematica Solution for the Curl of a Vector 3-12

4.1 Elongation of an Axial Rod 4-1

4.2 Elementary Definition of Strains in 2D 4-3

4.3 Position and Displacement Vectors 4-5

4.4 Position and Displacement Vectors, b = 0 4-6

4.5 Undeformed and Deformed Configurations of a Continuum 4-13

4.6 Physical Interpretation of the Strain Tensor 4-20

4.7 Relative Displacement du of Q relative to P 4-23

4.8 Mohr Circle for Strain 4-36

4.9 Bonded Resistance Strain Gage 4-39

4.10 Strain Gage Rosette 4-40

4.11 Quarter Wheatstone Bridge Circuit 4-41

4.12 Wheatstone Bridge Configurations 4-42

5.1 Physical Interpretation of the Divergence Theorem 5-3

6.1 Flux Through Area dS 6-2

6.2 Equilibrium of Stresses, Cartesian Coordinates 6-6

6.3 Flux vector 6-13

6.4 Flux Through Sides of Differential Element 6-14

6.5 *Flow through a surface r 6-14

9.1 Boundary Conditions in Elasticity Problems 9-2

9.2 Boundary Conditions in Elasticity Problems 9-3

9.3 Fundamental Equations in Solid Mechanics 9-4

9.4 St-Venant's Principle 9-6

9.5 Cylindrical Coordinates 9-7

9.6 Polar Strains 9-7

9.7 Stresses in Polar Coordinates 9-9

10.1 Torsion of a Circular Bar 10-2

10.2 Pressurized Thick Tube 10-9

10.3 Pressurized Hollow Sphere 10-10

10.4 Circular Hole in an Infinite Plate 10-10

11.1 Elliptical Hole in an Infinite Plate 11-1

11.2 Griffith's Experiments 11-2

11.3 Uniformly Stressed Layer of Atoms Separated by a0 11-3

11.4 Energy and Force Binding Two Adjacent Atoms 11-4

11.5 Stress Strain Relation at the Atomic Level 11-5

12.1 Types of Supports 12-3

12.2 Inclined Roller Support 12-4

12.3 Examples of Static Determinate and Indeterminate Structures 12-4

12.4 Geometric Instability Caused by Concurrent Reactions 12-5

12.5 Shear and Moment Sign Conventions for Design 12-6

12.6 Free Body Diagram of an Infinitesimal Beam Segment 12-7

12.7 Deformation of a Beam under Pure Bending 12-10

13.1 *Strain Energy and Complementary Strain Energy 13-2

13.2 Tapered Cantilivered Beam Analysed by the Vitual Displacement Method 13-8

13.3 Tapered Cantilevered Beam Analysed by the Virtual Force Method 13-10

13.4 Single DOF Example for Potential Energy 13-12

13.5 Graphical Representation of the Potential Energy 13-13

13.6 Uniformly Loaded Simply Supported Beam Analyzed by the Rayleigh-Ritz Method 13-15

13.7 Summary of Variational Methods 13-17

13.8 Duality of Variational Principles 13-18

C.1 Variational and Differential Operators C-2

NOTATION

Symbol

2 Yij X

0 0