Table of Contents
Acknowledgements xi
1 Introduction: Background and Motivation 1
1.1 What This Book Is All About 1
1.2 A Brief Historical Perspective 2
1.3 Symbiotic Structural Analysis 9
1.4 Linear Curved Beams and Arches 9
1.5 Geometrically Nonlinear Curved Beams and Arches 10
1.6 Geometrically Nonlinear Plates and Shells 11
1.7 Symmetry of the Tangent Operator: Nonlinear Beams and Shells 12
1.8 Road Map of the Book 14
References 15
Part I ESSENTIAL MATHEMATICS 19
2 Mathematical Preliminaries 21
2.1 Essential Preliminaries 21
2.2 Affine Space, Vectors and Barycentric Combination 33
2.3 Generalization: Euclidean to Riemannian Space 36
2.4 Where We Would Like to Go 40
3 Tensors 41
3.1 Introduction 41
3.2 Tensors as Linear Transformation 44
3.3 General Tensor Space 46
3.4 Tensor by Component Transformation Property 50
3.5 Special Tensors 57
3.6 Second-order Tensors 62
3.7 Calculus Tensor 74
3.8 Partial Derivatives of Tensors 74
3.9 Covariant or Absolute Derivative 75
3.10 Riemann–Christoffel Tensor: Ordered Differentiation 78
3.11 Partial (PD) and Covariant (C.D.) Derivatives of Tensors 79
3.12 Partial Derivatives of Scalar Functions of Tensors 80
3.13 Partial Derivatives of Tensor Functions of Tensors 81
3.14 Partial Derivatives of Parametric Functions of Tensors 81
3.15 Differential Operators 82
3.16 Gradient Operator: GRAD(∙) or ∇(∙) 82
3.17 Divergence Operator: DIV or ∇∙ 84
3.18 Integral Transforms: Green–Gauss Theorems 87
3.19 Where We Would Like to Go 90
4 Rotation Tensor 91
4.1 Introduction 91
4.2 Cayley’s Representation 100
4.3 Rodrigues Parameters 107
4.4 Euler – Rodrigues Parameters 112
4.5 Hamilton’s Quaternions 115
4.6 Hamilton–Rodrigues Quaternion 119
4.7 Derivatives, Angular Velocity and Variations 125
Part II ESSENTIAL MESH GENERATION 133
5 Curves: Theory and Computation 135
5.1 Introduction 135
5.2 Affine Transformation and Ratios 136
5.3 Real Parametric Curves: Differential Geometry 139
5.4 Frenet–Serret Derivatives 145
5.5 Bernstein Polynomials 148
5.6 Non-rational Curves Bezier–Bernstein–de Casteljau 154
5.7 Composite Bezier–Bernstein Curves 181
5.8 Splines: Schoenberg B-spline Curves 185
5.9 Recursive Algorithm: de Boor–Cox Spline 195
5.10 Rational Bezier Curves: Conics and Splines 198
5.11 Composite Bezier Form: Quadratic and Cubic B-spline Curves 215
5.12 Curve Fitting: Interpolations 229
5.13 Where We Would Like to Go 245
6 Surfaces: Theory and Computation 247
6.1 Introduction 247
6.2 Real Parametric Surface: Differential Geometry 248
6.3 Gauss–Weingarten Formulas: Optimal Coordinate System 272
6.4 Cartesian Product Bernstein–Bezier Surfaces 280
6.5 Control Net Generation: Cartesian Product Surfaces 296
6.6 Composite Bezier Form: Quadratic and Cubic B-splines 300
6.7 Triangular Bezier–Bernstein Surfaces 306
Part III ESSENTIAL MECHANICS 323
7 Nonlinear Mechanics: A Lagrangian Approach 325
7.1 Introduction 325
7.2 Deformation Geometry: Strain Tensors 326
7.3 Balance Principles: Stress Tensors 337
7.4 Constitutive Theory: Hyperelastic Stress–Strain Relation 351
Part IV A NEW FINITE ELEMENT METHOD 365
8 C-type Finite Element Method 367
8.1 Introduction 367
8.2 Variational Formulations 369
8.3 Energy Precursor to Finite Element Method 386
8.4 c-type FEM: Linear Elasticity and Heat Conduction 402
8.5 Newton Iteration and Arc Length Constraint 438
8.6 Gauss–Legendre Quadrature Formulas 446
Part V APPLICATIONS: LINEAR AND NONLINEAR 457
9 Application to Linear Problems and Locking Solutions 459
9.1 Introduction 459
9.2 c-type Truss and Bar Element 460
9.3 c-type Straight Beam Element 465
9.4 c-type Curved Beam Element 484
9.5 c-type Deep Beam: Plane Stress Element 498
9.6 c-type Solutions: Locking Problems 509
10 Nonlinear Beams 523
10.1 Introduction 523
10.2 Beam Geometry: Definition and Assumptions 530
10.3 Static and Dynamic Equations: Engineering Approach 534
10.4 Static and Dynamic Equations: Continuum Approach – 3D to 1D 539
10.5 Weak Form: Kinematic and Configuration Space 555
10.6 Admissible Virtual Space: Curvature, Velocity and Variation 560
10.7 Real Strain and Strain Rates from Weak Form 570
10.8 Component or Operational Vector Form 580
10.9 Covariant Derivatives of Component Vectors 587
10.10 Computational Equations of Motion: Component Vector Form 590
10.11 Computational Derivatives and Variations 596
10.12 Computational Virtual Work Equations 607
10.13 Computational Virtual Work Equations and Virtual Strains: Revisited 614
10.14 Computational Real Strains 627
10.15 Hyperelastic Material Property 630
10.16 Covariant Linearization of Virtual Work 639
10.17 Material Stiffness Matrix and Symmetry 655
10.18 Geometric Stiffness Matrix and Symmetry 658
10.19 c-type FE Formulation: Dynamic Loading 673
10.20 c-type FE Implementation and Examples: Quasi-static Loading 685
11 Nonlinear Shell 721
11.1 Introduction 721
11.2 Shell Geometry: Definition and Assumptions 727
11.3 Static and Dynamic Equations: Continuum Approach – 3D to 2D 746
11.4 Static and Dynamic Equations: Continuum Approach – Revisited 763
11.5 Static and Dynamic Equations: Engineering Approach 771
11.6 Weak Form: Kinematic and Configuration Space 783
11.7 Admissible Virtual Space: Curvature, Velocity and Variation 788
11.8 Real Strain and Strain Rates from Weak Form 799
11.9 Component or Operational Vector Form 810
11.10 Covariant Derivatives of Component Vectors 817
11.11 Computational Equations of Motion: Component Vector Form 820
11.12 Computational Derivatives and Variations 830
11.13 Computational Virtual Work Equations 841
11.14 Computational Virtual Work Equations and Virtual Strains: Revisited 851
11.15 Computational Real Strains 861
11.16 Hyperelastic Material Property 864
11.17 Covariant Linearization of Virtual Work 877
11.18 c-type FE Formulation: Dynamic Loading 891
11.19 c-type FE Formulation: Quasi-static Loading 914
11.20 c-type FE Implementation and Examples: Quasi-static Loading 930
Index 967
