Shell-like Structures (eBook)

Preface 6
Contents 8
Thin-Walled Structural Elements: Classification, Classical and Advanced Theories, New Applications 9
1 Introduction: Historical Remarks 10
2 Some Examples of New Applications 23
3 Main Directions in the Theory of Plates and Shells 27
3.1 Kirchhoff Theory 27
3.2 Mindlin Plate Theory 33
3.3 Reissner Plate Theory 38
3.4 Reddy Plate Theory 41
3.5 Föppl-von Kármán Plate Theory 41
4 Direct Approach to the Theory of Shells 47
5 Examples of Advanced Theories 54
5.1 Nanoeffects 54
5.2 Direct Approach to Viscoelastic Plates 58
References 66
Basics of Mechanics of Micropolar Shells 71
1 Introduction 71
2 On Rigid Body Dynamics 73
3 Kinematics of a Micropolar Shell 77
4 Euler's Motion Laws of a Micropolar Shell 78
5 Strain Energy Density and Strain Measures 81
6 Constitutive Equations of an Elastic Isotropic Shell 82
7 The Virtual Work Principle and Formulation of Boundary Value Problems 83
8 Compatibility Conditions 85
9 Variational Statements 87
9.1 Lagrange-Type Principle 87
9.2 Hu-Washizu-Type Principle 87
9.3 Hamilton-Type Principle 88
10 Linear Theory of Micropolar Shells 88
11 Linearized Boundary-Value Problems 90
12 Eigen-Vibrations of Prestressed Micropolar Shells 93
12.1 Rayleigh Principle 93
12.2 Influence of Initial (Residual) Stresses 96
13 Constitutive Restrictions for Micropolar Shells 97
13.1 Linear Theory of Micropolar Shell 98
13.2 Coleman-Noll Inequality for Elastic Shells 98
13.3 Strong Ellipticity and Hadamard Inequality 101
13.4 Strong Ellipticity Condition and Acceleration Waves 103
13.5 Ordinary Ellipticity 106
14 Applications 107
14.1 Surface Stresses 107
14.2 Thin-Walled Structures Made of Micropolar Materials 108
14.3 Thin-Walled Structures Made of Viscoelastic Materials 109
14.4 Shells and Plates with Phase Transitions (PT) 109
14.5 Beams and Rods 112
15 Conclusions 112
References 112
The Bending-Gradient Theory for Laminates and In-Plane Periodic Plates 120
1 Introduction 120
2 The Asymptotic Expansion Framework 122
2.1 Notations 122
2.2 The 3D Problem 123
2.3 Scaling 125
2.4 Properties of the Non-Dimensional Solution 127
2.5 Expansion 129
3 Explicit or Cascade Resolution 131
3.1 Low Order Displacement Fields 131
3.2 Zeroth-Order Plate Model (Kirchhoff-Love) 132
3.3 Higher-Order Analysis 134
4 The Bending-Gradient Theory 136
5 Application of the Bending-Gradient Theory to Laminates 142
5.1 Voigt Notations 142
5.2 Closed-Form Solution for Pagano's Configuration 143
5.3 Numerical Applications 146
6 Periodic Plates 150
References 154
Some Problems on Localized Vibrations and Waves in Thin Shells 156
1 Introduction 156
2 The Equivalent Single Layer Model for Thin Laminated Shells 158
2.1 Different Approaches in Modelling of Laminated Shells 158
2.2 Sandwich Structure 159
2.3 Basic Hypotheses 159
2.4 Governing Equations 161
3 Free Localized Vibrations of Thin Cylindrical Shells: Asymptotic Approach 164
3.1 Statement of a Problem 164
3.2 Asymptotic Method of Tovstik 165
3.3 Examples 171
4 On Localized Eigenmodes of Thin Laminated Shell Containing Magnetorheological Elastomer 173
4.1 Motivation 173
4.2 Setting a Problem 174
4.3 Asymptotic Solution 176
4.4 Circular Cylinder with Nonuniform Physical Properties of the MR Layer 178
5 Localized Eigenmodes of a Sandwich Cylindrical Shell Prestressed by Axial Forces 180
5.1 Setting a Problem 180
5.2 Asymptotic Solution 182
5.3 Reconstruction of Asymptotic Expansions 186
5.4 Examples 189
6 Eiegenmodes Localized Near Parallel in Long Axially Prestressed Cylindrical Shells 191
6.1 Setting a Problem 191
6.2 Asymptotic Solution 193
6.3 Examples 195
7 Wave Packets in Thin Cylindrical Shells 196
7.1 Localized Stationary and Quasi-Stationary Vibrations 196
7.2 Setting a Problem 197
7.3 Asymptotic Approach 198
7.4 Examples 208
8 Conclusions 214
References 214
Six Lectures in the Mechanics of Elastic Structures 217
1 Generalities on the Validation of Theories of Thin Elastic Structures 217
1.1 Prologue 218
1.2 Generalities on Theory Validation 218
1.3 Validation Methodologies 219
1.4 Computational Validation 220
1.5 Breaking a Lance for Simple Model Theories 220
2 Validation via Variational Convergence 221
2.1 Dimension-Reduction Methods 221
2.2 Standard ?--Convergence, Stripped to the Bone 223
2.3 Standard ?--Convergence Validation of an Approximate Problem 223
2.4 Improved ?--Convergence Validation of an Approximate Problem 225
3 The Virtual Power Principle 228
3.1 The VPP is not a Variational Statement 228
3.2 The Standard VPP 228
3.3 A Strenghtened Version of the VPP 229
3.4 The PVP as a Dimension-Reduction Tool 230
4 A Modicum of Continuum Mechanics 232
4.1 Kinematics 232
4.2 Internal Constraints 237
5 Plate Buckling, à la von Kármán, But Not Quite 240
5.1 The 3D Buckling Problem 241
5.2 The 2D von Kármán Model 242
5.3 Back to the 3D Buckling Problem 242
5.4 Compatibility and v. K's 1st Equation 244
5.5 Equilibrium and v. K's 2nd Equation 245
6 Mechanical Scaling 246
6.1 The Scaling Procedure in Summary 246
6.2 Preparatory Scalings 247
6.3 The Scaled Load Potential 248
6.4 The Scaled Total-Energy Functional 248
6.5 A Mechanical Principle of Convergence in Energy 249
6.6 Taxonomy of Energy Functionals 249
6.7 Reissner-Mindlin's Plates 251
Selected Topics on Mixed/Enhanced Four-Node Shell Elements with Drilling Rotation 252
1 Introduction 253
1.1 Drilling Rotation 253
1.2 Enhanced and Mixed/Enhanced Shell Elements 255
1.3 Algorithmic Treatment of Finite Rotations 256
2 Shell Kinematics and Drilling Rotation 256
2.1 Extended Configuration Space 256
2.2 Reissner-Mindlin Shell Kinematics 257
2.3 Drilling Rotation Constraint 258
3 Shell Hu-Washizu Functional with Rotations 260
3.1 3D HW Functional with Rotations 260
3.2 Complete (Pure) HW Functional for Shells 261
3.3 Incomplete (Partial) HW Functionals for Shells 262
4 Finite Rotations: Simple Algorithmic Treatment 264
4.1 Basic Definitions 264
4.2 Variations of Rotation Tensor 267
4.3 Operator T 268
4.4 Differential ?T 269
4.5 First Variation of Rotation Tensor for Canonical Parametrization 270
4.6 Simple Algorithm for Treating Finite Rotations 273
5 Enhanced/Mixed HW Shell Elements 275
5.1 Skew Coordinates 275
5.2 Assumed Stress or Couple Resultants and Assumed Shell Strains 278
5.3 Enhanced Assumed Displacement Gradient (EADG) Method 279
5.4 Approximation of Drilling RC 281
6 Numerical Tests 284
6.1 Intersection of Two Perpendicular Elements 285
6.2 Cylindrical Shell Under Wind Load 286
6.3 Long C-Beam 289
7 Final Remarks 290
References 291