Recent Approaches in the Theory of Plates and Plate-Like Structures

This book presents the various approaches in establishment the basic equations of one- and two-dimensional structural elements. In addition, the boundaries of validity of the theories and the estimation of errors in approximate theories are given. Many contributions contain not only new theories, but also new applications, which makes the book interesting for researcher and graduate students. Preface 7 Contents 13 List of Contributors 20 Chapter 1 On one Class of Spatial Problems of Layered Plates and Applications in Seismology 25 1.1 Introduction 25 1.2 Asymptotic Solutions of 3D Quasistatic Problem 27 1.3 Mathematically Precise Solutions 32 1.4 Investigation of Dynamic Processes 34 1.5 Conclusions 39 References 39 Chapter 2 Asymmetric Buckling of Heterogeneous Annular Plates 41 2.1 Introduction 41 2.2 Problem Formulation 42 2.3 Equations for Buckling 45 2.4 Numerical Results 46 2.5 Conclusion 48 References 48 Appendix 49 Chapter 3 Bending Stiffness of Multilayer Plates with Alternating Soft and Hard Layers 51 3.1 Introduction 51 3.2 Free Vibration and Bending of Multilayer Plate 52 3.3 Asymptotic Integration of Three-dimensional Equations 53 3.4 The Transverse Shear Stiffness 55 3.5 The Exact Value of the Shear Stiffness 55 3.6 About the TR Model for a Homogeneous Transversally Isotropic Plate 56 3.8 Numerical Results. Three-layer Plate Symmetrical in Thickness 57 3.9 Three-layer Plate Asymmetric in Thickness 58 3.10 Multilayer Plate 59 3.11 Buckling of a Multilayer Plate Under Uniform Compression 60 3.12 Discussion 61 References 62 Chapter 4 On the Bending of Multilayered Plates Considering Surface Viscoelasticity 63 4.1 Introduction 63 4.2 Surface Viscoelasticity 64 4.3 Bending of a Plate-like Body 66 4.4 Conclusions 69 References 69 Chapter 5 Buckling of a Ring-stiffened Cylindrical Shell Under the External Pressure 73 5.1 Introduction 73 5.2 Formulation of the Problem 74 5.3 Calculation of Eigenvalues of a Beam Stiffened by Springs 76 5.4 Shell Parameters Optimization 77 5.5 Minimization of Mass 79 5.6 Results 80 5.7 Conclusion 84 References 85 Chapter 6 Free Vibration Corrugated Open Cylindrical Shells 86 6.1 Introduction 86 6.2 Basic Relationships 88 6.3 Solution to the Test Problem 90 6.4 Construction of a Computational Model 91 6.5 Results of Numerical Calculations 92 6.6 Conclusion 96 References 96 Chapter 7 On a New Theory of the Cosserat Continuum with Applications in Electrodynamics 98 7.1 Introduction 98 7.2 Maxwell’s Equations for Anisotropic Materials 99 7.3 A Cosserat Continuum of a Special Type: a Nonlinear Theory 100 7.4 A Cosserat Continuum of a Special Type: the Linear Theory 101 7.5 Mechanical Analogies of Physical Quantities in the Linear Theory 102 7.6 The Nonlinear Theory and Maxwell’s Equations 102 7.7 On the Theories Based on Translational Degrees of Freedom 105 7.8 The Gauß Law for Gravitational Field 106 7.9 Conclusions 108 References 109 Chapter 8 Hierarchical Models of Conduction of Heat in Continua Contained in Prismatic Shell-like Domains 111 8.1 Introduction 111 8.2 Governing System of Conduction of Heat 112 8.3 Mathematical Moments 113 8.4 Construction of Hierarchical Models 118 8.5 The N = 0 Approximation 120 8.6 Case of Cusped Bodies 121 8.7 Conclusions 123 References 123 Chapter 9 Dynamic Sliding Contact for a Thin Elastic Layer 125 9.1 Introduction 126 9.2 Two-Sided Sliding 127 9.3 One-Sided Sliding 129 9.4 Concluding Remarks 134 References 134 Chapter 10 Analytical Approach to the Derivation of the Stress Field of a Cylindrical Shell with a Circular Hole under Axial Tension 137 10.1 Introduction 137 10.2 Problem Formulation 138 10.3 Solution 139 10.4 New Approach 141 10.5 Boundary Conditions 143 10.6 System Investigation 145 10.7 Results 147 References 148 Appendix 151 Chapter 11 Analysis of Solutions for Elliptic Boundary Layer in Cylindrical Shells at Edge Shock Loading 153 11.1 Introduction 153 11.2 Statement of the Problem 154 11.3 Equivalent Problem for the Infinite Shell 156 11.4 Solution of the Equivalent Problem for the Cylindrical Shell 158 References 161 Chapter 12 Dimension Reduction in the Plate with Tunnel Cuts 163 12.1 Introduction 163 12.2 Statement of the Problem 164 12.3 Problem 12.1 with Index i = 2 166 12.3.1 In-plane Shift 167 12.3.2 Torsion 167 12.4 Problem 12.2 with Indices i = ξ = 1,3 = x,z. Deformation in the Direction Perpendicular to the Fibers 169 12.4.1 Index AB = 22. Tension-compression and Bending Along the Fibers (in the 0xz-plane) 170 12.4.2 Index AB = 11. Tension-compression and Bending Perpendicular to the Fibers (in the 0yz-plane) 171 12.4.3 Index AB = 12,21. Shift/Torsion Perpendicular to the Fibers (in the 0yz-plane) 172 12.5 Numerical Solutions 172 12.5.1 The Boundary Layers 173 12.5.2 Wrinkling of the Top and Bottom Surfaces of the Plate 174 12.6 The Macroscopic SSS of General Form 174 12.7 Conlusions 174 References 175 Chapter 13 Topological Optimization of Multilayer Structural Elements of MEMS/NEMS Resonators with an Adhesive Layer Subjected to Mechanical Loads 176 13.1 Introduction 176 13.2 Statement of the Topological Optimization Problem 178 13.3 Case Study 1 179 13.4 Case Study 2 182 13.5 Concluding Remarks 185 References 186 Chapter 14 Forced Vibration Analysis of Laminated Piezoelectric Plates by a Strong Sampling Surfaces Formulation 188 14.1 Introduction 188 14.2 Basic Assumptions 189 14.3 Strong SAS Formulation 190 14.4 Free Vibrations of Simply Supported Piezoelectric Plates 192 14.5 Forced Vibrations of Simply Supported Piezoelectric Plates 194 14.6 Conclusions 195 References 198 Chapter 15 Asymptotic Analysis of Buckling of Layered Rectangular Plates Accounting for Boundary Conditions and Edge Effects Induced by Shears 200 15.1 Introduction 200 15.2 Governing Equations 202 15.3 Simply Supported Plate with the Edge Diaphragms 206 15.4 Buckling Modes Accounting for the Edge Effects 207 15.4.1 Layered Plates with the Reduced Young’s and Shear Moduli of the same Order 208 15.4.2 Layered Plates with Small Reduced Shear Modulus 216 15.5 Analysis of Influence of Boundary Conditions and Edge Effects on Critical Force 217 15.6 Conclusions 221 References 221 Chapter 16 Semi-analytical Model for the Close-range Stress Analysis of Transverse Cracks in Composite Plates 224 16.1 Introduction 224 16.2 Structural Situation 226 16.3 Semi-analytical Approach 227 16.3.1 CLT Solution 227 16.3.2 Internal Solution 228 16.4 Results 232 16.5 Summary and Conclusions 233 References 233 Appendix 236 Chapter 17 Shear Deformable Elastic Beam Models in Vibration and Sensitivity of Natural Frequencies to Warping Effects 238 17.1 Introduction 239 17.2 Kinematics, Stresses, and Warping Function 241 17.3 Equilibrium Equations and Governing Differential Equations 244 17.4 An Alternative Form of the Motion Equation 246 17.4.1 A Simplification: From Double Spectrum to Single Spectrum 247 17.5 Application to a Simply Supported Beam 248 17.5.1 Other Comments on the Obtained Numerical Results 252 17.6 Conclusion 253 References 255 Chapter 18 Conceptual Approaches to Shells. Advances and Perspectives 258 18.1 Introduction 258 18.2 Historical Perspective 260 18.2.1 On the Tension-torsion Test of Beams 260 18.2.2 Beam Bending and Neutral Line Detection History 261 18.2.3 To the History of the Derivation of the Shell Equations 261 18.3 Analytical Methods in the Shell Theory 263 18.4 A Mathematical View of Shells 265 18.5 On Forms of Shells 266 18.6 Problems of Modern Biomechanics 268 18.7 Micro- and Macro-scale Shells 269 18.8 The Role of Mechanics in the Development of Science 270 References 271 Chapter 19 Necessary Conditions for Energy Minimizers in a Cosserat Model of Fiber-reinforced Elastic Solids 274 19.1 Introduction 274 19.2 Cosserat Elasticity of Fiber-reinforced Materials 275 19.2.1 Kinematical and Constitutive Variables in Cosserat Elasticity 276 19.2.2 Virtual Power and Equilibrium 277 19.2.3 Fiber-matrix Interaction 278 19.3 Conservative Problems, Energy Minimizers and the Legendre-Hadamard Conditions 280 References 286 Chapter 20 Vibration Control of a Non-homogeneous Circular Thin Plate 288 20.1 Introduction 288 20.2 Statement of the Problem 289 20.3 Boundary Eigenvalue Problem 291 20.4 Results 292 20.4.1 Plate with the Variable Thickness 292 20.4.2 Plate with the Variable Stiffness 293 20.5 Conclusions 296 References 296 Appendix 297 Chapter 21 Modeling of an Inhomogeneous Circular Timoshenko Plate with an Elastically Supported Boundary 298 21.1 Introduction 298 21.2 Statement of the Problem 300 21.3 Direct Problem Solving Method 301 21.4 Computational Experiments 301 21.5 Conclusion 306 References 307 Chapter 22 Effect of Distributed Dislocations on Large Deformations of Cylindrical Tube made of Micropolar Elastic Material 308 22.1 Introduction 308 22.2 Input Relations 309 22.3 Cylindrical Tube with Distributed Dislocations 311 22.4 Distribution of Straight Edge Dislocations 314 22.5 Exact Solution 316 22.6 Conclusion 325 References 325