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

Description

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 Contents List of Contributors Chapter 1 On one Class of Spatial Problems of Layered Plates and Applications in Seismology 1.1 Introduction 1.2 Asymptotic Solutions of 3D Quasistatic Problem 1.3 Mathematically Precise Solutions 1.4 Investigation of Dynamic Processes 1.5 Conclusions References Chapter 2 Asymmetric Buckling of Heterogeneous Annular Plates 2.1 Introduction 2.2 Problem Formulation 2.3 Equations for Buckling 2.4 Numerical Results 2.5 Conclusion References Appendix Chapter 3 Bending Stiffness of Multilayer Plates with Alternating Soft and Hard Layers 3.1 Introduction 3.2 Free Vibration and Bending of Multilayer Plate 3.3 Asymptotic Integration of Three-dimensional Equations 3.4 The Transverse Shear Stiffness 3.5 The Exact Value of the Shear Stiffness 3.6 About the TR Model for a Homogeneous Transversally Isotropic Plate 3.8 Numerical Results. Three-layer Plate Symmetrical in Thickness 3.9 Three-layer Plate Asymmetric in Thickness 3.10 Multilayer Plate 3.11 Buckling of a Multilayer Plate Under Uniform Compression 3.12 Discussion References Chapter 4 On the Bending of Multilayered Plates Considering Surface Viscoelasticity 4.1 Introduction 4.2 Surface Viscoelasticity 4.3 Bending of a Plate-like Body 4.4 Conclusions References Chapter 5 Buckling of a Ring-stiffened Cylindrical Shell Under the External Pressure 5.1 Introduction 5.2 Formulation of the Problem 5.3 Calculation of Eigenvalues of a Beam Stiffened by Springs 5.4 Shell Parameters Optimization 5.5 Minimization of Mass 5.6 Results 5.7 Conclusion References Chapter 6 Free Vibration Corrugated Open Cylindrical Shells 6.1 Introduction 6.2 Basic Relationships 6.3 Solution to the Test Problem 6.4 Construction of a Computational Model 6.5 Results of Numerical Calculations 6.6 Conclusion References Chapter 7 On a New Theory of the Cosserat Continuum with Applications in Electrodynamics 7.1 Introduction 7.2 Maxwell’s Equations for Anisotropic Materials 7.3 A Cosserat Continuum of a Special Type: a Nonlinear Theory 7.4 A Cosserat Continuum of a Special Type: the Linear Theory 7.5 Mechanical Analogies of Physical Quantities in the Linear Theory 7.6 The Nonlinear Theory and Maxwell’s Equations 7.7 On the Theories Based on Translational Degrees of Freedom 7.8 The Gauß Law for Gravitational Field 7.9 Conclusions References Chapter 8 Hierarchical Models of Conduction of Heat in Continua Contained in Prismatic Shell-like Domains 8.1 Introduction 8.2 Governing System of Conduction of Heat 8.3 Mathematical Moments 8.4 Construction of Hierarchical Models 8.5 The N = 0 Approximation 8.6 Case of Cusped Bodies 8.7 Conclusions References Chapter 9 Dynamic Sliding Contact for a Thin Elastic Layer 9.1 Introduction 9.2 Two-Sided Sliding 9.3 One-Sided Sliding 9.4 Concluding Remarks References Chapter 10 Analytical Approach to the Derivation of the Stress Field of a Cylindrical Shell with a Circular Hole under Axial Tension 10.1 Introduction 10.2 Problem Formulation 10.3 Solution 10.4 New Approach 10.5 Boundary Conditions 10.6 System Investigation 10.7 Results References Appendix Chapter 11 Analysis of Solutions for Elliptic Boundary Layer in Cylindrical Shells at Edge Shock Loading 11.1 Introduction 11.2 Statement of the Problem 11.3 Equivalent Problem for the Infinite Shell 11.4 Solution of the Equivalent Problem for the Cylindrical Shell References Chapter 12 Dimension Reduction in the Plate with Tunnel Cuts 12.1 Introduction 12.2 Statement of the Problem 12.3 Problem 12.1 with Index i = 2 12.3.1 In-plane Shift 12.3.2 Torsion 12.4 Problem 12.2 with Indices i = ξ = 1,3 = x,z. Deformation in the Direction Perpendicular to the Fibers 12.4.1 Index AB = 22. Tension-compression and Bending Along the Fibers (in the 0xz-plane) 12.4.2 Index AB = 11. Tension-compression and Bending Perpendicular to the Fibers (in the 0yz-plane) 12.4.3 Index AB = 12,21. Shift/Torsion Perpendicular to the Fibers (in the 0yz-plane) 12.5 Numerical Solutions 12.5.1 The Boundary Layers 12.5.2 Wrinkling of the Top and Bottom Surfaces of the Plate 12.6 The Macroscopic SSS of General Form 12.7 Conlusions References Chapter 13 Topological Optimization of Multilayer Structural Elements of MEMS/NEMS Resonators with an Adhesive Layer Subjected to Mechanical Loads 13.1 Introduction 13.2 Statement of the Topological Optimization Problem 13.3 Case Study 1 13.4 Case Study 2 13.5 Concluding Remarks References Chapter 14 Forced Vibration Analysis of Laminated Piezoelectric Plates by a Strong Sampling Surfaces Formulation 14.1 Introduction 14.2 Basic Assumptions 14.3 Strong SAS Formulation 14.4 Free Vibrations of Simply Supported Piezoelectric Plates 14.5 Forced Vibrations of Simply Supported Piezoelectric Plates 14.6 Conclusions References Chapter 15 Asymptotic Analysis of Buckling of Layered Rectangular Plates Accounting for Boundary Conditions and Edge Effects Induced by Shears 15.1 Introduction 15.2 Governing Equations 15.3 Simply Supported Plate with the Edge Diaphragms 15.4 Buckling Modes Accounting for the Edge Effects 15.4.1 Layered Plates with the Reduced Young’s and Shear Moduli of the same Order 15.4.2 Layered Plates with Small Reduced Shear Modulus 15.5 Analysis of Influence of Boundary Conditions and Edge Effects on Critical Force 15.6 Conclusions References Chapter 16 Semi-analytical Model for the Close-range Stress Analysis of Transverse Cracks in Composite Plates 16.1 Introduction 16.2 Structural Situation 16.3 Semi-analytical Approach 16.3.1 CLT Solution 16.3.2 Internal Solution 16.4 Results 16.5 Summary and Conclusions References Appendix Chapter 17 Shear Deformable Elastic Beam Models in Vibration and Sensitivity of Natural Frequencies to Warping Effects 17.1 Introduction 17.2 Kinematics, Stresses, and Warping Function 17.3 Equilibrium Equations and Governing Differential Equations 17.4 An Alternative Form of the Motion Equation 17.4.1 A Simplification: From Double Spectrum to Single Spectrum 17.5 Application to a Simply Supported Beam 17.5.1 Other Comments on the Obtained Numerical Results 17.6 Conclusion References Chapter 18 Conceptual Approaches to Shells. Advances and Perspectives 18.1 Introduction 18.2 Historical Perspective 18.2.1 On the Tension-torsion Test of Beams 18.2.2 Beam Bending and Neutral Line Detection History 18.2.3 To the History of the Derivation of the Shell Equations 18.3 Analytical Methods in the Shell Theory 18.4 A Mathematical View of Shells 18.5 On Forms of Shells 18.6 Problems of Modern Biomechanics 18.7 Micro- and Macro-scale Shells 18.8 The Role of Mechanics in the Development of Science References Chapter 19 Necessary Conditions for Energy Minimizers in a Cosserat Model of Fiber-reinforced Elastic Solids 19.1 Introduction 19.2 Cosserat Elasticity of Fiber-reinforced Materials 19.2.1 Kinematical and Constitutive Variables in Cosserat Elasticity 19.2.2 Virtual Power and Equilibrium 19.2.3 Fiber-matrix Interaction 19.3 Conservative Problems, Energy Minimizers and the Legendre-Hadamard Conditions References Chapter 20 Vibration Control of a Non-homogeneous Circular Thin Plate 20.1 Introduction 20.2 Statement of the Problem 20.3 Boundary Eigenvalue Problem 20.4 Results 20.4.1 Plate with the Variable Thickness 20.4.2 Plate with the Variable Stiffness 20.5 Conclusions References Appendix Chapter 21 Modeling of an Inhomogeneous Circular Timoshenko Plate with an Elastically Supported Boundary 21.1 Introduction 21.2 Statement of the Problem 21.3 Direct Problem Solving Method 21.4 Computational Experiments 21.5 Conclusion References Chapter 22 Effect of Distributed Dislocations on Large Deformations of Cylindrical Tube made of Micropolar Elastic Material 22.1 Introduction 22.2 Input Relations 22.3 Cylindrical Tube with Distributed Dislocations 22.4 Distribution of Straight Edge Dislocations 22.5 Exact Solution 22.6 Conclusion References