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

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