Shell-like Structures

Name: Shell-like Structures | Non-classical Theories and Applications
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Non-classical Theories and Applications

Holm Altenbach Victor A. Eremeyev(Editor)

Springer (Publisher)

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Published on 3. July 2011

XI, 750 pages

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Content

Intro
Shell-like Structures
Preface
Contents
Mathematical Problems
1Nonclassical Spatial Boundary Value Problemsof Statics and Dynamics of Shellsand the Asymptotic Method of Their Solution
1.1 Introduction
1.2 Second and Mixed Static Boundary Value Problem ofAnisotropic Thermoelastic Shells
1.3 Solutions of Illustrative Problems
References
2 Analytical Solution for the Bending of a Plateon a Functionally Graded Layer of ComplexStructure
Introduction
2.1 Interaction of a Thin Circular Plate with aNon-Homogeneous Half-Space. Statement and Formulationof the Problem
2.2 Construction of the Solution
2.3 Determination of the Surface Displacement Outsideof the Plate
2.4 Numerical Examples
References
3 Analysis of the Deformation of Multi-layeredOrthotropic Cylindrical Elastic Shells Usingthe Direct Approach
3.1 Introduction
3.2 Basic Equations for Shells. General Geometry
3.3 Cylindrical Shells with Arbitrary Cross-Section Shape.Formulation of the Problem
3.4 Solution Procedure
3.4.1 Solutions Depending Only on the Circumferential Coordinate
3.4.2 Basic Solutions to the Equilibrium Equations
3.4.3 Cylindrical Shells Subjected to Terminal Resultant Forcesand Moments
3.4.4 Deformation of Loaded Cylindrical Shells
3.5 Circular Cylindrical Shells
3.6 Three-Layered Shells
References
4 Asymptotic Integration of One NarrowPlate Problem
4.1 Introduction
4.2 Oscillations of Narrow Orthotropic Rectangular Plate onElastic Foundation. Governing Equations
4.3 First Iteration Process
4.4 Second Iteration Process
4.4.1 Solution of Generalized Bi-Harmonic Problem
4.4.2 One More Boundary Value Problem
4.5 Estimation of the Remainder of Series in the Linear Case
4.6 Closing Remarks
References
5 On Cusped Shell-like Structures
5.1 Introduction
5.2 Geometry of Structures Under Consideration
5.3 Hierarchical Models
5.4 Cusped Shells and Plates
5.5 Cusped Beams
5.6 Relations of 3D, 2D, and 1D Problems
5.7 Cusped Prismatic Shell-Fluid Interaction Problems
References
6 Effect of the Tangential Loads on the Bendingof Elastic Plates
6.1 Problem Formulation
6.2 Solutions for Lightweight Restraint Conditions at x = 0
6.3 Solutions for Straitened Sliding Contact Conditions at x = 0
6.4 Conclusions
References
7 On the Convergence of an Iteration Methodin Timoshenko's Theory of Plates
7.1 Problem Formulation
7.2 Reduction of the Problem
7.3 The Numerical Algorithm
7.4 The Jacobi Matrix
7.5 The Iteration Method Convergence and Error
7.6 An Alternative Form of Requirements
References
8 Mathematical Models of Micropolar ElasticThin Shells
8.1 Introduction
8.2 Problem Statem
8.3 Model of Micropolar Elastic Thin Shells with IndependentFields of Displacements and Rotations
8.4 Model of Micropolar Elastic Thin Shells with ConstrainedRotation
8.5 Model of Micropolar Shells with "Small TransverseStiffness"
Conclusion
Dynamics and Stability
9 Closed-Form Approximate Solutionfor the Postbuckling Behavior of OrthotropicShallow Shells Under Axial Compression
9.1 Introduction
9.2 Governing Equations
9.2.1 Classical Formulations
9.2.2 Non-Dimensional Quantities
9.2.3 In-Plane Displacements
9.3 Closed-Form Solution
9.3.1 Boundary Conditions
9.3.2 Linear Buckling Loads
9.3.3 Compatibility Condition - Stress Function
9.3.4 Equilibrium Condition - Load-Deflection-Relation
9.4 Results and Discussion
9.5 Conclusions
References
10Nonlinear Magnetoelastic Waves in a Plate
10.1 Introduction
10.2 Equations of Magnetoelasticity
10.3 Beam of Longitudinal Waves Propagation
10.4 Conclusion
References
11 Basic Concepts in the Stability Theoryof Thin-Walled Structures
11.1 Introduction
11.2 Some Historical Remarks
11.3 Notions and Definitions for Stability in the Theory of Thin-Walled Structures
11.4 Why Do We See Discrepancies between Theoretical andExperimental Results on the Stability of Thin-WalledStructures?
11.5 Conclusion
References
12 High-Frequency Free Vibrations of Platesin the Reissner's Type Theory
12.1 Introduction
12.2 Summary of the Basic Equations of Free Vibrations ofReissner's Plate
12.3 Asymptotic Analysis of the Equations of Reissner's PlateTheory
12.4 Approximate Formulation of the Problem ofHigh-Frequency Free Vibrations
12.5 Formulation of the Problem of High-Frequency FreeVibrations of the Reissner's Plate without Taking Accountof Rapidly Varying Function
12.6 Asymptotic and Numerical Analysis of High-Frequency Free Vibrations of Rectangular Plates
12.7 Discussion of the Physical Meaning of Obtained Results
References
13 On the Reconstruction of Inhomogeneous InitialStresses in Plates
13.1 Introduction
13.2 Formulation of Out-of-Plane Vibrations of the ThinPrestressed Plate
13.3 Derivation o
13.4 Solution of the Direct Problem
13.5 Analysis of the Prestress Level Influence on the FrequencyResponse Functions of the Plate Points
13.6 Formulation of the Inverse Problem of theNon-homogeneous Prestress Reconstruction in the Solid
13.7 Formulation of the Inverse Problem of the UniaxialNon-Homogeneous Prestress Reconstruction in theRectangular Plate
13.8 Numerical Experiments on the Prestress Identification
13.9 Conclusion
References
14 Dynamic Response of Pre-Stressed SpatiallyCurved Thin-Walled Beams of Open ProfileImpacted by a Falling ElasticHemispherical-Nosed Rod
14.1 Introduction
14.2 Problem Formulation and Governing Equations
14.2.1 Dynamic Response of Axially Prestressed Spatially CurvedThin-Walled Rods of Open Profile
14.2.2 Construction of the Desired Wave Fields in Terms of theRay Series
14.3 Impact Response of a Thin-Walled Beam of Open Profile
14.3.1 Impact of an Elastic Hemispherical-Nosed Rod against aThin-Walled Beam of Open Profile
14.3.1.1 Numerical Example
References
15 On Stability of Elastic Rectangular SandwichPlate Subject to Biaxial Compression
15.1 Introduction
15.2 Equilibrium of the Rectangular Sandwich Plate Subject toBiaxial Compression
15.3 The Perturbed State
15.4 Conclusion
References
Nonlinear Models and Coupled Fields
16 On the Nonlinear Theory of Two-Phase Shells
16.1 Introduction
16.2 Kinematics of 6-Parametric Theory of Shells
16.3 Integral Balance Equations
16.4 Local Shell Equations and Constitutive Relations
16.5 Continuity Conditions Along Phase Interface and KineticEquation
16.6 Variational Statement of Thermodynamic Equilibrium ofTwo-Phase Shell
16.7 Conclusions
References
17 A Gradient-Enhanced Damage Modelfor Viscoplastic Thin-Shell Structures
17.1 Introduction
17.2 Dynamic Thin Shell Model
17.2.1 Basic Kinematics of Shells
17.2.1.1 Shell Configurations
17.2.1.2 Motion of Shell Director
17.2.2 Strain Measures
17.2.1.1 Shell Configurations
17.2.1.2 Motion of Shell Director
17.2.3 Effective Stress Resultants
17.3 Gradient-Enhanced Damage of Shell Structures
17.3.1 Kinetic Energy
17.3.2 Potential Energy
17.3.3 Weak Formulation of Motion Equations
17.4 Local Constitutive Laws
17.4.1 Hyperelasticity Law on the Mid-surface
17.4.2 Viscoplastic Damage
17.5 Examples
17.5.1 Example 1
17.6 Conclusions
References
On Constitutive Restrictionsin the Resultant Thermomechanics of Shellswith InterstitialWorking
18.1 Introduction
18.2 Local Laws of Refined Resultant Thermomechanicsof Shells
18.3 Restrictions on the Form of Constitutive Equations
18.3.1 Viscous Shells with Heat Conduction
18.3.2 Thermoelastic Shells
18.4 Kinetic Constitutive Equations
18.5 Conclusions
References
19 Free Finite Rotations in Deformationof Thin Bodies
19.1 Separation of Local Rotations in Cauchy Continuum
19.2 A Rod Deformation Model with One-Dimensional Fieldof Finite Rotations
19.3 A Shell Deformation Model with Two-Dimensional Fieldof Finite Rotations
19.4 Axisymmetric Bending Modes of Plates and Shells
19.4.1 Circular Plate Under Radial Compression
19.4.2 Conical Shell Under Radial Compression
19.4.3 A Dome Axisymmetric Deformation in Thermal Cycleof Phase Transformations
Concluding Remark
References
On Universal Deformations of NonlinearIsotropic Elastic Shells
20.1 Basic Statements
20.2 Families of Non-Uniform Deformations
20.3 Uniform Deformations of an Isotropic Plate
20.4 Equilibrium of an Isotropic Cosserat Membrane
20.5 Conclusion
References
Numerical Analysis
21 Application of Genetic Algorithms to the ShapeOptimization of the Nonlinearly ElasticCorrugated Membranes
21.1 Introduction
21.2 Membrane Modeling
21.3 Genetic Algorithm
21.4 Numerical Results
21.5 Conclusion
References
22 Advances in Quadrilateral Shell Elementswith Drilling Degrees of Freedom
22.1 Introduction
22.1.1 Element Configuration, Shape Functions and Rigid BodyProjection
22.2 Linear Element Formulation
22.2.1 Membrane Element
22.2.2 Plate Bending Element
22.2.2.1 The Bending Part of the Functional
22.2.2.2 The Shear Part of the Functional
22.2.2.3 The Stiffness Matrix
22.3 Generalization to Geometrical Nonlinearities
22.3.1 The Constant Part of the Internal Force Vector
22.3.2 The Internal Stabilization Force Vector
22.4 The Mass Matrix
22.5 Numerical Examples
22.5.1 Slab Supported by a Central Column
22.5.2 Hemispherical Shell
22.5.3 Roll-Up of a Clamped Beam
22.6 Summary
References
23 Invariant-Based Geometrically NonlinearFormulation of a Triangular Finite Elementof Laminated Shells
23.1 Introduction
23.2 Basic Requirements and Assumptions
23.3 Natural Components of Strain
23.4 Invariants of Strain Tensor
23.5 Templates for Invariants of Tensors
23.6 Strain Energy Density for Plane Stress
23.6.1 Isotropic Material
23.6.2 Six-Constant Anisotropic Material
23.7 Transverse Shear Strain Energy Density
23.7.1 Isotropic Material
23.7.2 Transversely Isotropic Material
23.7.3 Anisotropic Material
23.8 Total Strain Energy of Laminated Shell
23.9 Approximation of Natural Strains
23.9.1 Normal Strains
23.9.2 Curvature Changes and Transverse Shear Strains
23.10 Strain Energy of Triangular Shell Element
23.11 Variations of the Strain Energy and Algorithm of Solution
23.12 Numerical Testing
23.12.1 Linear Analysis of Orthotropic Square Plates UnderUniformly Distributed Transverse Load
23.12.2 Linear Analysis of Laminated Square Plates UnderUniformly Distributed Transverse Load
23.12.3 Pure Bending of a Plate
23.12.4 Nonlinear Analysis of Laminated Square
23.12.5 Snap-through of a Hinged Cylindrical Laminated Roof
23.12.6 Circular Ring Pinched by Four Loads
23.13 Concluding Remarks
References
24 Consistency Issues in Shell Elementsfor Geometrically Nonlinear Problems
24.1 Introduction
24.2 Thorough Micropolar Setting of the Surface Mechanics
24.2.1 Constitutive Characterization of the 3D Non-Polar Medium
24.2.2 A Linear Example
24.2.3 Micropolar Surface Mechanics
24.2.4 A Nonlinear Example
24.3 Consistent Approximation of the Element Kinematic Field
24.4 Consistent Linearization and Discrete Approximation
24.5 Consistent Modeling of the Displacement Field on CurvedSurfaces
24.5.1 Helicoidal Parameterization and Helicoidal Modeling
24.5.2 The Helicoidal Shell Element
24.6 Conclusion
References
25 An Algorithm for the Automatisation of PseudoReductions of PDE Systems Arising fromthe Uniform-Approximation Technique
25.1 Introduction
25.1.1 The Uniform-Approximation Technique
25.1.2 The General Aim of the Pseudo-Reduction Process
25.1.3 The Aim of the Presented Algorithm
25.1.4 The Main Idea of the Presented Algorithm
25.2 The Setting
25.2.1 The One-Dimensional Case
25.2.2 Homogeneous Material
25.2.3 Regularity
25.2.4 The Original PDE System
25.2.5 Differentiability
25.2.6 Multiplication with Characteristic Parameters
25.2.7 Division by Characteristic Parameters
25.3 The Algorithm
25.3.1 Selection of Main Variable and Main PDE
25.3.2 cd-Variables
25.3.3 Right-Hand Sides
25.3.4 D-Factors
25.3.5 Generating Additional Equations
25.3.5.1 A Simple Approach for Generating All Additional Equations
25.3.5.2 Extension by Divisions by Characteristic Parameters
25.3.5.3 Extension by Multiplications by Characteristic Parameters
25.3.6 Computation of Pseudo Reduction Possibilities
25.4 Summary
References
26 Recent Improvements in Hu-Washizu ShellElements with Drilling Rotation
26.1 Introduction
26.2 Three-Dimensional Formulation with Rotations
26.3 Shell with Drilling Rotation
26.4 Features of Enhanced HWShell Elements
26.4.1 Skew Coordinates
26.4.2 Assumed Stress and Strain
26.4.3 Enhanced Assumed Displacement Gradient (EADG)Method
26.4.5 Approximation of Lagrange Multiplier for Drill RC
26.5 Numerical Tests
26.5.1 Cook's Membrane
26.5.2 Pinched Hemisphere with Hole
26.6 Final Remarks
References
Engineering Design
27 Dynamic Analysis of Debonded Sandwich Plateswith Flexible Core - Numerical Aspects andSimulation
27.1 Introduction
27.2 FE Model Developments
27.2.1 Face Sheet FE Model
27.2.2 Core FE Model
27.2.3 General 3-D FE Model
27.3 Aspects of FE Modeling
27.3.1 FE Equations of Motion
27.3.2 Contact Model
27.3.3 Integration Rule
27.3.4 FE Analysis
27.4 Numerical Results
27.4.1 Validation
27.4.2 Free Vibration Analysis
27.4.3 Forced Vibration Analysis
27.5 Conclusions
References
28 On Elasto-Plastic Analysis of Thin Shellswith Deformable Junctions
28.1 Introduction
28.2 Notation and Basic Relations
28.3 Constitutive Elasto-Plastic Modelling in Thin Shells
28.4 Deformation and Stress States in Axisymmetric Casing
28.5 Conclusions
References
29 Thermal Stress and Strain of Solar Cellsin Photovoltaic Modules
29.1 Introduction
29.2 Photovoltaic Modules
29.2.1 Structure
29.2.2 Thermal Cycling
29.3 Experimental
29.4 Material Models
29.4.1 Silicon
29.4.2 Glass
29.4.3 Back Sheet
29.4.4 Encapsulant
29.4.4.1 Linear Elasticity
29.4.4.2 Temperature-Dependent Linear Elasticity
29.4.4.3 Viscoelasticity
29.5 Finite-Element-Simulations
29.5.1 Evaluation of Material Models
29.5.2 Photovoltaic Module
29.5.2.1 Geometry and Boundary Conditions
29.5.2.2 Simulation Results
29.5.2.3 Discussion
29.6 Conclusion
References
30 Computational Models of Laminated Glass Plateunder Transverse Static Loading
30.1 Introduction
30.2 Experiments
30.2.1 Compressive Shear Test
30.2.2 Cylindrical Bending Test
30.3 Computational Models
30.3.1 3-D Brick Element Model
30.3.2 Shell Element Model
30.4 Triplex Laminated Glass (TLG) Plate Element
30.4.1 Stress and Strain in Laminated Glass Plate
30.4.2 Potential Strain Energy of Laminated Glass Plate
30.4.3 Stiffness Matrix Derivation
30.4.4 TLG Plate Element Validation
30.5 Conclusions
References
31 Unbending of Curved Tube by Internal Press
31.1 Introduction
31.2 Pure Bending Deformation
31.3 Cylindrical Membrane
31.4 Circular Cross Section
31.5 Conclusions
References
32 Characterization of Polymeric Interlayersin Laminated Glass Beams for PhotovoltaicApplications
32.1 Introduction
32.1.1 First Order Shear Deformation Theory
32.1.2 Layerwise Beam Theory
32.2 Discussion
References
33 On the Determination of Edge ReinforcementProperties for Optimum Lightweight Designof Composite Stiffeners
33.1 Introduction
33.2 Analysis
33.3 Determination of Optimum Lightweight Design
33.4 Conclusion
References
Micro- and Nanomechanical Applications
34 Evaluation of the Mechanical Parametersof Nanotubes by Means of Nonclassical Theoriesof Shells
34.1 Introduction
34.2 Problem Definition
34.3 Correlations of the Rodionova-Titaev-Chernykh and thePaliy-Spiro Shell Theory
34.4 Numerical Method
34.5 Numerical Results
References
35 Gauss-Codazzi Equations for Thin Filmsand Nanotubes Containing Defects
35.1 Introduction
35.2 Equations of the Nonlinear Shell Theory
35.3 Dislocations and Disclinations in Shell-Like Structures
35.4 Gauss-Codazzi Equations in Presence of Defects
35.5 Conclusions
References
36 Effective Mechanical Properties of Closed-CellFoams Investigated with a MicrostructuralModel and Numerical Homogenisation
36.1 Introduction
36.2 Finite Element Analysis
36.3 Results
36.3.1 Imperfections
36.3.1.1 Curved Cell Walls
36.3.1.2 Random Geometrical Irregularity
36.3.2 Linear Buckling Analysis
36.3.3 Postbuckling Analysis
36.4 Conclusion
References
37 What Shell Theory Fits Carbon Nanotubes?
37.1 Introduction
37.2 An Abridged Presentation of the Shell Theory We Propose
37.2.1 Kinematics
37.2.2 Balance Equations
37.2.3 Constitutive Equations
37.2.3.1 General Orthotropic Response
37.2.3.2 The Case of MWCNTs
37.2.3.3 The Case of SWCNTs
37.2.4 Field Equations
References
38 A Variationally Consistent Derivation ofMicrocontinuum Theories
38.1 Introduction
38.2 Basic Equations
38.3 Equilibrium Equations
38.3.1 Micromorphic Theory
38.3.2 Micropolar Theory
38.3.3 Uncoupled Micromorphic Theory
38.3.4 Evaluation of the Residuum
38.4 Nonlinear Strain Measures
38.4.1 Evaluation of the Residuum
38.5 Constitutive Equations
38.6 Summary
References
39 Shell-Models for Multi-Layer CarbonNano-Particles
39.1 Introduction
39.2 Continuum Mechanics Modeling of Carbon Nanostructures
39.2.1 Layer Properties
39.2.2 Covalent Interlayer Bonding
39.2.3 Van der Waals Interactions
39.2.3.1 Planar Multi-Layer Carbon Nanostructures
39.2.3.2 Curved Multi-Layer Carbon Nanostructures
39.2.4 Surface Energy
39.3 Carbon Crystallites
39.3.1 Shell Model
39.3.2 Results
39.4 Carbon Onions
39.4.1 Shell Model
39.4.2 Growth Process and Results
39.5 Conclusion
Appendix
References
Biomechanics
40 Mechanics of Biological Membranes fromLattice Homogenization
40.1 Introduction
40.2 Beam Equations in the Geometrically NonlinearFramework
40.3 Discrete Homogenization of Networks
40.3.1 Simplified Beam Model
40.3.2 Stress and Couple Stress Vectors
40.3.3 Stress Tensor and Micromoment Tensor
40.4 Non-linear Problem
40.5 Application: Equivalent Properties of the PeptidoglycanCellWall
40.6 Conclusion and Discussion
References
41 Biological and Synthetic Membranes: Modelingand Experimental Methodology
41.1 Introduction
41.2 Elasticity of Perfectly Flexible Membranes
41.2.1 Equilibrium Equations and Constitutive Laws
41.2.2 Experimentations with Isotropic Membranes
41.2.3 Quantification of Anisotropic Responses
41.3 Membranes with Bending Stiffness
41.3.1 Elastic Response Functions
41.3.2 Linearly Elastic Anisotropic Membranes
41.4 Inelastic Effects in Membranes
41.4.1 Spherical Inflation of Balloons
41.4.2 Stress Softening of Membranes
41.5 Discussions
References
42 Nonclassical Theories of Shells in Applicationto Soft Biological Tissues
42.1 Introduction
42.2 Problem Formulation
42.3 The Paliy-Spiro Theory
42.4 The Rodionova-Titaev-Chernykh Theory
42.5 Results
42.5.1 Relationships for the Deflections and Stresses
42.5.2 Numerical Examples
42.6 Conclusions
References
FGM and Laminated Plates and Shells
43 Axisymmetric Bending Analysis of TwoDirectional Functionally Graded Circular PlatesUsing Third Order Shear Deformation Theory
43.1 Introduction
43.2 Formulation of the Problem
43.3 Results and Discussion
43.4 Conclusions
References
44 Stability Analysis of Functionally Graded PlatesSubject to Thermal Loads
44.1 Introduction
44.2 Functionally Graded Materials
44.3 Stability Equations
44.4 Buckling Analysis
44.4.1 Uniform Temperature Rises
44.4.2 Linear Temperature Rise
44.4.3 Sinusoidal Temperature Rise
44.5 Numerical Results and Discussion
44.5.1 Comparisons
44.5.2 Buckling Analysis of FGM Plates
44.6 Conclusions
References
45 A Best Theory Diagram for Metallic andLaminated Shells
45.1 Introduction
45.2 Carrera Unified Formulation
45.2.1 Governing Differential Equations
45.3 Method to Build the Best Plate/Shell Theories
45.4 Results and Discussion
45.4.1 Plates
45.4.2 Shells
45.4.3 The Best Theory Diagram
45.5 Conclusions
References
46 In-Plane Strain and Stress Fields in Theoriesof Shearable Laminated Plates Subjectto Transverse Loads
46.1 Introduction
46.2 A Model of Shearable Multilayered Plate
46.3 Equilibrium Solutions
46.4 Two Other Models
46.5 Comparison
46.5.1 Uniform Load
46.5.2 Bessel Loads
46.5.3 Concluding Remarks
References
47 On the Use of a New Concept of SamplingSurfaces in Shell Theory
47.1 Introduction
47.2 Kinematic Description of Undeformed Shell
47.3 Kinematic Description of Deformed Shell
47.4 Displacement and Strain Approximations in ThicknessDirection
47.5 Variational Equation
47.6 Finite Element Formulation
47.7 Numerical Examples
47.7.1 Square Plate Under Sinusoidal Loading
47.7.2 Cylindrical Shell Under Sinusoidal Loading
47.8 Conclusions
References
48 Theory of Thin Adaptive Laminated ShellsBased on Magnetorheological Materials and ItsApplication in Problems on VibrationSuppression
48.1 Introduction
48.2 Sandwich Structure and Properties of Damping Layers
48.3 Governing Equations
48.3.1 Basic Hypotheses
48.3.2 Strain-Displacement Relations
48.3.3 Constitutive Equations
48.3.4 Stresses and Moments
48.3.5 Equations of Motion in Stress Terms
48.3.6 Governing Equations in Terms of Displacement and StressFunctions
48.4 Free and Forced Vibrations under Stationary MagneticField
48.4.1 Free Vibrations
48.4.2 Forced Vibrations
48.5 Soft Suppression of Running Low-Frequency Vibrations inMR Adaptive Shells
48.6 Conclusions
References

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