Nonlinear Wave Dynamics of Materials and Structures (Advanced Structured Materials, 122)

✍ Scribed by Holm Altenbach (editor), Victor A. Eremeyev (editor), Igor S. Pavlov (editor), Alexey V. Porubov (editor)

• Publisher: Springer
• Year: 2020
• Tongue: English
• Leaves: 473
• Category: Library

✦ Synopsis

This book marks the 60th birthday of Prof. Vladimir Erofeev – a well-known specialist in the field of wave processes in solids, fluids, and structures. Featuring a collection of papers related to Prof. Erofeev’s contributions in the field, it presents articles on the current problems concerning the theory of nonlinear wave processes in generalized continua and structures. It also discusses a number of applications as well as various discrete and continuous dynamic models of structures and media and problems of nonlinear acoustic diagnostics.

✦ Table of Contents

Preface
Contents
Contributors
1 Elimination of the Flutter Phenomenon in a Forced and Self-excited Nonlinear Beam Using an Improved Saturation Controller Algorithm
1.1 Introduction
1.2 Model of Structure
1.3 Mathematical Analysis
1.3.1 Multiple Scale Analysis
1.3.2 Equilibrium Solution
1.3.3 Stability Analysis
1.4 Results and Discussions
1.4.1 Nonlinear Beam Without Control
1.4.2 Effects of Velocity Feedback Controller
1.4.3 The Effects of Various Controller Parameters and the Time Margins of Various Time Delays
1.5 Conclusions
References
2 Use of the Modified Method of Parameter Continuation in Nonlinear Dynamics
2.1 Introduction
2.2 Asymptotic Method for Estimation of Free Vibrations of Beams and Rectangular Plates
2.2.1 Explanation of Main Ideas of the Method on the Base of Calculation of Beam Oscillations
2.2.2 Asymptotic Estimation of Free Vibrations of Nonlinear Plates
2.3 Using Asymptotic Method for Estimation of Parameter-Dependent Vibrations of Beams and Rectangular Plates
2.3.1 Explanation of the Method on the Base of Parameter-Dependent Beam Oscillations
2.3.2 Calculation of Parameter-Dependent Plates Vibrations
2.4 Nonlinear Vibration of Integrally Stiffened Cylindrical Shell
2.5 Using MMPC for Investigation of Systems with a Finite Number of Degrees of Freedom
2.5.1 Duffing Pendulum
2.5.2 Application of MMPC for Two Coupled Oscillators
2.6 Conclusions
References
3 A Mathematically Consistent Vector-Matrix Representation of Generalized Hooke's Law for Shear-Rigid Plates
3.1 Introduction
3.1.1 Motivation
3.1.2 Organization of the Paper
3.1.3 Preliminaries and Notation
3.2 Linear Elastic Shear-Rigid Plates
3.2.1 Generalized Hooke's Law
3.2.2 Properties of Tensorial Quantities
3.3 Vector-Matrix Notation
3.3.1 Derivation
3.3.2 Mathematical Consistency
3.4 Conclusion
References
4 Mathematical Simulation of the Plate–Beam Interaction Affected by Colored Noise
4.1 Introduction
4.2 Problem Statement
4.3 Solution Methods
4.4 Numerical Experiment
4.5 Conclusion
References
5 Dynamic Homogenization of a Chain with Bistable Springs. Statistical Approach
5.1 Introduction
5.1.1 Some Previous Results
5.1.2 Structure of the Paper
5.2 Preliminaries: Single-Spring Dynamics
5.2.1 Regimes
5.3 Dynamics of a Multi-spring Chain
5.3.1 Differential Equations and Numerics
5.3.2 Results of Numerical Simulation
5.4 Model for Dynamic Homogenization
5.4.1 Ansatz: Random Transitions
5.4.2 Properties
5.5 Validation of the Model
5.5.1 Numerical Issues
5.5.2 Relationship of Excitation Energy and Dispersion
5.6 Conclusions
References
6 Analysis of Resistance to Penetration of a Cone into Frozen Sand Based on Data from Inverted Experiments
6.1 Introduction
6.2 Measuring Bars Methodology in Inverse Experiment
6.3 Mathematical Formulation of the Impact and Penetration Problem
6.4 The Data from Inverted Experiments and Calculations
6.5 Conclusion
References
7 Some Solutions of Dynamic and Static Nonlinear Nonautonomous Klein-Fock-Gordon Equation
7.1 Introduction
7.2 Methods of Construction of Exact Analytical Solutions of Dynamic and Static Nonautonomous Nonlinear Klein-Fock-Gordon Equations
7.3 Ansatzes for Solution of Dynamic Equation
7.4 Particular Solutions
7.5 Ansatzes for Solution of Static Equation
7.6 Conclusion
References
8 A Comparison Between Heterogeneous and Homogeneous Layers for Nonlinear Bright Solitary SH Waves in Terms of Heterogeneous Effect
8.1 Introduction
8.2 A Review Part for Some Materials in Homogeneous Media
8.2.1 Compressible Materials
8.2.2 Incompressible Materials
8.2.3 Generalized Neo-Hookean Materials
8.3 Comparison of Nonlinear Shear Horizontal Waves
8.4 Conclusions with Some New Results
References
9 On Surface Kinetic Constitutive Relations
9.1 Introduction
9.2 Kinematics and Surface Strain Energy
9.3 Kinetic Constitutive Equation
9.4 Generalized Young–Laplace Equation
9.5 Conclusions
References
10 Reduced Linear Viscoelastic Isotropic Cosserat Medium with Translational Viscosity: A Double Negative Acoustic Metamaterial
10.1 Introduction
10.2 Dynamic Equations and Dispersion Relation
10.3 Absence of the Band Gap for an Arbitrary Translational Dissipation n
10.4 Asymptotical Approximation for Infinitesimal n
10.4.1 Case |Ω2-Ω1 D2| ggn
10.4.2 Case |Ω2-Ω1 D2| lln
10.4.3 Case (Ω2-Ω1 D2) / n = O(1)
10.5 Comparison of Analytical and Numerical Results and Discussion
References
11 On Dynamic Model of Structural Transformations in Solids
11.1 Introduction
11.2 Basic Equations of Two-Component Medium
11.3 Statement of the Problem. Dispersion Curves
11.4 On the Method of Variable Interval
11.5 Discrete Model
11.6 Continuous Model
11.7 Conclusion
References
12 Dynamic Penetration into Water Saturated and Frozen Sand: Numerical Analysis of the Inverse Experimental Methodology
12.1 Introduction
12.2 Grigoryan’s Mathematical Model of the Dynamics of Soil Media
12.3 Formulation of Numerical Modeling Problems
12.4 Results of Numerical Computations
12.5 Conclusion
References
13 Extended Model of Surface-Related Effects in Second-Gradient Elasticity. Surface Waves Related to the Nature of Adhesion
13.1 Introduction
13.2 Formulation of Boundary Value Problems for Elastic Bodies with Adhesion-Active Surfaces
13.3 Qualitative Analysis of the Elastic Moduli of Surface Interactions
13.4 Extended Continual Adhesion Models of Classical Body
13.4.1 Symmetry Conditions
13.4.2 Frame-Indifference Condition
13.4.3 Weak Frame-Indifference Condition
13.5 Surface Waves Related to the Nature of Adhesion
13.5.1 Longitudinal Surface U-waves
13.5.2 Transverse Surface V-Waves
13.5.3 Transverse Surface W-Waves
13.5.4 Surface Θ-Waves and Plane Surface Ω-Waves
13.6 Conclusion
References
14 Generalized Space–Time Fractional Dynamics in Networks and Lattices
14.1 Introduction
14.2 Renewal Process and Continuous-Time Random Walk
14.3 Poisson Process
14.4 Fractional Poisson Process
14.5 Generalization of the Fractional Poisson Process
14.6 Continuous-Time Random Walk on Networks
14.7 Generalized Space–Time Fractional Diffusion in  mathbbZd
14.8 Diffusion Limit
14.9 Conclusions
Appendix: Laplace Transforms and Fractional Operators
References
15 Analytical Method for Describing the Dynamics of Mechanical Systems in Variable Time Intervals
15.1 Introduction
15.2 Problem of Analytical Description of the Dynamics of Mechanical Systems in Variable Time Intervals
15.3 Method of Time Intervals
15.4 Relationship of Time Intervals with the Angle of Rotation During Rotational Motion
15.5 Relationship of the Period with the Displacement During Oscillatory Motion
15.6 Relationship of Time Intervals with Displacement During Reciprocating Motion
15.7 Estimation of the Scope of Linear Relations
15.8 Conclusion
References
16 Propagation of Non-stationary Axisymmetric Perturbations from a Spherical Cavity in Cosserat Medium
16.1 Introduction
16.2 Statement of the Problem
16.3 Presentation of the Solution in the Form of Series
16.4 General Solution Images
16.5 Problem-Solving Images
16.6 Linear Approximation of the Solution
16.7 Originals of the Solution
16.8 Examples
16.9 Conclusion
References
17 The Equations of Coupled Dynamics of Electromagnetoelastic Thin Shells
17.1 Introduction
17.2 Equations of Motion of the Elastic Shell at Given Loads
17.3 Closed-Form Solution for System of Equations of an Electromagnetoelastic Shell
17.4 Equations for an Isotropic Conductor Shell
17.5 Equations of Motion of an Electromagnetoelastic Plate
References
18 Nonlinear Dynamics of Two-Dimensional Lattices with Complex Structure
18.1 Introduction
18.2 Two-Dimensional Waves in a Generalized Square Lattice
18.2.1 Statement of the Problem
18.2.2 Auxetic Behavior in the Linearized Model
18.2.3 Continuum Nonlinear Equations
18.2.4 Shear Waves
18.3 Two-Dimensional Nonlinear Waves Propagation in Graphene Lattice
18.3.1 Continuum Limit for Weakly Transversely Perturbed Waves
18.4 Two-Dimensional Dynamical Strain Processes
18.4.1 Exact Solutions
18.4.2 Transverse Instability of Longitudinal and Shear Waves
18.4.3 Numerical Solutions
18.5 Conclusions
References
19 Influence of First to Second Gradient Coupling Tensors Terms with Surface Effects on the Wave Propagation of 2D Network Materials
19.1 Introduction
19.2 Second Order Discrete Homogenization for Viscoelastic Network Materials
19.3 Wave Propagation Analysis in Non-centrosymmetric Architectures
19.4 Conclusion
Appendix: Transition from Curvilinear to Cartesian Coordinates
References
20 A Short Review of Rotations in Rigid Body Mechanics
20.1 Introduction
20.2 Rotation and Change of Base
20.2.1 Rotations of Tensors
20.2.2 Representation of the Rotation Tensor
20.3 Time Derivatives in Rotating Systems
20.4 Analysis of Sequential Rotations
20.4.1 Simplification for Attached Axes
20.4.2 Summary for Sequential Rotations
20.5 Treatment of Successive Rotations in the Literature
20.6 Conclusion
20.7 The Levi-Civita Tensor
20.8 Angular Velocity in Terms of Orientation Parameters
References
21 A Variant of the Description of the Acoustic and Optical Branches of the Dispersion Law of High-Frequency Waves in an Elastic Medium
21.1 Introduction
21.2 General Provisions and Assumptions
21.2.1 An Elastic Medium Model
21.2.2 Expansion Feature of Elastic Material Model
21.3 The Optical and Acoustic Branches Model for Dispersion Law
21.4 Results and Experimental Data Comparison
21.5 Conclusion
References
22 Supercomputer Modeling of Wave Propagation in Blocky Media Accounting Fractures of Interlayers
22.1 Introduction
22.2 Blocky Medium with Elastic-Plastic Interlayers
22.3 Simulation of Cracks in Interlayers
22.4 Elastic-Plastic Cosserat Continuum
22.5 Results of Computation
22.6 Concluding Remarks
References
23 Structural and Micropolar Beam Models of Nanocrystalline Materials (One-Dimensional Case)
23.1 Introduction
23.2 Discrete-Moment Model of the Atom Chain. Hamilton’s Principle
23.3 One-Dimensional (“Bending”) Continual Model of Linear Chain of Atoms. Hamilton’s Principle for the Continual Model
23.4 Equations of the Simple Applied Theory of Micropolar Elastic Thin Beams with Free Fields of Displacements and Rotations. Comparison of the Constructed Models and Determination of the Micropolar Elastic Parameters
23.5 Conclusion
References
24 Circuit Analogies in the Search for New Metamaterials: Phenomenology of a Mechanical Diode
24.1 Introduction
24.2 Continuum Model of Pantographic Structures
24.2.1 Deformation Energy of a Pantographic Sheet
24.3 A Mechanical Diode
24.4 Conclusion
References
25 Damping of Oscillations by a Vibro-Impact System with Serial Magnetic Impact Pairs
25.1 Introduction
25.2 Problem Statement
25.3 Motion Equations
25.4 Frequency Response
25.5 Conclusion
Appendix
References
26 Exact Solutions of Cubic-Quintic Modified Korteweg-de-Vries Equation
26.1 Introduction
26.2 Painlevé Analysis
26.3 Case 1. a1 = 0,a0 =0. Periodic and Soliton Solutions
26.4 Case 2. a1 = 0,a0 = 0. Periodic and Soliton Solutions
26.5 Case 3. Kink-Shaped Solution
26.6 Case 4. a1 =0,a0 =0. Approximate Solution
References
27 Modelling of Unsteady Elastic Diffusion Oscillations of a Timoshenko Beam
27.1 Introduction
27.2 Problem Formulation
27.3 Integral Representation of the Solution
27.4 Solution Algorithm
27.5 Example of Computation
27.6 Conclusions
References