Problems of Nonlinear Mechanics and Physics of Materials (eBook)

Prof. Yuri V. Mikhlin has a position at Dept. of Applied Mathematics, National Technical University “Kharkov Polytechnical Institute”, Ukraine. He has near 50 years of experience in Nonlinear Dynamics and Applied Mathematics. His priority scientific results were published in leading int. journals. Organizer and chair for few int. conferences and symposia. Memberships in Editorial Board of the journals: Nonlinear Dynamics; Int. Journal of Nonlinear Mechanics; Journal of Mechanical Engineering Science. Dr. Sci. Arkadiy I. Manevich is a Professor at Theoretical and Computational Mechanics Department of Dniepr National University (Ukraine). The main areas of his scientific work are dynamics, stability and optimal design of linear and nonlinear multidimensional systems, in particular, thin-walled structures. He is author and coauthor of six monographs published in England, Poland and Ukraine, and approximately of 300 papers, mainly in famous scientific journals and proceedings of international congresses and conferences

Preface 7
Contents 9
Contributors 12
Stationary and Non-stationary Dynamics of Oscillators and Oscillatory Chains 16
Wide Frequency Higher-Order Dynamic Model for Transient Waves in a Lattice 17
1 Introduction 17
2 Monatomic Lattice and the Higher-Order Dynamic Equation 19
3 Dynamic Response to External Loads 21
4 Conclusions 25
References 25
Analysis of the Beating States in the System of Nonlinearly Coupled Parametrically Forced Oscillators 27
1 Introduction 27
2 Model 28
3 Slow Flow Model 28
4 Non-dissipative Case (?= 0) 29
5 Dissipative Case 33
6 Conclusions 36
References 37
Is Energy Localization Possible in the Conditions of Non-local Acoustic Vacuum? 38
1 Introduction 38
2 The Model and Equations of Motion 39
3 Nonlinear Normal Modes 40
4 Two-Mode Approximation 42
5 Cluster Variables 45
6 Phase Plane 47
6.1 Analytical Description of LPT 48
7 Conclusions 50
References 51
Phase Dynamics of Intrinsic Localized Modes in Two Weakly Coupled Nonlinear Chains and Correspondence Between Periodic Tunneling of Classical and Quantum Objects 52
1 Introduction 53
2 Model 54
3 Tunneling Dynamics of Weakly Coupled ILMs 55
4 Conclusions 62
References 63
Non-linear Beatings as Non-stationary Synchronization of Weakly Coupled Autogenerators 66
1 Introduction 66
2 Empirical Model of Self-localization with Dissipative Coupling 67
3 The Model of Non-stationary Synchronization and Its Reduction 68
4 The NNM and LPT Symmetries 70
5 Phase Plane Analysis and Effect of Dissipation 71
6 The Role of Nonlinearity 75
7 The Effect of Soft and Stiff Nonlinearities 79
8 Synchronization in the Presence of Detuning 80
9 Non-symmetric Case 82
10 Energy Balance Approach 84
11 Model with Coupling via Linear Oscillator 86
12 Quantum Analogy 90
13 Conclusions 93
Appendix: Symmetry Analysis 94
References 95
Normal Modes of Chaotic Vibrations and Transient Normal Modes in Nonlinear Systems 97
1 Introduction 97
2 Forced Nonlinear Normal Modes of Chaotic Vibrations 99
3 Transient Nonlinear Normal Modes in Dissipative System with a Limited Power-Supply Coupled with Nonlinear Absorber 100
4 Transient Nonlinear Normal Modes in Dissipative Spring-Pendulum System Under Resonance Conditions 107
5 Conclusion 110
References 111
Advanced Nonlinear System Identification for Modal Interactions in Nonlinear Structures: A Review 113
1 Introduction 114
2 Preliminary Concepts and the Proposed Method 115
2.1 Proper Orthogonal Decomposition 115
2.2 Rayleigh Quotient 116
2.3 The Wavelet-Bounded Empirical Mode Decomposition 117
2.4 The Proposed Method 118
3 Detection of Strongly Nonlinear Modal Interactions 119
4 Nonlinear System Identification of a Strongly Nonlinear Attachment 124
5 Concluding Remarks 127
References 129
Non-smooth Spatial and Temporal Substitutions in Impact Dynamics 131
1 Introduction 131
2 Nonsmooth Coordinates and Velocities 133
2.1 Systems with Delta-Pulses Included as Summands 133
2.2 Distributions as Parametric Inputs 134
2.3 Continualization of Impulsively Loaded Systems 136
2.4 Nonsmooth Positional Coordinates 137
2.5 Nonsmooth Transformation of Dynamic States 140
3 Nonsmooth Temporal Arguments 142
3.1 Positive Time 143
3.2 Triangular Wave Time Substitution 146
3.3 Modeling Energy Losses at Perfectly Stiff Barriers 147
4 Concluding Remarks 149
References 150
Revolution of Pendula: Rotational Dynamics of the Coupled Pendula 153
1 Introduction 153
2 Rotation of Single Pendulum 154
3 Rotation of Two Weakly Coupled Pendulums 160
3.1 In-Phase Rotation of Coupled Pendula 161
3.2 Out-of-Phase Rotation of Coupled Pendula 164
4 Conclusion 166
References 167
Plane Motion of a Rigid Body Suspended on Nonlinear Spring-Damper 169
1 Introduction 169
2 Problem Formulation and Equations of Motion 170
3 Multiple Scales Method 172
4 Non-resonant Vibration 175
5 Resonant Vibration 177
5.1 Modulation Problem Near Resonances 178
6 Conclusions 181
References 181
Molecular Dynamics of Polymer Crystals and Nanostructures 183
Supermolecular Structure Formation During Electrospinning, and Its Effect on Electrospun Polymer Nanofiber Unique Features 184
1 Introduction 184
2 Theoretical Modelling of Polymer Dynamics During Electrospinning 186
2.1 Velocity and Radius of an Electrospinning Jet 186
2.2 Polymer System Structure 187
2.3 Axial Stretching of an Entangled Polymer Network During Electrospinning 188
2.4 Radial Contraction 192
2.5 System State Depending on Network Strain 194
3 Experimental 197
3.1 Fast X-Ray Phase-Contrast Imaging 197
3.2 “Bead-on-a-String Structure” 198
4 Size-Dependent Behaviour of Electrospun Polymer Nanofibers and Their Internal Structure 201
4.1 The Structure of an Amorphous Polymer Matrix of Electrospun Nanofibers 201
4.2 The Mathematical Model for Polymer Nanofiber Elongation 205
4.3 Scaling of the Size-Dependent Elastic Modulus of Electrospun Polymer Nanofibers 209
5 Conclusions 212
References 213
Recent Developments in Theory and Modeling of Polymer-Based Nanocomposites 216
1 Introduction 216
2 Polymer-Based Nanocomposites: Morphology and Structure Prediction 219
2.1 Molecular-Level Simulations 219
2.2 Mesoscale Modeling Approaches 220
3 Summary and Outlook 228
References 228
BA Transition in a Short DNA Molecule 236
1 Introduction 236
2 Description of the Sugar CG DNA Model 238
3 The Influence of Water Viscosity, the Order and the Location of the A-B Transition 244
4 The Structure of the DNA-Ions Conglomerate in the `Inviscid' Water 248
5 Discussion and Conclusions 249
References 250
2D Chain Models of Nanoribbon Scrolls 252
1 Introduction 252
2 Chain Model of Molecular Nanoribbon 253
3 Stationary States of Nanoribbon Scrolls 258
4 Scrolled Packing of Graphane Nanoribbons 260
5 Scrolled Packing of Fluorographene Nanoribbons 263
6 Scrolled Packing of Fluorographane Nanoribbons 265
7 Scrolled Packing of Graphone Nanoribbons 267
8 Conclusions 271
References 272
Interaction Between DNA Molecule and Nanosize Pore 274
1 Introduction 274
2 Interaction Simulation 275
References 281
Condensed Matter Mechanics and Physics 282
Wave-Particle Duality and Quantum-Classical Analogy 283
1 Introduction 283
1.1 Genesis of Wave-Particle Duality 283
1.2 Duality and Statistical Dependence 285
1.3 On the Asymptotic Interpretation of Duality 285
1.4 Coherence and Quantum-Classical Analogy 286
2 Two-Level Quantum Systems 288
3 The Classical System of Two Weakly Coupled Oscillators 289
4 The Quantum-Classical Analogy 290
5 Analysis of Two-Component Classical and Quantum Models 290
6 Linear Multi-level System 292
7 Nonlinear Systems 294
8 Conclusion 301
Appendix A 303
Appendix B 304
Appendix C 304
Appendix D 305
Appendix E 307
References 308
Molecular Simulation of Plastic Deformation of Oligomer Systems 312
1 Introduction 312
2 Molecular Model and Modeling Techniques 313
3 True Stress—True Strain Curve 315
4 Density During Deformation 315
5 Local Density and Local Rearrangements 316
6 Relaxation 318
References 320
Plastic Deformation in Disordered Solids: The State of the Art and Unresolved Problems 322
1 Introduction 322
2 Key Features of Inelastic Deformation of DSs 323
3 The Mechanism of Plasticity in DSs 325
3.1 The Shear Transformations (STs) Nucleation Model [1, 3, 20] 326
3.2 Shear Transformation Zone (STZ) Model [2, 5, 7] 328
3.3 Deformation of 2D Lennard–Jones (LD) Glass [33, 34] 329
4 Rearrangements in DSs by Structural Quantity “Softness” 332
4.1 Relationship of Softness to Rearrangements 333
5 Some Unresolved Problems in Plasticity of DSs 334
5.1 Is It Necessary for DSs to Have the Structural Precursors for Development of Plastic Deformation? 335
5.2 Sizes of the Unit Plastic Events 335
5.3 Free and Activation Volumes in GSs 336
5.4 Enhancement of Molecular Motions in DSs During Deformation 337
References 339
Shockwaves and Kinks in Exothermic Nonlinear Chains 342
1 Introduction 342
2 Conservative Bi-Stable Chains 345
2.1 The Model 345
2.2 The Dynamics of Propagation in the Case of a Subsonic Kink 347
2.3 The Supersonic Shockwave Dominated by Gradient Nonlinearities 350
2.4 Polynomial Coupling Potential in the Form of Fermi-Pasta-Ulam (FPU) 356
2.5 LJ Coupling Potential 361
2.6 Validity of the Shock Wave Dynamics 363
3 Transition Fronts in the Chain with On-Site Linear Damping 365
3.1 The Case of Low Damping 366
3.2 The Case of Large Damping 369
References 373
Theory of Beams, Plates and Shells 376
Local Buckling of Cylindrical Shells. Pogorelov’s Geometrical Method 377
1 Introduction and a Brief Historical Excurse 377
2 Pogorelov’s Geometrical Method 380
3 The Mathematical Model 382
4 Validation and Analysis of the Model 387
4.1 Axially Compressed Cylindrical Shells Under Local External Perturbations 389
4.2 Lower and Upper Local Buckling Loads 392
4.3 Energy Barrier and Design Buckling Load 394
5 Conclusion 397
References 398
Stretching of Reinforced Orthotropic Plate 400
1 Introduction 400
2 Statement of the Problem and Solution 407
3 Conclusion 416
References 417
Features of Deformation of Smooth and Stringer Cylindrical Shells at Axial Compression and Statistical Properties of Their Critical Loads 418
1 Introduction and Statement of the Problem 418
2 Specimens and Technique of the Experiment 420
3 Test Results 422
3.1 Smooth Shells 422
3.2 Stringer-Stiffened Shells 424
4 Discussion of the Results 428
References 430
Discontinuities in Viscoelastic Timoshenko Beam Under Moving Concentrated Loads 432
1 Introduction 432
2 Governing Equations 433
3 Elastic Timoshenko Beam 435
4 Viscoelastic Timoshenko Beam 437
5 Conclusions 438
References 438
Theory of Elasticity and Thermo-elasticity 440
Analytical Study of a Nonlinear Beam Including a Piezoelectric Patch 441
1 Introduction 441
2 Beam and Piezoelectric Material Model 442
2.1 Beam Model 442
2.2 Modelling of the Piezoelectric Material 444
2.3 Governing Equations of the Multi-physics Beam 445
3 Treatments of System Equations 446
3.1 The Problem in Space 446
3.2 The Problem in Time 448
4 Numerical Results 450
5 Conclusion 452
References 453
On Higher Order Effective Boundary Conditions for a Coated Elastic Half-Space 454
1 Introduction 454
2 Statement of the Problem 455
3 Asymptotic Analysis 457
4 Comparison with the Exact Solution of a Plane Strain Problem 462
5 Conclusion 465
References 466
Electrically Plane and Mechanically Antiplane Problem for an Inclusion with Stepwise Rigidity Between Piezoelectric Materials 468
1 Introduction 469
2 Basic Equations for a Piezoelectric Material Under Out-of-Plane Mechanical Loading and In-Plane Electric Loading 470
3 Bimaterial Case 471
4 Formulation of the Problem for an Electrically Insulated Inclusion with Stepwise Changing Rigidity 474
5 Absolutely Rigid Inclusion Along All Its Length 475
6 Solution of the Problem for an Electrically Insulated Inclusion with Stepwise Changing Rigidity 476
7 Determination of the Stress and the Electric Fields at the Interface 479
8 Numerical Illustration 481
9 Conclusion 484
References 485
Thermomechanical Coupling and Transient to Steady Global Dynamics of Orthotropic Plates 487
1 Introduction 487
2 Reduced Order Models and Their Comparative Outcomes 489
3 Transient and Steady Global Dynamics Under Thermal Excitations 497
3.1 Bending 497
3.2 Membrane 498
4 Conclusions 501
References 502
Appendix: Professional Life of Professor Leonid Isakovich Manevitch 504
Biographical Sketch of Leonid I. Manevitch 504
List of Books of Professor Leonid I. Manevitch 505
The Scientific Contribution of Professor L. I. Manevitch 506
I. Mechanics of Solids 507
II. Nonlinear Normal Vibration Modes 512
III. Nonstationary Processes in Essentially Nonlinear Systems 515
IV. Physics of Solid Polymers 518
V. Nonlinear Chains: Quasi-One-Dimensional System 522
VI. Methodology of Science 529