Tractable Models of Solid Mechanics (eBook)

Preface 4
Abbreviations 7
Contents 8
1 Introduction: Historical Development of Tractable Models 11
References 21
2 Discrete Finite Systems 22
2.1 Linear Oscillators 22
2.1.1 Linear Conservative Oscillator 23
2.1.2 Linear Oscillator with Viscous Damping 24
2.1.2.1 Strong Energy Dissipation (e > >
2.1.2.2 Weak Energy Dissipation (e< <
2.1.2.3 Exercise 30
2.1.3 Linear Oscillator with Viscous Damping and Periodic (Harmonic) Forcing 30
2.1.3.1 Exercise 32
2.1.4 Two Coupled Oscillators 33
2.1.4.1 Weakly Coupled Oscillators with Strongly Different Frequencies 33
2.1.4.2 Exercise 36
2.1.4.3 Weakly Coupled Oscillators with Close Frequencies 36
2.1.4.4 Exercise 40
2.1.4.5 Strongly Coupled Oscillators with Essentially Different Frequencies 40
2.1.4.6 Exercise 42
2.1.4.7 Strongly Coupled Oscillators with Close Frequencies 43
2.1.4.8 Exercise 43
2.2 Single-DOF Nonlinear Oscillator 43
2.2.1 Quasilinear Oscillator 45
2.2.2 Strongly Nonlinear Oscillator and Vibro-Impact Approximation 48
2.2.2.1 Special Solutions for Vibro-Impact Motions 48
2.2.2.2 Vibro-Impact Systems Treatable by Analytic Functions 51
2.2.3 Oscillator with Multiple Equilibriums 55
2.3 Forced Nonlinear Oscillator 56
2.3.1 General Remarks 56
2.3.2 Governing Equations 57
2.3.3 The Dynamics of the Oscillator without Dissipation and the LPT 59
2.3.4 The Transient Dynamics of a Weakly Damped Oscillator 62
2.3.4.1 Non-smooth Temporal Transformations 63
2.3.4.2 The Construction of a Generating Solution 64
2.3.4.3 Calculation of A0(t0) and .0(t0) 67
2.3.5 Quasi-Linear Oscillations 70
2.4 Entrainment, Synchronization and Resonance Capture 73
2.4.1 Pendulum with Constant External Torque 74
2.4.2 Entrainment of the Van der Pol Oscillator by External Harmonic Force 77
2.4.3 Synchronization of Oscillators and Related Models 84
2.4.4 Resonance Capture 86
2.4.4.1 Exercises 89
2.4.5 Forced Oscillator with Multiple States of Equilibrium 90
2.5 Symmetric Systems of Coupled Nonlinear Oscillators Beating Phenomena 93
2.5.1 Exercise 102
2.6 2DOF Systems of Nonlinear Oscillators with Essential Asymmetry Targeted Energy Transfer (TET) 103
2.6.1 Targeted Energy Transfer in an Unforced 2DOF System 103
2.6.2 Targeted Energy Transfer in Forced 2DOF System 114
2.7 Coupled Nonlinear Oscillators with Time Delays 142
2.7.1 Analytic Model 142
2.7.2 Numeric Verification -- Straight Modes 150
2.7.3 Numeric Verification -- "Oval"-- Modes and Phase -- Locked Solutions 153
2.8 Low-DOF Discrete Nonlinear Systems 156
2.9 Concluding Remarks 169
References 170
3 Infinite Discrete Systems 175
3.1 Dynamics of Infinite Nonlinear Chains 175
3.1.1 Long-Wavelength Approximation. Equation of Supersonic Extension Solitons in an Infinite FPU Chain 175
3.1.2 Zigzag Chain and Long-Wave Solitons 177
3.1.3 Envelope Solitons 179
3.1.4 Optical Breathers in a Zigzag Chain 184
3.1.5 Torsional Solitons 194
3.1.6 Approximation of Immobile Neighbour Chains 196
3.2 Dynamics of Essentially Nonlinear and Vibro-Impact Chains 200
3.2.1 Oscillatory Chain with Rigid Barriers 201
3.2.2 Discrete Breathers in a Vibro-Impact Chain 206
3.2.2.1 System of Klein-Gordon (KG) Type 207
3.2.2.2 System of Fermi-Pasta-Ulam (FPU) Type 211
3.3 The Problem of Heat Conduction in Dielectrics 215
3.4 Solitons in Energetically Nondegenerate Quasi-One-Dimensional Models 225
3.4.1 Quasi-One-Dimensional Model of a Molecular Crystal: Soliton Modes of Motion in a Bistable Nonlinear System 227
3.5 Dynamics of Ensembles of Interacting Nonlinear Chains 233
3.6 Concluding Remarks 241
References 241
4 Continuous Systems 245
4.1 One-Dimensional Models 245
4.1.1 Bolotin Model 245
4.1.1.1 Exercises 249
4.1.2 Simplification of the Timoshenko Beam 249
4.2 The Planar Dynamical Problem and Tractable One-Dimensional Models of an Elastic Solid 256
4.2.1 Exercises 261
4.3 The Two-Dimensional Orthotropic Model and Its Application to a Complex Contact Problem 261
4.3.1 Basic Asymptotic Decomposition of the Orthotropic Plate Problem 261
4.3.2 The Contact Problem for a Planar Orthotropic Strip 267
4.4 Models of Elastic Foundation 272
4.4.1 General Equations and Asymptotic Analysis 272
4.4.2 Example -- Dynamical Problem 276
4.4.3 Example -- An Axisymmetric Stamp 278
4.5 On the Concept of Solids 289
4.6 Models of Non-Fourier Heat Conduction 292
4.7 Concluding Remarks 299
References 299
Afterword 302
Index 304