Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials (eBook)

Prof. Mezhlum Sumbatyan is a head of the Department of Computational and Theoretical Hydrodynamics, as well as a head of the Ultrasonic Laboratory, both in the Southern Federal University, Rostov-on-Don, Russia. He is a member of the National Russian Committee on Theoretical and Applied Mechanics, and a foreign member of the Armenian National Academy of Sciences.His scientific interests are related to ultrasonic methods in evaluation of acoustic metamaterials, to identification of defects in solids on the basis of inverse problems, to numerical algorithms in direct and inverse problems of diffraction theory, as well as to mathematical modeling of acoustic filters of a periodic internal structure.He has published more than 250 papers, 108 of them in the Scopus data base. He  has processed as teams’ leader in more than10 Russian and international grants.

Foreword 6
Contents 9
Contributors 11
Mathematical Models and Finite Element Approaches for Nanosized Piezoelectric Bodies with Uncoulped and Coupled Surface Effects 13
1 Introduction 14
2 Model of Piezoelectric Materials with Damping and Surface Effects 15
3 Week Formulations of Dynamic Problem 18
4 Finite Element Approaches 21
4.1 Nonstationary Problems 21
4.2 Static Problems 22
4.3 Steady-State Oscillation Problems 23
4.4 Modal Problems 23
4.5 Mode Superposition Method for Steady-State Oscillation Problems 24
4.6 Mode Superposition Method for Nonstationary Problems 25
4.7 The Newmark Scheme for Solving Non-stationary Problems 26
5 Concluding Remarks 27
References 28
On the Theory of Acoustic Metamaterials with a Triple-Periodic System of Interior Obstacles 31
1 Introduction 31
2 Mathematical Formulation of the Problem 32
3 The Properties of the Basic Integral Equation 38
4 Calculation of the Wave Characteristics 41
5 Conclusions 43
References 44
3 Analytical and Computer Methods to Evaluate Mechanical Properties of the Metamaterials Based on Various Models of Polymeric Chains 46
Abstract 46
1 Introduction 47
2 Simulation of the Adsorption Process 50
3 Model “Polymer-Substrate” 55
4 Simulation of the Simplest Types of Mechanical Deformation 59
5 Simulation of Ferroelectric Properties of Thin Films 66
6 Application to Surface Elasticity 74
7 Conclusion 76
Acknowledgements 76
References 77
Identification of Arrays of Cracks in the Elastic Medium by the Ultrasonic Scanning 81
1 Introduction 81
2 Basic Equations of the Elasticity Theory for Anti-plane Mode of Deformation 83
3 Mutual Influence of the Elementary Cracks 84
4 Identification Problem and Some Details of the Numerical Algorithm 89
5 Conclusions 95
References 98
5 Short-Wave Diffraction of Elastic Waves by Voids in an Elastic Medium with Double Reflections and Transformations 100
Abstract 100
1 Introduction 100
2 Problem Statement 101
3 Method of Solution 102
4 Single Reflection of the Longitudinal Wave by the Cavity Surface 103
5 Single Reflection of the Transverse Wave by the Cavity Surface 108
6 The Double Re-Reflection of Elastic Waves in View of All Possible Transformations 109
7 Conclusion 114
Acknowledgements 114
References 114
Finite Element Modeling and Computer Design of Anisotropic Elastic Porous Composites with Surface Stresses 116
1 Introduction 117
2 Homogenization of Two-Phase Elastic Nanocomposite by Effective Moduli Method 118
3 Finite Element Approximations 121
4 Modeling of Representative Volumes 123
5 Numerical Results and Discussion 125
References 129
Acceleration Waves in Media with Microstructure 132
1 Introduction 132
2 Basic Equations of a Hyperelastic Media with Internal Variables 133
3 Acceleration Waves 135
4 Example: Micropolar Media 138
5 Conclusions 139
References 139
Models of Active Bulk Composites and New Opportunities of the ACELAN Finite Element Package 142
1 Introduction 143
2 Effective Moduli Method for Thermopiezomagnetoelectric Composites 144
3 Finite Element Technologies 151
4 Modelling of Inhomogeneous Polarization 153
5 New Opportunities of ACELAN 159
6 Examples of Computer Design of Active Bulk Composites in ACELAN 163
7 Conclusion 164
References 165
On the Models of Three-Layered Plates and Shells with Thin Soft Core 167
1 Introduction 167
2 Layer-Wise Models of Plates 168
2.1 Equilibrium Conditions 168
2.2 Constitutive Equations 170
2.3 Compatibility Conditions and Further Assumptions 170
2.4 Differential Equations for Relative In-Plane Displacements and Deflection 171
2.5 Maximal Deflection of the Three-Layered Strip 172
3 On Layer-Wise Models of Shells 174
4 First-Order Shear Deformable Plate Model 175
4.1 Equilibrium Conditions 175
4.2 Constitutive Equations 176
4.3 Stiffness Parameters 177
5 Conclusions 177
References 178
Ray Tracing Method for a High-Frequency Propagation of the Ultrasonic Wave Through a Triple-Periodic Array of Spheres 180
1 Introduction 180
2 Formulation of the Problem 181
3 The Applied Method of Solution 182
4 High-Frequency Analytical Representations 182
5 Application of the Ray Tracing Method 189
6 Conclusions 193
References 194
11 An Experimental Model of the Ultrasonic Wave Propagation Through a Doubly-Periodic Array of Defects 195
Abstract 195
1 Introduction 195
2 Problem Statement 196
3 Numerical Treatment and the Solution 199
4 Numerical Results 201
5 Full-Scale Experiment 205
6 Conclusions 209
Acknowledgements 209
References 210
Finite Element Simulation of Thermoelastic Effective Properties of Periodic Masonry with Porous Bricks 211
1 Introduction 212
2 Effective Moduli Method for Linear Thermoelastic Composite Materials 214
3 Computer Design of the Effective Properties of Masonry Made of Porous, Hollow and Porous--Hollow Bricks 218
4 Discussion and Conclusions 224
References 225
Spectral Properties of Nanodimensional Piezoelectric Bodies with Voids and Surface Effects 227
1 Introduction 228
2 Classical Formulation of Eigenvalue Problems for Piezoelectric Media with Voids and Surface Effects 229
3 Generalized Problem Formulations 233
4 Theorems about Changes of Eigenfrequencies 238
5 Concluding Remarks 241
References 242
A Review on Models for the 3D Statics and 2D Dynamics of Pantographic Fabrics 244
1 Introduction 244
2 Standard 3D Cauchy Continuum 246
2.1 Numerical Identification 247
3 Wave Propagation 251
4 Elastic Surfaces 255
5 Conclusions 258
References 259