Progress in Continuum Mechanics

ISBN-13: 9783031437359 / Angielski / Twarda / 2023

Opis

This book gives an insight into the current developments in the field of continuum mechanics. Twenty-five researchers present new theoretical concepts, e.g., better inclusion of the microstructure in the models describing material behavior. At the same time, there are also more applications for the theories in engineering practice.In addition to new theoretical approaches in continuum mechanics and applications, the book puts an emphasis on discussing multi-physics problems.

Spis treści

1 A Semi-Empirical Fluid Force Model for Vortex-Induced Vibration of an Elastic Structure 

Andrei K. Abramian and Sergey A. Vakulenko

1.1 Introduction 

1.2 A Fluid Force Model 

1.2.1 Formulation of the Problem 

1.2.2 Solution of the Structure Motion Equation 

1.2.2.1 Pure Resonance Case 

1.2.2.2 Near Resonance Case 

1.2.2.3 Non-Resonant Case 

1.2.3 Effect of Variation of an Added Mass with the Reduced Velocity 

1.3 An Example 

1.4 Conclusion and Discussion 

References 

2 Nonlinear Buckling and Equilibria of Layered Shallow Parabolic Arches with Interlayer Slip 

Christoph Adam, Ivan Paulmichl, and Thomas Furtmüller

2.1 Introduction 

2.2 Basic Equations 

2.3 Solution 

2.4 Buckling and Post-Buckling Analysis 

2.4.1 Primary Equilibrium Path 

2.4.2 Limit Loads and Limit Points 

2.4.3 Bifurcation Loads and Bifurcation Points 

2.4.4 Post-Bifurcation Equilibrium Path 

2.5 Application 

2.5.1 Example Problem 1 

2.5.2 Example Problem 2 

2.5.3 Critical Loads 

2.5.4 Parabolic Shallow Arch vs. Circular Shallow Arch 

2.6 Summary and Conclusions 

References 

3 On the General Strategies to Formulate Shell and Plate Theories 

Holm Altenbach and Victor A. Eremeyev

3.1 Introduction 

3.2 Classification Principles 

3.2.1 Classification of Structural Models 

3.2.2 Classification of Theories for Two-Dimensional Structures 

3.3 Direct Approach 

3.3.1 General Cosserat Surface Theory 

3.3.2 12-Parameter Theory 

3.3.3 6-Parameter Theory 

3.3.4 5-Parameter Theory 

3.3.5 3-P arameter Theory 

3.4 Conclusions 

References 

4 Conceptual Generalizations of the Kapitsa Problem 

Alexey V. Babenko, Oksana R. Polyakova, and Tatyana P. Tovstik

4.1 Introduction 

4.2 Mathieu Equation 

4.3 Model of the Flexible Rod of the Kapitsa Pendulum

4.4 Asymptotic Expansion

4.5 Pade Approximation 

4.6 Discussion of Results 

4.6.1 Resonances of Longitudinal Vibrations 

4.6.2 General Picture of Stability by Asymptotic Formulas 

4.6.3 Comparison with the Exact Solution 

4.6.4 Conclusions

4.7 Some Hypotheses Regarding the Possible Application of the Kapitsa Pendulum Effect in Modern and Advanced Technology 

References 

5 Dynamic Properties of Periodic Structures with Symmetric Inclusions

Ludmila Ya. Banakh and Igor S. Pavlov

5.1 Introduction 

5.2 Oscillations of Symmetric ����-gon Frames 

5.3 Oscillations of Periodic Systems Containing Symmetric Subsystems

5.4 An Example of Calculation of a 3-Section System with Symmetric Subsystems 

5.5 Structure of the Spectrum of Natural Frequencies of a Multisection Structure 

5.6 Conclusions 

Appendix A. Application of the Group Representation Theory for Mechanical Systems 

A.1. Basic Concepts of the Representation Theory of Symmetry Groups 

A.2. Matrix Symmetry Operators 

References

 

6 Mathematical Model for Myopia Correction with MyoRing Implants 

Svetlana M. Bauer, Liudmila A. Venatovskaya, Eva B. Voronkova, Vladimir V. Kornikov, Larisa A. Avershina, and Anna E. Terenteva

6.1 Introduction 

6.2 Problem Statement 

6.3 Results and Discussion 

6.4 Conclusion 

References 

7 Numerical Modeling the Stresses in Incompressible and Rigid Bodies 

Nikolai M. Bessonov and Yaroslava I. Litvinova

7.1 Introduction 

7.2 Numerical Modeling of the Flow of the Incompressible Micropolar Liquids 

7.3 Numerical Modeling of the Deformation of a Rubber-Like Incompressible Solid Body 

7.4 Numerical Modeling of Stresses in the Rigid Body 

7.4.1 First Example 

7.4.2 Second Example 

Appendix A: 3D Iterative Alternative Direction Implicit Method 

References 

8 Three-Dimensional Numerical Analysis of Natural Vibrations and Stability of Cylindrical Shells Interacting with Fluid 

Sergey A. Bochkarev, Sergey V. Lekomtsev, Valerii P. Matveenko, and Alexander N. Senin

8.1 Introduction 

8.2 Mathematical and Numerical Formulations 

8.3 Single Cylindrical Shells 

8.3.1 Circular Cylindrical Shells 

8.3.2 Elliptical Cylindrical Shells 

8.4 System of two Circular Cylindrical Shells 

8.4.1 Coaxial Shells 

8.4.2 Eccentric shells 

8.5 Conclusion 

References 

9 On the Problem of Modeling the Influence of Ice Cover and Surface Waves of a Liquid on the Dynamics of a Floating Body 

Anastasiia A. Chevrychkina, Nikolai M. Bessonov, and Andrei K. Abramian

9.1 Introduction 

9.2 Statement of the Problem 

9.3 Numerical Method 

9.4 Results 

9.5 Conclusion 

References 

10 Nonlinear Stationary Waves in a Thin-Walled Bar Affected by Deplanation of Its Cross-Section in Torsion 

Vladimir Erofeev, Boris Lampsi (Jr.), Anna Leonteva, and Nadezhda Semerikova

10.1 Introduction 

10.2 Differential Equation for Torsional Vibrations of a Bar Taking into Account the Nonlinearity and Deplanation of the Bar Cross Section

10.3 Wave Processes in a Thin-Walled Bar Taking into Account the Quadratic Nonlinearity 

10.4 Wave Processes in a Thin-Walled Bar Taking into Account the Cubic Nonlinearity 

10.5 Wave Processes in a Thin-Walled Bar with Simultaneous Consideration to the Quadratic and Cubic Nonlinearities 

10.6 Conclusions 

References 

11 Linear Reduced Elastic Isotropic Cosserat Medium Subjected to the External Follower Viscoelastic Torque as a Smart Acoustic Metamaterial 

Elena F. Grekova and Sabina M. Isaeva

11.1 Introduction and Notation 

11.2 Equations of the Reduced Elastic Linear Isotropic Cosserat Medium Subjected to a Viscoelastic Follower Body Torque Spectral Problem 

11.3 Isotropic Linear Elastic Reduced Cosserat Medium Subjected to an Elastic Follower Torque 

11.4 Isotropic Elastic Reduced Cosserat Medium Subjected to a Viscous or Viscoelastic Follower Torque 

11.4.1 Dispersion Relation for the Shear–Rotational Wave 

11.4.2 Small Dissipation far from Characteristic Frequencies Ω = Ω1 and Ω = 1 

11.4.2.1 Real Part of the Wave Number 

11.4.2.2 Imaginary Part of the Wave Number 

11.4.2.3 Logarithmic Decrement 

11.4.3 Small Dissipation near the Lower Characteristic Frequency Ω1 

11.4.4 Small Dissipation near the Upper Characteristic Frequency Ω = 1 

11.5 Conclusions 

References 

12 Nonlinear Vibrations of Bimodular Continua by Means of Isogeometric Analysis 

Rudolf Heuer and Galeb El Chabaan

12.1 Introduction 

12.2 Mechanical Modeling 

12.2.1 Kinematic Relations 

12.2.2 Governing Equations of Elastic Bimodular Beams 

12.3 Considered Geometries - Isosceles Triangle Cross-Section 

12.4 Application of Isogeometric Analysis to the Bimodular Beam 

12.4.1 B-Splines 

12.4.2 Isogeometric Analysis of Bimodular Beam Vibration 

12.5 Numerical Studies 

12.6 Conclusions 

References 

13 On the Equivalence Between Singular Waves Propagating in Force Loaded Viscoelastic Bodies and in Elastic Bodies Additionally Loaded by Eigenstrains 

Hans Irschik, Michael Krommer, and Astrid S. Pechstein

13.1 Introduction 

13.2 Basic Relations 

13.3 An Equivalence Problem and its Solution 

13.4 1D Shock Wave Propagating in a Semi-Infinite Half-Space or Rod 

References 

14 Influence Tensors for the Analytical Mechanics of Anisotropic Eigenstressed Composites with Inclusions of Various Shapes and Orientations  

Nabor Jiménez Segura, Bernhard L.A. Pichler, and Christian Hellmich

14.1 Introduction 

14.2 Fundamentals of Continuum Micromechanics and Composite Mechanics 

14.2.1 Representative Volume Element, Average Rules, and Scale Transition Relations 

14.2.2 Characteristics of Strain Concentration and Microeigenstress-to-Microstrain Influence Tensors 

14.3 Derivation of Influence Tensors for Inclusions in Anisotropic Multishape Composites, from Eigenstressed Matrix-Inclusion Problems 

14.4 Determination of Matrix Influence Tensor Q�������� 

14.5 Check of Influence Tensor Expressions 

14.5.1 Fulfillment of Influence Tensor Average Rule 

14.5.2 Consistency of Influence Tensor Expressions with Levin’s Theorem 

14.5.3 Consistency of Influence Tensor Expressions with Elastic Reciprocal Theorem 

14.6 Benchmark Examples 

14.6.1 Benchmark I: Multishape Composite with Isotropic Phase Properties 

14.6.2 Benchmark II: Multishape Composite with Transversely Isotropic Matrix Phase 

14.7 Conclusions 

References 

15 Computation of Eigenstrains for Static Shape Control of Arbitrarily Shaped Sub-Domains of Force-Loaded Elastic Bodies 

Michael Krommer, Astrid S. Pechstein, and Hans Irschik

15.1 Introduction 

15.2 Static Shape Control of Material Bodies - a Brief Introduction 

15.2.1 Displacement Tracking 

15.2.2 Numerical Example 

15.3 Static Shape Control of Sub-Domains 

15.3.1 Strain Tracking 

15.3.1.1 Variational Formulation 

15.3.1.2 Numerical Example

15.3.2 Displacement Tracking 

15.3.2.1 Optimization 

15.3.2.2 Numerical Example 

References 

16 Flexural Deformations and Vibrations of a Three-Layer Beam-Strip with a Stiff Core and Soft Skins 

Gennadi Mikhasev, Marina Botogova, and Nguyen Le

16.1 Introduction 

16.2 Statement of the Problem 

16.3 Asymptotic Integration of Boundary-Value Problem 

16.3.1 Leading Approximation 

16.3.2 First-Order Approximation 

16.3.3 Governing Equation 

16.4 Free Vibrations

16.5 Conclusions 

References 

17 Maxwell’s Equations Through the Ages 

Wolfgang H. Müller and Elena N. Vilchevskaya

17.1 Introduction and Scope of the Paper 

17.2 Physical Principles of Electromagnetism 

17.2.1 Conservation of Electromagnetic Flux

17.2.2 Conservation of Electric Charge 

17.3 Early Forms of Maxwell’s Equations 

17.3.1 Maxwell’s Treatise on Electromagnetism 

17.3.2 Maxwell’s Disciples 

17.4 World-Tensor Form of Maxwell’s Equations 

17.4.1 A Naive Introduction of the World-Tensors of Electrodynamics and Maxwell’s Equation in Space-Time Formulation 

17.4.2 Absolute Space-Time Notations

17.5 Conclusions and Outlook 

References 

18 Multi-Objective Optimization of the Helix Shape of Cylindrical Milling Tools 

Chigbogu Ozoegwu and Peter Eberhard

18.1 Introduction 

18.2 Milling Flute Cutting Force 

18.3 An Analytical Cutting Force Model 

18.3.1 The Integrated Force Components 

18.3.2 Force Dependence on the Variable Helix Angle �������� 

18.4 Optimization 

18.4.1 The Objective Functions 

18.4.2 The Optimization Problem 

18.5 Numerical Results 

18.6 Conclusions 

References 

19 Experimental and Numerical Studies on the Tensile Strength of Lap Joints of PEEK Plates and CF Fabric Prepregs Formed by Ultrasonic Welding 

Sergey V. Panin, Svetlana A. Bochkareva, Iliya L. Panov, Vladislav O. Alexenko, Anton V. Byakov, and Boris A. Lyukshin

19.1 Introduction 

19.2 Experimental Investigation of the Influence of the Prepreg ‘Design’ and its Properties on the Tensile Strength of the USW Lap Joints 

19.2.1 Samples’ Fabrication 

19.2.2 Tensile Tests of the USW Joints 

19.3 A Parametric Study of the Tensile Deformation Behavior of the USW Joints Based on Numerical Simulation

19.3.1 The Problem Statement for the Parametric Studies on the Tensile Deformation Behavior of the USW Lap Joints 

19.3.2 Criteria for the Damage Simulation 

19.3.3 A Study of the Tensile Strength of the USW Joints in the 3D Formulation 

19.3.3.1 The Effect of the Prepreg Thickness on the Tensile Strength of the USW Lap Joints 

19.3.3.2 The Effect of the Adhesion Level Between the Prepreg and the PEEK Adherends on the Tensile Strength of the USW Lap Joints 

19.3.3.3 The Effect of the Adhesion Level Between the CF Fabric and the PEEK ‘Facing’ Layer on the Tensile Strength of the USW Lap Joints

19.3.3.4 The Effect of the Adhesion Heterogeneity Between the Prepreg and the PEEK Adherends on the Tensile Strength of the USW Lap Joints 

19.3.4 The Effect of the Dimensions of the PEEK Adherends,Their Properties and the Loading Type on the Tensile Strength of the USW Lap Joints (2D Formulation) 

19.3.4.1 The Effect of the Prepreg’s Elastic Modulus on the Tensile Strength of the USW Lap Joints

19.3.4.2 The Effect of the Partial Interlayer Contact on the Tensile Strength of the USW Lap Joints in the Case of the Short PEEK Adherends 

19.4 Conclusions 

References 

20 On two Approaches for Determination of the Effective Conductivity of a Polycrystalline Material by Homogenization Methods 

Dmitry Pashkovsky, Ksenia Frolova, and Elena Vilchevskaya

20.1 Introduction 

20.2 Problem Statement 

20.3 Results

20.4 Conclusion 

References 

21 The Functionally Invariant Solutions and Nonlinear Wave Equations 

Yuri V. Pavlov

21.1 Introduction 

21.2 Functionally Invariant Solutions 

21.3 Solving the Nonlinear Equation 

21.4 Conclusion 

References 

22 Hydrogen Skin Effect vs. Hydrogen Diffusion 

Vladimir A. Polyanskiy, Dmitry G. Arseniev, Anastasiia A. Chevrychkina, and Yuri A. Yakovlev

22.1 Introduction 

22.2 The Theoretical Observation 

22.2.1 Hydrogen Transport Model 

22.2.2 Model Description

22.2.3 Computational Algorithm of the Model 

22.3 Simulation Results

22.4 The Discussion of the Results 

22.5 Conclusions 

References 

23 Bending Waves in Mass-in-Mass Metamaterial 

Alexey V. Porubov and Yuting Zhao

23.1 Introduction 

23.2 Bending in Mass-in-Mass Chain 

23.3 Dispersion Analysis 

23.4 Discussion 

References 

24 Numerical Investigations of Large Amplitude Oscillations of Planar Parametrically Excited Beams 

Alois Steindl, Roman Buchta, Michael Ruttmann, and Yury Vetyukov

24.1 Introduction and Model Description 

24.1.1 Model of a Parametrically Excited Beam 

24.1.2 Finite Element Formulation

24.1.3 Parameter Values and Non-Dimensionalization 

24.2 Stability of the Trivial Solution 

24.2.1 Analytical Approximation of the Stability Limit (Bolotin’s Method) 

24.2.2 Numerical Determination of the Stability Boundaries 

24.2.2.1 Simulation with Fixed Values of Frequency ���� 

24.2.2.2 Simulation of the FE Equations with Slowly Varying Frequency 

24.3 Calculation of Periodic Solutions Bifurcating from the Primary Resonance 

24.3.1 FE Simulations and Use of a Boundary Value Problem Solver 

24.3.2 Application of the Galerkin Method 

24.3.3 Local Analytical Investigation of the Oscillation Equation Close to the 2 : 1-Resonance 

24.3.4 Approximation by the Ritz Method 

24.4 Conclusions and Further Research Goals 

References 

25 Continuum Mechanics Applied for Studying Instabilities in Nanoparticles 

Melanie Todt, Markus A. Hartmann, and Franz G. Rammerstorfer

25.1 Introduction 

25.2 Methods and Models 

25.2.1 Atomistic Approaches 

25.2.2 Continnuum Mechanics Approaches 

25.3 Mechanical Properties Used in Continuum Shell Models

25.3.1 Effective Elastic Properties and Thickness 

25.3.2 Continuum Mechanics Models for van der Waals Interaction and the Intrinsic Curvature Induced Excess Surface Energy

25.4 Some Examples for Application of Continuum Mechanics to Nanoparticles 

25.4.1 Carbon Crystallites 

25.4.2 Carbon Onions 

25.5 Conclusion 

References 

26 Spectral Domain Approach for the Numerical Modeling of Elastodynamic Fields in Layered Structures 

Thomas Voglhuber-Brunnmaier and Bernhard Jakoby

26.1 Introduction 

26.2 Modeling 

26.2.1 Governing Equations 

26.2.1.1 Linear Elastodynamics 

26.2.1.2 Electrodynamics 

26.2.1.3 Piezoelectric Media 

26.2.2 Dimensional Analysis and Scaling 

26.2.2.1 Non-Dimensionalization 

26.2.2.2 Scaling of Physical Units 

26.2.3 Conversion to Ordinary Differential Equations 

26.2.4 Equation System and Green’s Function 

26.2.5 Electrical Field Calculation 

26.2.5.1 Method of Moments 

26.3 Examples 

26.3.1 Electrical Capacitance Calculation of Interdigital Capacitors 

26.3.2 Vibrating Fluid Sensor 

26.3.3 Piezoelectric Fluid Sensor 

References