Sixty Shades of Generalized Continua: Dedicated to the 60th Birthday of Prof. Victor A. Eremeyev

ISBN-13: 9783031261855 / Angielski / Twarda / 2023 / 745 str.

Holm Altenbach; Arkadi Berezovski; Francesco Dell'isola

Opis

In this book, well-known scientists discuss modern aspects of generalized continua, in order to better understand modern materials and advanced structures. They possess complicated internal structure, and it requires the development of new approaches to model such structures and new effects caused by it. This book combines fundamental contributions in honor of Victor Eremeyev and his 60th birthday.

Spis treści

1 Effects of 3-D Printing Infill Density Parameter on the Mechanical Properties of PLA Polymer 

Reza Afshar, Simon Jeanne, and Bilen Emek Abali

1.1 Introduction 

1.2 Additive Manufacturing 

1.3 Materials and Methods 

1.3.1 Fused Deposition Modeling (FDM) 

1.3.2 Unidirectional Tensile Tests 

1.3.3 Digital Image Correlation Method 

1.4 Results and Discussions 

1.5 Conclusions 

References 

2 Advance Approximate Analytical Solutions of the Contact Problem for an Inhomogeneous Layer 

Sergei M. Aizikovich, Polina A. Lapina, and Sergei S. Volkov

2.1 Introduction 

2.2 Statement of the Problem of a Shear of the Surface of an Inhomogeneous Layer 

2.3 Integral Equations of Contact Problems 

2.4 Numerical Analysis 

2.5 Closure 

References 

3 The Direct Approach for Plates Considering Hygrothermal Loading and Residual Kinetics 

Marcus Aßmus, Zia Javanbakht, and Holm Altenbach

3.1 Introduction 

3.2 Frame of Reference  

3.3 Thermal Effects and Hygroscopic Impact 

3.4 Residual Kinetics 

3.5 Conclusion 

References 

4 A Technique for Determining True Deformation Diagrams Under Dynamic Tension Using DIC 

Artem V. Basalin, Anatoly M. Bragov, Aleksandr Yu. Konstantinov,

and Andrey R. Filippov

4.1 Introduction 

4.2 Pneumatic Dynamic Installation for Testing Materials at a Deformation Rate of the Order of 10-100 s−1 

4.2.1 Installation Scheme 

4.2.2 Methods of Obtaining and Processing Information in the Experiment 

4.3 Test Results of Sheet (3 mm) Steel 09G2S in a Wide Range of Strain Rates 

4.4 Conclusion 

References 

5 Strain Gradient Elasticity and Dual Internal Variables 

Arkadi Berezovski

5.1 Introduction 

5.2 Dual Internal Variables 

5.2.1 Evolution Equations 

5.2.2 Quadratic Free Energy 

5.3 Concluding Remarks 

References 

6 On the Coercivity of Strain Energy Functions in Generalized Models of 6-Parameter Shells 

Mircea Bîrsan and Patrizio Neff

6.1 Introduction 

6.2 General 6-Parameter Elastic Shells. Governing Equations 

6.3 The Order ℎ3 Model of 6-Parameter Shells made of Cosserat Material 

6.3.1 Coercivity Results for the Model of Order ����(ℎ3) 

6.3.2 Existence of Minimizers 

6.4 The Higher Order Model of Cosserat 6-Parameter Shells 

References 

7 Solving the Equations of Nonlinear Model of Crystalline Media with Complex Lattice and Some Structures of Plane Deformation 

Anatolii N. Bulygin and Yuri V. Pavlov

7.1 Introduction 

7.2 Nonlinear Model of Deformation of Crystal Media 

7.3 General Solution of Dynamic Equations of Plane Deformation of the Nonlinear Model 

7.4 Solving the Micro-Field Equations 

7.5 Conclusion 

References 

8 Modal Analysis of a Second-Gradient Annular Plate made of an Orthogonal Network of Logarithmic Spiral Fibers 

Alessandro Ciallella, Francesco D’Annibale, Francesco dell’Isola,

Dionisio Del Vescovo, and Ivan Giorgio

8.1 Introduction 

8.2 The Model for a Fiber Net Arranged in Logarithmic Spirals 

8.3 Modal Analysis 

8.4 Conclusions and Future Perspectives 

References 

9 Non-Linear Simplest Reduced Kelvin’s Medium in the Vicinity of the Spherical Stress State: Waves and Instabilities 

Mikhail A. Drepin and Elena F. Grekova

9.1 Introduction and Notation

9.2 Basic Equations for the Isotropic Elastic Reduced Kelvin’s Medium in the Vicinity of a Preliminary Stress State 

9.2.1 Dynamic Laws of a Nonlinear Reduced Kelvin Medium 

9.2.2 Constitutive Relations for a Nonlinear Reduced Kelvin Medium

9.2.3 The Simplest Nonlinear Reduced Kelvin Medium. Strain Energy

9.2.4 Medium in the Vicinity of a Homogeneous Nonlinear Spherical Deformation State 

9.3 Dispersion Relations of the Simplest Elastic Reduced Kelvin Medium in the Vicinity of Preliminary Spherical Strain State for Special Directions of Perturbation Propagation. Waves and Stability

9.3.1 Harmonic Waves 

9.3.2 Propagation of Harmonic Waves Along the Body Point Axis (ˆ������������ = ±������������0) 

9.3.3 Propagation of Harmonic Waves in the Direction Orthogonal to the Body Point Axis (ˆ������������⊥������������0) 

9.4 Conclusion 

References 

10 On the Spectrum of Relaxation Times of Coupled Diffusion and Rheological Processes in Media with Microstructure 

Dmitrii S. Dudin and Ilya E. Keller

10.1 Introduction 

10.2 Deformations 

10.3 Balance Equations 

10.4 The Helmholtz Free Energy 

10.5 Thermodynamic Inequality 

10.6 Constitutive Equations 

10.7 Model Problem and its Equations 

10.8 Diffusion Coefficients in Coupled System 

10.9 Conclusion 

References 

11 Representative Volume Element Size and Length Scale Identification in Generalised Magneto-Elasticity 

Sinan Eraslan, Inna M. Gitman, Mingxiu Xu, Harm Askes, and René de Borst

11.1 Introduction 

11.2 Formulation 

11.2.1 Homogenisation and Macroscopic Characteristic Length Scale Parameters

11.2.2 Determination of RVE Sizes and Identification of Characteristic Length Scale Parameters 

11.3 Numerical Results and Discussion

11.4 Conclusions 

References 

12 Rayleigh Waves in the Cosserat Half-Space (Reduced Model) and Half-Space of Damaged Material 

Vladimir Erofeev, Artem Antonov, Anna Leonteva, and Alexey Malkhanov

12.1 Introduction 

12.2 Rayleigh Waves in the Cosserat Half-space (Reduced Model) 

12.3 Rayleigh Waves in the Half-space of Damaged Material 

12.4 Conclusion 

References 

13 Validation of a Hemi-Variational Block-Based Approach to the Modelling of Common In-plane Failures in Masonry Structures 

José Manuel Torres Espino, Jaime Heman Espinoza Sandoval, Chuong Anthony Tran, Roberto Fedele, Emilio Turco, Francesco dell’Isola, Luca

Placidi, Anil Misra, Francisco James León Trujillo, and Emilio Barchiesi

13.1 Introduction 

13.2 Mathematical Formulation 

13.2.1 Vertex Springs and Stiffnesses 

13.2.2 Deformation Energy and Impenetrability Potential 

13.2.3 Damage Laws 

13.2.4 Principle of Minimum Potential Energy 

13.2.5 Numerical Model 

13.2.6 Stiffness Matrix

13.2.7 Algorithm 

13.3 Results 

13.3.1 Comparative Result 

13.3.2 Influence of Mortar Thickness on Masonry Performance 

13.3.3 Bending and Shear Sliding 

13.3.4 Rocking 

13.4 Conclusions and Future Challenges 

References 

14 Size Effects in Cosserat Crystal Plasticity 

Samuel Forest and Flavien Ghiglione

14.1 Introduction 

14.2 Problem Setting 

14.2.1 Field Equations 

14.2.2 Constitutive Equations

14.2.3 Studied Boundary Value Problem 

14.3 Cosserat Elastoplasticity Based on a Quadratic Potential

14.3.1 Simple Glide in Isotropic Elasticity 

14.3.2 Crystal Plasticity Based on the Full Stress Tensor 

14.3.3 Schmid Law Limited to the Symmetric Part of the Stress Tensor 

14.3.4 Comparison with the Curl ���� ���� Model 

14.4 Rank One Energy Potential 

14.4.1 Elasticity Solution

14.4.2 Crystal Plasticity 

14.4.3 Comparison with the Curl ���� ���� Model

14.5 Combined Potential 

14.6 Application to Grain Boundary Behaviour 

14.6.1 Cosserat-Phase Field Model of Grain Boundaries 

14.6.2 Analytical Solution of a Single Flat Grain Boundary 

14.6.3 Grain Boundary Energy 

References 

15. On the Influence of Poisson’s Ratio on Phase Transformations Limiting Surfaces 

Alexander B. Freidin and Leah L. Sharipova

15.1 Introduction

15.2 Phase Equilibrium and Phase Transition Zones for Phases with Positive and Negative Poisson’s Ratios

15.3 Optimal Laminates and Phase Transformations Limiting Surfaces

15.4 Results 

15.5 Conclusions 

References 

16. Application of Nonlocal Fick’s Law Within Micropolar Approach 

Ksenia Frolova, Nikolay Bessonov, and Elena Vilchevskaya

16.1 Introduction 

16.2 Diffusion in Media Modeled by Micropolar Continuum 

16.3 Axially Symmetric Problem 

16.4 Results and Discussion 

16.5 Conclusions 

Appendix. Some Remarks on Numerical Approximation 

References 

17 Geometrically Nonlinear Cosserat Elasticity with Chiral Effects Based upon Granular Micromechanics 

Ivan Giorgio, Anil Misra, and Luca Placidi

17.1 Introduction 

17.2 Discrete and Continuous Models for Granular Systems 

17.2.1 Identification via Piola’s Ansatz 

17.2.2 Relative Intergranular Displacement and Related Continuum Deformation Measures 

17.2.3 On the Objective (Macro and Micro-macro)Displacement Vectors 

17.2.4 On the Objective Tensor 

17.2.5 The Objective Scalar Deformation Measures

17.3 Elastic Energy Function 

17.4 Identification of the Undamaged Isotropic Case 

17.4.1 Characterization of the Undamaged Isotropic Case 

17.4.2 Macroscopic Isotropic Stiffness Matrices 

17.4.3 Identification of the Macroscopic Isotropic Stiffness Matrices 

17.5 Conclusion 

References 

18 Study of the Dynamic Properties of Reinforced Concrete Under High-Speed Compression 

Mikhail E. Gonov, Vladimir V. Balandin, Anatoly M. Bragov,

and Aleksandr Yu. Konstantinov

18.1 Introduction 

18.2 Test Method 

18.3 Characteristics of the Tested Materials 

18.4 Results of Dynamic Tests for Uniaxial Compression 

18.5 Conclusion 

References

19 Multistability of Convective Flows in a Porous Enclosure 

Vasily Govorukhin, Mezhlum Sumbatyan, and Vyacheslav Tsybulin

19.1 Introduction 

19.2 Mathematical Formulation of the Problem

19.3 Numerical Methods and Extreme Multistability 

19.3.1 Spectral Global Galerkin Method 

19.3.2 Cosymmetry Preserving Finite-Difference Approximations

19.3.3 Continuation on the Hidden Parameter Method 

19.4 Multistability 

References 

20 Hydrogen Transport in Framework of Linear Non-Equilibrium Thermodynamics Approach 

Polina M. Grigoreva and Vladimir A. Polyanskiy

20.1 Introduction 

20.2 Problem Statement and Governing Equations 

20.3 Plane Boundary Value Problem 

20.4 Conclusion

References 

21 Classical and Non-Classical Models of Changes in the Young Modulus of Geomaterials Under Alternating Loads 

Mikhail A. Guzev, Evgenii P. Riabokon, Mikhail S. Turbakov,

Vladimir V. Poplygin, Evgenii V. Kozhevnikov, and Evgenii A. Gladkikh

21.1 Introduction 

21.2 Experimental Studies 

21.3 Classical Model of Changes in the Young Modulus Under an Alternating Load 

21.4 Non-classical Model of Change in the Young Modulus Under the Alternating Load 

21.4.1 Formulation of the Non-classical Model 

21.4.2 Building the Solution 

21.5 Conclusion 

References

22 Two Approaches to Modeling Viscoelastic Cosserat Continua 

Elena A. Ivanova

22.1 Introduction 

22.2 Kinematics and Balance Equations 

22.3 Differential Equations Relating the Strain Tensors to the Velocity Vector and the Angular Velocity Vector 

22.4 The Reduced Energy Balance Equation and the Heat Conduction Equation: Zhilin’s Method 

22.5 Integral Equations Relating the Strain Tensors to the Velocity Vector and the Angular Velocity Vector 

22.6 Source Terms in the Strain Balance Equations

22.7 The Reduced Energy Balance Equation and the Heat Conduction Equation: A new Method 

22.8 A Comparison of two Approaches 

22.9 Discussion 

References 

23 Porous Media Models Based on Generalized State Equations with Simple Examples 

Anna Knyazeva and Nelli Nazarenko

23.1 Introduction 

23.2 Definitions and General Relationships 

23.3 Thermodynamical Relations 

23.4 Examples of Particular Problems 

23.4.1 Compressible Nonviscous Gas 

23.4.2 Compressible Fluid with Bulk Viscosity 

23.4.3 Non-ideal Gas Under Non-isothermal Conditions 

23.4.4 Binary Non-viscous Imperfect Mixture 

23.4.5 Diffusion and Filtration in Media with Double Porosity 

23.4.6 Viscous Two-component Fluid in Media with Double

Porosity

23.4.7 Nonviscous Two-component Fluid in Porous media with Nano- and Micro-pores 

23.5 Conclusion 

References 

24 Ball Indentation of Perforated Circular Hyperelastic Membranes 

Alexey M. Kolesnikov

24.1 Introduction 

24.2 Mathematical Model 

24.2.1 Axisymmetric Problem of Non-linear Elastic Membranes 

24.2.2 Ball Indentation of Perforated Circular Hyperelastic Membranes 

24.3 Numerical Results 

24.4 Conclusions 

References

25 Integrated Asymptotic Approach to the Structural Mechanics Models 

Alexander G. Kolpakov and Sergey I. Rakin

25.1 Motivation of the Research 

25.2 Modern State of Structural Mechanics 

25.2.1 Deformation of the “Main” Part of Frame 

25.2.2 Deformation of Connecting Units of Frame 

25.2.3 “Infinite Rigidity” of Connecting Units 

25.2.4 Structural Mechanics Achievements and Limitations 

25.3 Integrated Approach to the Computation of Thin-walled Structures 

25.3.1 Displacements in the “Main” Part of the Beams 

25.3.2 “Rigid” Displacements of the Connecting Unit 

25.3.3 Conjugation of the Displacements in the Main Part of a Beam with the “Rigid Body” Displacements of the Connecting Unit 

25.3.4 Assembling the Single-beam Domain Functions into the Function in 2-D Frame 

25.3.5 Supplement of the Functions (25.5) with the Local Perturbations 

25.4 Local Stress-strain State in the Connecting Units 

25.5 Representative Fragment of Joint 

25.6 Integrated Procedure for Computation of Framework 

25.7 Actual Problems 

25.8 Conclusions 

References 

26 The Homogenized Delamination Criterion for Fiber-reinforced Plate 

Alexander G. Kolpakov, Sergey I. Rakin, and Igor V. Andrianov

26.1 Introduction 

26.2 Boundary Layer 

26.3 Numerical Computation Results 

26.3.1 Extension Along ��������- or ��������-axis 

26.3.2 Shift in ��������2����3-plane 

26.4 The Asymptotic Homogenized Strength Criterion of the Interface Zone 

26.5 Constructing the Delaminating Strength Criterion

26.6 Conclusions 

References 

27 Lightly Loaded Hydrodynamic Thrust Bearing Lubricated by a Non-Newtonian Fluid 

Ilya I. Kudish, Sergei S. Volkov, and Andrey S. Vasiliev

27.1 Introduction 

27.2 Formulation of the Lubrication Problem 

27.3 Asymptotic Analysis of the Rheological and Motion Equations 

27.4 Examples of Some Specific Lubrication Problem Solutions and Discussion 

27.5 Closure 

References 

28 The Idea of Using Adhesive Bonds in Shaping of Cold-formed Thin-walled Beam-columns

Marcin Kujawa, Antonio Cazzani, Lukasz Smakosz, Violetta

Konopińska-Zmysłowska, Karol Winkelmann, Faizullah Jan, and

Czesław Szymczak

28.1 Introduction 

28.2 State of the Art

28.2.1 Static, Dynamics, and Stability Analysis of Thin-walled Members 

28.2.2 Influence of Imperfections 

28.2.3 Failure Analysis with Attention to Creep in Adhesive Bonded Metal Members/Structures 

28.3 The Rationale for Addressing the Research Problem 

28.4 Conclusions 

References 

29 Dissipative Mechano-Electro-Magnetism Simulations in Electronic Components 

Yiming Liu, Wolfgang H. Müller, and Bilen Emek Abali

29.1 Introduction

29.2 Governing Equations 

29.3 Constitutive Equations 

29.4 Generating Weak Forms 

29.5 Implementation and Results 

29.6 Conclusion 

References 

30 On Possible Reduction of Gradient Theories of Elasticity 

Sergey A. Lurie, Petr A. Belov, and Yury O. Solyaev

30.1 Introduction

30.2 Variational Formulation of Gradient Theories 

30.3 Structure of Sixth Rank Tensors 

30.4 Special Structure of Gradient Part of Potential Energy. Hypothesis on Absence of Divergent Terms

30.5 Features of Variational Formulation of Vector-Type Gradient Models 

30.6 System of Governing Equilibrium Equations 

30.7 On Uniqueness of Reduced Vector-Type Models 

30.8 Conclusions 

References 

31 Dynamics of a Rectangular Rigid Body on a Movable Base 

Vladimir S. Metrikin, Leonid A. Igumnov, and Elena I. Komarova

31.1 Introduction 

31.2 Mathematical Model

31.3 Construction of a Point Mapping of the Poincaré Surface 

31.3.1 The Coordinates of the Fixed Point Corresponding to the Symmetrical Periodic Motion. Sustainability 

31.3.2 Equations for Determining the Coordinates of a Fixed Point Corresponding to Asymmetric Periodic Motions.Sustainability 

31.4 Numerical Results for ¥����(����) = ����sin(��������) 

31.5 Numerical Results 

31.6 Conclusion 

References 

32 Asymptotically Correct Analytical Model for Flexural Response of a Two-Layer Strip with Contrast Elastic Constants 

Gennadi Mikhasev and Nguyen Le

32.1 Introduction 

32.2 Statement of the Problem 

32.3 Bernoulli-Euler Type Model for Strip Consisting of Layers with Close Material Constants 

32.4 Timoshenko-Reissner Type Model for Strip Consisting of Layers with High-Contrast Elastic Properties 

32.4.1 Leading Approximation 

32.4.2 First-Order Approximation 

32.4.3 One-Dimensional Governing Equation

32.5 Free Vibrations 

32.6 Conclusions

References 

33 On Analytical Modeling of Tension-Assisted Winding of Flexible Sheets 

Aleksandr Morozov, Wilhelm Rickert, and Sergei Shubin

33.1 Introduction 

33.2 General Assumptions for the Winding Problem 

33.3 Winding the First Layer 

33.4 Winding the Subsequent Layers 

33.5 Conclusions 

References 

34 On Using Rotations as Primary Variables in the Nonlinear Theory of Thin Elastic Shells 

Wojciech Pietraszkiewicz

34.1 Introduction 

34.2 Geometry and Deformation of the Reference Surface 

34.3 Equilibrium Conditions 

34.4 Boundary Value Problem with Independent Rotations 

34.5 Constitutive Equations of Rubber-Like Shells 

References 

35 Continuum Description of Extended Mass-in-Mass Metamaterial Models 

Alexey V. Porubov

35.1 Introduction 

35.2 Classic Mass-in-Mass Chain 

35.3 Chain with Extra Attached Masses

35.4 Chain with Extra Internal Attached Masses 

35.5 Attached Masses Through one Element of the Main Chain 

35.6 Conclusions 

References 

36 Multi-Element Metamaterial’s Design Through the Relaxed Micromorphic Model 

Leonardo A. Perez Ramirez, Gianluca Rizzi, and Angela Madeo

36.1 Introduction 

36.2 The Relaxed Micromorphic Model: Constitutive laws,Equilibrium Equations, and Boundary Conditions 

36.2.1 Equilibrium Equations 

36.2.2 Boundary and Interface Conditions 

36.2.3 Numerical Results 

36.3 Parametric Study on the Thickness of a Shielding Device: Capability Limit for the Relaxed Micromorphic Model 

36.4 Design of a Double Shield Device 

36.5 Multiple-Shields Optimization 

36.6 Conclusions 

References

37 On Magnetically Induced Motion of Micropolar Ferrofluids

Wilhelm Rickert, Margarita Dementeva, Gregor Ganzosch, Elena N.

Vilchevskaya, and Wolfgang H. Müller

37.1 Introduction 

37.2 A Flow Problem Coupled to Electromagnetism 

37.3 Simplifications and Normalization 

37.4 Electromagnetic Force Density 

37.5 Initial Values, Boundary Conditions and Implementation 

37.6 Results 

37.6.1 Slip 

37.6.2 No Slip 

37.6.3 Homogeneous Magnetic Field 

37.7 Summary 

References 

38 Manufacturing Quality Evaluation of Photopolymer Resin 3D-Printed Scaffolds Using Microtomography 

Evgeniy V. Sadyrin, Andrey L. Nikolaev, Sergei V. Chapek, Dmitry V.

Nazarenko, Sergei M. Aizikovich, and Yun-Che Wang

38.1 Introduction 

38.2 Materials and Methods 

38.3 Results and Discussion 

38.4 Conclusion 

References 

39 Comparison of Homogenization Techniques in Strain Gradient Elasticity for Determining Material Parameters 

Bekir Cagri Sarar, M. Erden Yildizdag, and Bilen Emek Abali

39.1 Introduction 

39.2 Determining Strain Gradient Parameters 

39.3 Microscale Structure 

39.4 Results and Discussion 

39.5 Conclusion 

References 

40 Buckling of Rectangular Micropolar Plate with Prestressed Coatings 

Denis N. Sheydakov

40.1 Introduction 

40.2 Micropolar Plate with Prestressed Coatings

40.3 Equations of Neutral Equilibrium 

40.4 Micropolar Plate with Identical Coatings 

40.5 Conclusion 

References 

41 A Cosserat Model for Fiber-Reinforced Elastic Plates 

David J. Steigmann, Mircea Bîrsan, and Milad Shirani

41.1 Introduction 

41.2 The Three-Dimensional Cosserat Model for Fiber-Reinforced Elastic Solids 

41.2.1 Kinematical Variables and Strain Measures in Cosserat Elasticity 

41.2.2 Fiber-Reinforced Materials 

41.2.3 Equilibrium and Constitutive Equations 

41.3 Plate Theory for Fiber-Reinforced Laminae

41.4 Specific Constitutive Assumptions 

41.4.1 Coercivity 

41.4.2 Minimum Property 

Appendix 

References 

42 On the Structure of Solutions in the Vicinity of Discontinuity of Boundary Conditions for Gradient Models

Alexander O. Vatulyan, Sergey A. Nesterov, Victor O. Yurov, and Oksana

V. Yavruyan

42.1 Introduction 

42.2 Constitutive Relations of the Gradient Theory of Elasticity and Electroelasticity 

42.3 Problem for a Strip with Delamination 

42.4 Problem for an Electroelastic Strip with a Surface Electrode

42.5 Conclusion 

References 

43 A Damaged Medium Model for Assessing Life Characteristics of Polycrystalline Structural Alloys with Joint Mechanical Fatigue and Long-Term Strength of Material 

Ivan A. Volkov, Leonid A. Igumnov, Aleksandr A. Belov, and Andrey I.

Volkov

43.1 Introduction

43.2 Constitutive Relations of the Mathematical Model of Mechanics of Damaged Medium 

43.2.1 Constitutive Relations of Thermoviscoplasticity 

43.2.2 Evolutionary Equations for Damage Accumulation

43.3 Numerical Results 

43.4 Conclusion

References

44 Bandgap Properties of a Class of Chiral and Achiral Metamaterials 

Yun-Che Wang, Tse-Chun Liao, Kai-Wen Tan, and Sergey M. Aizikovich

44.1 Introduction 

44.2 Theoretical Considerations 

44.3 Numerical Considerations 

44.4 Results and Discussion 

44.5 Conclusions 

References

45 Large Strains of a Spherical Shell with Distributed Dislocations and Disclinations 

Leonid M. Zubov and Mikhail I. Karyakin

45.1 Introduction 

45.2 The Model of a Nonlinear Elastic Micropolar Shell 

45.3 Continuously Distributed Dislocations in an Elastic Micropolar

Shell 

45.4 Transformation of Incompatibility Equations and Equilibrium

Equations 

45.5 Equilibrium of a Closed Spherical Shell with Distributed

Dislocations and Disclinations 

45.6 Numerical Results 

45.7 Conclusions 

References