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

Preface
Contents
List of Contributors
Chapter 1 Effects of 3-D Printing Infill Density Parameter on the Mechanical Properties of PLA Polymer
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
Chapter 2 Advance Approximate Analytical Solutions of the Contact Problem for an Inhomogeneous Layer
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
Chapter 3 The Direct Approach for Plates Considering Hygrothermal Loading and Residual Kinetics
3.1 Introduction
3.2 Frame of Reference
3.3 Thermal Effects and Hygroscopic Impact
3.4 Residual Kinetics
3.5 Conclusion
References
Chapter 4 A Technique for Determining True Deformation Diagrams Under Dynamic Tension Using DIC
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
Chapter 5 Strain Gradient Elasticity and Dual Internal Variables
5.1 Introduction
5.2 Dual Internal Variables
5.2.1 Evolution Equations
5.2.2 Quadratic Free Energy
5.3 Concluding Remarks
References
Chapter 6 On the Coercivity of Strain Energy Functions in Generalized Models of 6-Parameter Shells
6.1 Introduction
6.2 General 6-Parameter Elastic Shells. Governing Equations
6.3 The Order h^3 Model of 6-Parameter Shells made of Cosserat Material
6.3.1 Coercivity Results for the Model of Order O(h^3)
6.3.2 Existence of Minimizers
6.4 The Higher Order Model of Cosserat 6-Parameter Shells
References
Chapter 7 Solving the Equations of Nonlinear Model of Crystalline Media with Complex Lattice and Some Structures of Plane Deformation
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
Chapter 8 Modal Analysis of a Second-Gradient Annular Plate made of an Orthogonal Network of Logarithmic Spiral Fibers
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
Chapter 9 Non-Linear Simplest Reduced Kelvin’s Medium in the Vicinity of the Spherical Stress State: Waves and Instabilities
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 (k = ±m0)
9.3.3 Propagation of Harmonic Waves in the Direction Orthogonal to the Body Point Axis (k⊥m0)
9.4 Conclusion
References
Chapter 10 On the Spectrum of Relaxation Times of Coupled Diffusion and Rheological Processes in Media with Microstructure
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
Chapter 11 Representative Volume Element Size and Length Scale Identification in Generalised Magneto-Elasticity
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
Chapter 12 RayleighWaves in the Cosserat Half-Space (Reduced Model) and Half-Space of Damaged Material
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
Chapter 13 Validation of a Hemi-Variational Block-Based Approach to the Modelling of Common In-plane Failures in Masonry Structures
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
Chapter 14 Size Effects in Cosserat Crystal Plasticity
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 CurlH^p Model
14.4 Rank One Energy Potential
14.4.1 Elasticity Solution
14.4.2 Crystal Plasticity
14.4.3 Comparison with the CurlH^p 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
Chapter 15 On the Influence of Poisson’s Ratio on Phase Transformations Limiting Surfaces
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
Chapter 16 Application of Nonlocal Fick’s Law Within Micropolar Approach
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
References
Chapter 17 Geometrically Nonlinear Cosserat Elasticity with Chiral Effects Based upon Granular Micromechanics
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
Chapter 18 Study of the Dynamic Properties of Reinforced Concrete Under High-Speed Compression
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
Chapter 19 Multistability of Convective Flows in a Porous Enclosure
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
Chapter 20 Hydrogen Transport in Framework of Linear Non-Equilibrium Thermodynamics Approach
20.1 Introduction
20.2 Problem Statement and Governing Equations
20.3 Plane Boundary Value Problem
20.4 Conclusion
References
Chapter 21 Classical and Non-Classical Models of Changes in the Young Modulus of Geomaterials Under Alternating Loads
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
Chapter 22 Two Approaches to Modeling Viscoelastic Cosserat Continua
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
Chapter 23 Porous Media Models Based on Generalized State Equations with Simple Examples
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
Chapter 24 Ball Indentation of Perforated Circular Hyperelastic Membranes
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
Chapter 25 Integrated Asymptotic Approach to the Structural Mechanics Models
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
Chapter 26 The Homogenized Delamination Criterion for Fiber-reinforced Plate
26.1 Introduction
26.2 Boundary Layer
26.3 Numerical Computation Results
26.3.1 Extension Along Oy- or Oy-axis
26.3.2 Shift in Ox2x3-plane
26.4 The Asymptotic Homogenized Strength Criterion of the Interface Zone
26.5 Constructing the Delaminating Strength Criterion
26.6 Conclusions
References
Chapter 27 Lightly Loaded Hydrodynamic Thrust Bearing Lubricated by a Non-Newtonian Fluid
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
Chapter 28 The Idea of Using Adhesive Bonds in Shaping of Cold-formed Thin-walled Beam-columns
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
Chapter 29 Dissipative Mechano-Electro-Magnetism Simulations in Electronic Components
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
Chapter 30 On Possible Reduction of Gradient Theories of Elasticity
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
Chapter 31 Dynamics of a Rectangular Rigid Body on a Movable Base
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 Ü(τ) = Asin(ωτ)
31.5 Numerical Results
31.6 Conclusion
References
Chapter 32 Asymptotically Correct Analytical Model for Flexural Response of a Two-Layer Strip with Contrast Elastic Constants
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
Chapter 33 On Analytical Modeling of Tension-Assisted Winding of Flexible Sheets
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
Chapter 34 On Using Rotations as Primary Variables in the Nonlinear Theory of Thin Elastic Shells
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
Chapter 35 Continuum Description of Extended Mass-in-Mass Metamaterial Models
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
Chapter 36 Multi-Element Metamaterial’s Design Through the Relaxed Micromorphic Model
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
Chapter 37 On Magnetically Induced Motion of Micropolar Ferrofluids
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
Chapter 38 Manufacturing Quality Evaluation of Photopolymer Resin 3D-Printed Scaffolds Using Microtomography
38.1 Introduction
38.2 Materials and Methods
38.3 Results and Discussion
38.4 Conclusion
References
Chapter 39 Comparison of Homogenization Techniques in Strain Gradient Elasticity for Determining Material Parameters
39.1 Introduction
39.2 Determining Strain Gradient Parameters
39.3 Microscale Structure
39.4 Results and Discussion
39.5 Conclusion
References
Chapter 40 Buckling of Rectangular Micropolar Plate with Prestressed Coatings
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
Chapter 41 A Cosserat Model for Fiber-Reinforced Elastic Plates
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
References
Chapter 42 On the Structure of Solutions in the Vicinity of Discontinuity of Boundary Conditions for Gradient Models
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
Chapter 43 A Damaged Medium Model for Assessing Life Characteristics of Polycrystalline Structural Alloys with Joint Mechanical Fatigue and Long-Term Strength of Material
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
Chapter 44 Bandgap Properties of a Class of Chiral and Achiral Metamaterials
44.1 Introduction
44.2 Theoretical Considerations
44.3 Numerical Considerations
44.4 Results and Discussion
44.5 Conclusions
References
Chapter 45 Large Strains of a Spherical Shell with Distributed Dislocations and Disclinations
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