Theoretical Analyses, Computations, and Experiments of Multiscale Materials: A Tribute to Francesco dell’Isola (Advanced Structured Materials, 175)

Preface
Foreword
Francesco dell’Isola: a majhtikìc of Magna Graecia
Contents
List of Contributors
Part I Theoretical Analysis
Chapter 1 A Different Catch for Poisson
	1.1 Introduction
		1.1.1 Key Objectives
		1.1.2 Notation and Definitions
		1.1.3 Organization of the Manuscript
	1.2 Poisson’s Ratio in Classical Continuum Mechanics
		1.2.1 Poisson’s Ratio for Small-Strain Linear Elasticity
		1.2.2 Poisson’s Ratio for Large Deformations Nonlinear Elasticity
	1.3 Poisson’s Ratio in Peridynamics
	1.4 Conclusion
	References
Chapter 2 Nonlinear Deformation of a Clamped-Edge Strip-Like Nano-Film
	2.1 Introduction
	2.2 Problem Formulation
	2.3 One-Dimensional Model of Deformation of Strip-Like Nano-Film
		2.3.1 Nonlinear Bending under the Eigenform-Like External Loading
		2.3.2 Post-Critical Buckling under Compression
		2.3.3 Nonlinear Free Transverse Vibration
	2.4 Conclusion
	References
Chapter 3 Closed-form Analytic Solutions of the Problem of a Hollow Sphere Made of Second Gradient Plastic Porous Material and Subjected to Hydrostatic Loading
	3.1 Introduction
	3.2 Description of the Hollow Sphere Problem
	3.3 Solution to the Hollow Sphere Problem When the Porosity is Neglected
	3.5 Analytic Results in the Presence of Porosity in the Matrix
		3.5.1 Derivation of Cauchy Stress Components
		3.5.2 Moment Components Derivation
		3.5.3 Discussion
	3.6 Conclusion
	References
Chapter 4 Quantum Dynamics Effects on Amplitude-Frequency Response of Superharmonic Resonance of Second-Order of Electrostatically Actuated NEMS Circular Plates
	4.1 Introduction
	4.2 Differential Equation of Motion
	4.3 Superharmonic Resonance of Second-Order
	4.4 Method of Multiple Scales: First-Order Hard Excitations Model
	4.5 Stability Testing
	4.6 Method of Multiple Scales: Second-Order Hard Excitations Model
	4.7 Electrostatic Reduced Order Model
	4.8 Casimir Reduced Order Model
	4.9 Van der Waals Reduced Order Model
	4.10 Numerical Simulations
		4.10.1 Electrostatic Model of Microelectromechanical Systems Clamped Circular Plates
		4.10.2 Casimir Force Effect on Nanoelectromechanical Systems Plates
		4.10.3 Van der Waals Force Effect on Nanoelectromechanical Systems Plates
		4.10.4 Stability
	4.11 Discussion and Conclusions
	References
Chapter 5 Propagation of Chaos for a Stochastic Particle System Modelling Epidemics
	5.1 Introduction
	5.2 Model
	5.3 Kinetic Limit
	5.4 Particle Approximation
	5.5 Concluding Remarks
	References
Chapter 6 On the Constitutive Assumptions for a Continuum Model of Scintillating Crystals
	6.1 Introduction
	6.2 A Continuum with Microstructure Model for Scintillators
	6.3 Balance Laws
	6.4 Thermodynamics. Constitutive Relations
	6.5 Reaction-Diffusion-Drift Equations for Scintillators: Constitutive Assumptions
		6.5.1 The Gibbs Entropy
		6.5.2 The Fermi–Dirac Integrals
	References
Chapter 7 Strong Ellipticity within the Strain Gradient Elasticity: Elastic Bar Case
	7.1 Introduction
	7.2 Nonlinear Elasticity
	7.3 Strain Gradient Elasticity
	7.4 Conclusions
	References
Chapter 8 Two Thermodynamic Laws in Phenomenological Mechanics of Continuum: Postulates or Definitions ?
	8.1 VariousWays of Axiomatization
	8.2 The General Form of Postulates in Mechanics of Continuum
	8.3 Postulate IV. The Law on the Change of Internal Energy
	8.4 Postulate V. The Law on the Change of Entropy
	8.5 Introduction to Mathematical Model of Interactions with a New Nature
	References
Chapter 9 On an Extended Family of Quasi-Equivalent Models of the Gradient Elasticity Theory
	9.1 Introduction
	9.2 Basic Definitions
	9.3 Family of Quasi-Equivalent Models
		9.3.1 Kinematic Restrictions for Displacement Vector Components
		9.3.2 Gradient Static Friction Model
		9.3.3 Kinematic Restrictions for Components of Derivatives of Displacements
	9.4 Examples of Gradient Models without Edge Conditions
		9.4.1 Variational Gradient Dilation Model
		9.4.2 Variant of Vector Gradient Elasticity Model
	9.5 General Case of Kinematic Restrictions. Generalized Pinching
	9.6 The Theorem on the Self-Balance of Meniscus Forces
	9.7 Conclusions
	References
Chapter 10 Continuum Models via Granular Micromechanics
	10.1 Introduction
	10.2 Granular Micromechanics Approach
		10.2.1 Piola Ansatz for Micro-Macro Kinematic Identification
		10.2.2 Micro-Macro Kinematic Identification Using Prescribed Micromotion
	10.3 Outlook
	References
Chapter 11 Some Variational Principles in the Three-Dimensional Micropolar Theories of Solids and Thin Solids
	11.1 Introduction
	11.2 Some Definitions and Integral Identities
	11.3 Lagrange Variational Principle (Lagrange Theorem)
		11.3.1 On Compatibility Conditions in Linear Micropolar Theory
		11.3.2 The Mixed Boundary Value Problems and the New Statement of the Boundary Value Problem with Respect to the Tensors of Stresses and Couple Stresses in Micropolar Solid Mechanics
	11.4 Castigliano’s Variational Principle (Castigliano’s Theorem)
	11.5 Generalized Reissner-Type Variational Principle
	11.6 The Generalized Reissner-Type Variational Principle in the Micropolar Theory of Thin Bodies with One Small Size under the New Parameterization of the Body Domain
	11.7 Generalized Reissner-Type Variational Principle in the Micropolar Theory of Thin Bodies with One Small Size in Moments under the New Parameterization of a Body Domain
	11.8 Generalized Variational Principle of Reissner-Type in the Micropolar Theory of Multilayer Thin Bodies with One Small Size with Full Contact of the Layers
	11.9 Generalized Reissner-Type Variational Principle in the Micropolar Theory of Multilayer Thin Bodies with One Small Size in the Case of Domains of Weakened Adhesion
		11.9.1 Jump-Type Model. Interphase (Interlayer) Displacements and Rotation Vectors. Vectors of Generalized Interfacial Forces and Moments
		11.9.2 Generalized Reissner-Type Variational Principle in the Theory of Multilayer Thin Bodies in Moments if There Are Domains of Weakened Adhesion
	11.10 Conclusion
	References
Chapter 12 Asymptotic Comparison of the Strain-Gradient and Micromorphic Models when Loading Forces Are Widely Spread
	12.1 Introduction
	12.2 Notation
	12.3 Spread Loads
	12.4 Fourier Expansion
	12.5 Comparison of Different Models at Large Scale
	12.6 Examples
	12.7 Conclusion
	References
Chapter 13 Quasiconvexity and Rank-One Convexity in Cosserat Elasticity Theory
	13.1 Introduction
	13.2 Cosserat Elasticity,
		13.2.1 Kinematical and Constitutive Framework
		13.2.2 Virtual Power and Equilibrium
	13.3 Conservative Problems and Potential Energy
		13.3.1 Example: Dead-Load Problems
	13.4 The Quasiconvexity Condition
	13.5 Rank-One Convexity
	13.6 Conclusion
	References
Chapter 14 Models of Viscoelastic Materials: a Review on Historical Development and Formulation
	14.1 Introduction
	14.2 The Simplest Models of Viscoelasticity
		14.2.1 Three-Element Models
		14.2.2 Four-Element Models
		14.2.3 Models with Large Numbers of Elements
	14.3 Viscoelastic Models with Fractional Derivatives
		14.3.1 First Applications of Fractional Calculus in Viscoelasticity
		14.3.2 The Simplest Fractional Calculus Viscoelastic Models
		14.3.3 Viscoelasticity Models with Several Different Fractional Parameters and One Relaxation (Retardation) Time
		14.3.4 Viscoelastic Models with One or More Fractional Parameters and Several Relaxation (Retardation) Times
		14.3.5 Models of Viscoelastic Fluids with Two or More Scott-Blair Fractional Derivative Elements
	14.4 Viscoelastic Models with Variable Viscosity
	14.5 Nonlinear Viscoelasticity Models with Fractional Derivatives
	14.6 Conclusion
	References
Chapter 15 Invariance Aspects of F = FeFi Representations in Coupled-Field Problems
	15.1 Introduction
	15.2 Basic Equations and Free Energy Expressions
	15.3 Modeling Representation
		15.3.1 First Modeling Representation: ψ(c;C)
		15.3.2 Second Modeling Representation: ψ(c;Ĉe)
		15.3.3 Relation Between Both Representations
	15.4 Discussion and Example
	15.5 Summary and Conclusion
	References
Part II Computations
Chapter 16 Strain-Gradient Modeling and Computation of 3-D Printed Metamaterials for Verifying Constitutive Parameters Determined by Asymptotic Homogenization
	16.1 Introduction
	16.2 Asymptotic Homogenization
	16.3 Computation
	16.4 Results and Verification
		16.4.1 Reference Solution Based on 1st-Gradient Theory
		16.4.2 Determination of Constitutive Parameters in the Strain-Gradient Model
		16.4.3 Simulations with 2nd-Gradient Theory and Validation
	16.5 Conclusion
	References
Chapter 17 On Boundary Layers Observed in Some 1D Second-Gradient Theories
	17.1 Introduction
	17.2 Euler–Bernoulli Beam
	17.3 Pantographic Beam
		17.3.1 Longitudinal Pantographic Beam with Nonlinear First Gradient Term
		17.3.2 Longitudinal Pantographic Beam with Nonlinear Second Gradient Term
	17.4 Concluding Remarks
	References
Chapter 18 Design and Parametric Enhancement of a Flexible Planar TEG - Numerical Study
	18.1 Introduction
	18.2 Materials and Methodology
		18.2.1 Theoretical Analysis
		18.2.2 Numerical Analysis
		18.2.3 Geometrical Parameters
		18.2.4 Boundary Conditions
		18.2.5 Material Choice
		18.2.6 Mesh Convergence Analysis
		18.2.7 Sensitivity Analysis
		18.2.8 Enhancement Methodology
	18.3 Results and Discussion
		18.3.1 Parametric Theoretical Enhancement
		18.3.2 Parametric Numerical Enhancement
		18.3.3 Validation with the Literature
		18.3.4 Final TEG design
	18.4 Conclusions
	References
Chapter 19 Implementation and Comparison of Non-Newtonian Viscosity Models in Hemodynamic Simulations of Patient Coronary Arteries
	19.1 Introduction
	19.2 Methodology
		19.2.1 Theoretical Analysis
		19.2.2 Patients Data and Artery Models
		19.2.3 Boundary Conditions
		19.2.4 Mesh Convergence Analysis
		19.2.5 Numerical Simulation Definitions
		19.2.6 Validation of Numerical Method with the Literature
	19.3 Results and Discussion
		19.3.1 Steady-State Flow of Newtonian Blood
		19.3.2 Transient Flow of Newtonian Blood
		19.3.3 Steady-State Flow of Non-Newtonian Blood
		19.3.4 Transient Flow of Non-Newtonian Blood
	19.4 Conclusions
	19.5 Appendix I
	References
Chapter 20 Bending/Tension of Plate Reinforced by a System of Parallel Fiber
	20.1 Introduction
	20.2 Reduction of 3-D PCP (20.2) to 2-D problems
	20.3 Numerical Computations
	20.4 Conclusion
	References
Chapter 21 Semi-Automatic Method of Stent Development for Hemodynamic Simulations in Patient Coronary Arteries with Disease
	21.1 Introduction
	21.2 Methodology
		21.2.1 Left Coronary Artery Geometry and Stent Design
		21.2.2 Mesh Generation
		21.2.3 Blood Properties
		21.2.4 Boundary Conditions
		21.2.5 Numerical Method
		21.2.6 Hemodynamic Descriptors
	21.3 Results and Discussion
	21.4 Conclusion
	References
Chapter 22 The Efficient Trabecular Bone Remodeling Numerical Tool Enabling Multiple Load Case Simulation
	22.1 Introduction
	22.2 The Trabecular Bone Remodeling Regulatory Model with the Lazy Zone Concept
	22.3 Multiple Loading Conditions
	22.4 The Simulation Approach Including the Postulates based on Shape Optimization Studies
	22.5 The Numerical Implementation and Mesh Generation Parallelization
	22.6 The Sample Simulation Results
	22.7 The System Efficiency and Scalability
	22.8 Conclusions
	References
Chapter 23 Modeling the Magnetic Relaxation Behavior of Micropolar Ferrofluids by Means of Homogenization
	23.1 Introduction
	23.2 Problem Setup and Homogenization Procedure
	23.3 The Governing Equations
	23.4 Solution to the Microscopic Problem
	23.5 Solution to the Macroscopic Problem
	23.6 Homogenization and Parameter Identification
	23.7 Results and Conclusion
	References
Chapter 24 Numerical Homogenisation of Gradient Materials
	24.1 Introduction
	24.2 Basic Notation
	24.3 Second Gradient Macroscopic Continuum and First Gradient Microscopic Continuum
		24.3.1 Macroscopic Boundary Value Problem
		24.3.2 Microscopic Boundary Value Problem (First Gradient)
	24.4 Continuous Strain Energy Formulation for Fibers
	24.5 Second Gradient Macroscopic Continuum and Second Gradient Microscopic Continuum
		24.5.1 Microscopic Boundary Value Problem (Second Gradient)
	24.6 Numerical Example: Cook Membrane
	24.7 Conclusion
	References
Chapter 25 Modeling the Slow Crack Growth of an Edge Crack within the Cohesive Zone Model Approach
	25.1 Introduction
	25.2 An Edge Crack with Cohesive Zone
	25.3 Solving a Problem on an Edge Crack with Cohesive Zone by the Regularization of Singular Integral Equation
	25.4 Subcritical State of a Crack
	25.5 Modeling the Slow Crack Growth
	25.6 Conclusion
	References
Chapter 26 An Insight into Computational Challenges in Damage Mechanics: Analysis of a Softening Hooke’s Spring
	26.1 Introduction
	26.2 Formulation of the Problem
		26.2.1 Linearly Softening Hooke’s Spring
	26.3 Analytical Solutions
		26.3.1 Displacement Control Solution
		26.3.2 Force Control Solution
	26.4 Numerical Integration Algorithms
		26.4.1 Displacement Control Integration
		26.4.2 Load Control Integration
		26.4.3 Arc-Length Control Integration
	26.5 Numerical Results
	26.6 Conclusions
	References
Chapter 27 Thermodynamic Compatibility of the HystereticPoly Uniaxial Material Implemented in OpenSees
	27.1 Introduction
	27.2 A Review of the HystereticPoly Constitutive Model
	27.3 The Thermodynamic Compatibility
		27.3.1 Negative Softening
		27.3.2 Hysteresis Crossing Paths
	27.4 Numerical Applications
	27.5 Discussion and Conclusions
	References
Chapter 28 Studying the Higher-Order Inertia in the Second-Order Theory of Elasticity for Modeling Metamaterials
	28.1 Introduction
	28.2 Model Implementation
	28.3 Results and Discussion
	28.4 Conclusion
	References
Chapter 29 Structural Analysis of Doubly-Curved Shells with General Boundary Conditions
	29.1 Introduction
	29.2 Geometrical Representation of Shells in Principal Coordinates
	29.3 ESL Assessment of Kinematic Quantities
	29.4 Generalized Constitutive Equations
	29.5 External Loads
	29.6 Equation of Motion
	29.7 Isogeometric Mapping of the Physical Domain
	29.8 Numerical Implementation via GDQ Method
	29.9 Equilibrium-Based Recovery Procedure
	29.10 General Boundary Conditions
	29.11 Applications and Results
		29.11.1 Free Vibration Analysis
		29.11.2 Static Analysis
	29.12 Conclusion
	References
Part III Experiments
Chapter 30 Characterisation of Mechanical Properties of Wood: Size Effect
	30.1 Introduction
		30.1.1 Wood Properties and Importance
		30.1.2 Size Effect
	30.2 Materials and Method
		30.2.1 Digital Image Correlation (DIC) Method
		30.2.2 Steps for Stiffness Parameter Calculation
	30.3 Results and Discussions
	30.4 Conclusions and Future Work
	References
Chapter 31 Covering a Surface with Pre-Stressed Ribbons: From Theory to Nano-Structures Fabrication
	31.1 Introduction
	31.2 Geometric and Kinematical Settings
	31.3 Weak Transversal Homogenenity and the Moment Equations
	31.4 Small Strain Deformation of a Ribbon
	31.5 From Theory to Fabrication of a Nano-Sphere
		31.5.1 Optimal Covering with Constant Parallel Ribbons
		31.5.2 Elastic Layers with Pre-Stress: Material Parameters
		31.5.3 Design and Fabrication
		31.6 Conclusions and Perspectives
	References
Chapter 32 Experimental and Theoretical Investigations of Auxetic Sheet Metal
	32.1 Introduction
	32.2 Material and Methods
		32.2.1 Sample Material
		32.2.2 Perforated Aluminium Sheet
		32.2.3 Experimental Setup
		32.2.4 Sample Preparation
	32.3 Results and Discussion
		32.3.1 Mechanical Properties of Bulk AlMg3
		32.3.2 Experiment on the Auxetic Sheet
		32.3.3 Simulation
	32.4 Conclusion
	References
Correction to: Theoretical Analyses, Computations, and Experiments of Multiscale Materials