Size-Dependent Continuum Mechanics Approaches: Theory and Applications

Description

This book offers a comprehensive and timely report of size-dependent continuum mechanics approaches. Written by scientists with worldwide reputation and established expertise, it covers the most recent findings, advanced theoretical developments and computational techniques, as well as a range of applications, in the field of nonlocal continuum mechanics. Chapters are concerned with lattice-based nonlocal models, Eringen’s nonlocal models, gradient theories of elasticity, strain- and stress-driven nonlocal models, and peridynamic theory, among other topics. This book provides researchers and practitioners with extensive and specialized information on cutting-edge theories and methods, innovative solutions to current problems and a timely insight into the behavior of some advanced materials and structures. It also offers a useful reference guide to senior undergraduate and graduate students in mechanical engineering, materials science, and applied physics. Foreword Preface Contents Editors and Contributors Lattice-Based Nonlocal Elastic Structural Models 1 Introduction 2 Discrete and Nonlocal Rods 2.1 Axial Lattices 2.2 Lattice Formulation: Governing Equations 2.3 Lattice Formulation: Resolution 2.4 Nonlocal Continualized Model and Eringen's Model 2.5 Nonlocal Solutions 2.6 Lattice with Direct and Indirect Interactions 3 Discrete and Nonlocal Beams 3.1 Hencky-Bar-Chain Model 3.2 Continualised Nonlocal Beam Model 3.3 Buckling and Vibrations Analyses of Discretized Beam 4 Discrete and Nonlocal Plates 4.1 Hencky-Bar-Chain Net 4.2 Eringen's Nonlocal Plate Model 4.3 Microstructure-Based Nonlocal Plate Model 5 Conclusions References Eringen's Nonlocal Integral Elasticity and Applications for Structural Models 1 Introduction 2 Eringen's Formulation 2.1 Governing Equations 2.2 A Discussion About Eringen's Nonlocal Stress Models 2.3 The Correlation Between Eringen's Nonlocal Model and Mindlin's Gradient Model 3 Applications for One-Dimensional Problems 3.1 Beam Equilibrium Equations 3.2 Static Problems 3.3 Dynamical Problems 4 Conclusions References Nonlocal Mechanics in the Framework of the General Nonlocal Theory 1 Introduction 2 Eringen's Nonlocal Theory 2.1 Nonlocal Mechanics of Particles 2.2 Nonlocal Continuum Mechanics 2.3 Eringen's Constitutive Equations 2.4 Nonlocal Field Equation 2.5 Dispersion Relations 2.6 Limitations of Eringen's Nonlocal Theory 3 General Nonlocal Theory 3.1 Equilibrium and Constitutive Equations 3.2 Nonlocal Moduli 3.3 Propagation of Dispersive and Aggregative Waves 3.4 Comparison to Eringen's Nonlocal Theory 4 Relation to Strain Gradient and Couple Stress Theories 4.1 Strain Gradient Theory 4.2 Couple Stress Theory 4.3 Relation to Mindlin's Strain Gradient Theories 4.4 Relation to Couple Stress Theory 4.5 Wave Propagation 5 Infeasibility of Nonlocal Strain Gradient Theory 6 Identification of Nonlocal Parameters and Length Scales of Strain Gradient and Couple Stress Theories 7 Conclusions References Displacement Based Nonlocal Models for Size Effect Simulation in Nanomechanics 1 Introduction 2 An Overview on the Nonlocal Models 2.1 Integral Models 2.2 Gradient Models 2.3 Mixture Models 3 Displacement Based Nonlocal Model 3.1 Nonlocal Rod 3.2 Dynamical Problem of Nonlocal Beam Model 4 Concluding Remarks References One-Dimensional Well-Posed Nonlocal Elasticity Models for Finite Domains 1 Introduction 2 One-Dimensional Well-Posed Nonlocal Elasticity Theory for Finite Domains 2.1 Nonlocal Integral Constitutive Equation 2.2 Differential Constitutive Equation and Its Boundary Conditions 3 Equilibrium Equation and Boundary Conditions 3.1 Governing Equation of a Nanorod 3.2 Governing Equation of a Nanobeam 4 Numerical Results 4.1 Static Deformation of Nanorods 4.2 Bending of Nanobeams 4.3 Buckling of Nanobeams 5 Conclusions References Iterative Nonlocal Residual Elasticity 1 Introduction 2 Nonlocal Residual Elasticity 2.1 Nonlocal Mechanics of Particles 2.2 Nonlocal Continuum Mechanics: Balance Laws 2.3 Nonlocal Continuum Mechanics: Constitutive Model 2.4 Boundary Value Problem and Solution Procedure: Iterative Nonlocal Residual Elasticity 3 Application to Euler-Bernoulli Beams 4 Conclusions References Nonlocal Gradient Mechanics of Elastic Beams Under Torsion 1 Introduction 2 Local Elastic Beams 3 Nonlocal Gradient Elastic Beams 3.1 Nonlocal Strain-Driven Gradient (NstrainG) Elasticity 3.2 Nonlocal Stress-Driven Gradient (NstressG) Elasticity 4 Nonlocal Gradient Nano-Structures Under Torsion 4.1 Elastostatic Torsion 4.2 Torsional Free Vibrations 5 Conclusions References Reformulation of the Boundary Value Problems of Nonlocal Type Elasticity: Application to Beams 1 Introduction 2 Problem Formulation 2.1 Nonlocal Elasticity 2.2 Asymptotic Theory for Nonlocal Elasticity 3 BVPs for Euler–Bernoulli Beam Model 3.1 Reformulation 3.2 Dynamic Behavior of Nonlocal Cantilevers: Euler–Bernoulli Beam 4 BVPs for Timoshenko Beam Model 4.1 Reformulation 4.2 Dynamic Behavior of Nonlocal Cantilevers: Timoshenko Type 5 Conclusions References Application of Combined Nonlocal and Surface Elasticity Theories to Vibration Response of a Graded Nanobeam 1 Introduction 2 Classical Euler–Bernoulli Beam Theory with Surface Effects 3 Nonlocal Euler–Bernoulli Beam Theory with Surface Effects 4 Free Vibration Analysis 4.1 Problem Formulation 4.2 Analysis Using the Method of Multiple Scales 5 Forced Vibration Analysis 5.1 Problem Formulation 5.2 Analysis Using Method of Multiple Scales 6 Numerical Results and Discussion 6.1 Validation Studies 6.2 Parametric Study 7 Conclusions References Finite Element Nonlocal Integral Elasticity Approach 1 Introduction 2 Nonlocal Integral Theory 2.1 Elastic Constitutive Equations 2.2 Viscoelastic Constitutive Equations 2.3 Notes on the Kernel Type 3 Nonlocal Integral Finite Element Method 3.1 Variational Equations 3.2 Finite Element Formulations 3.3 Element Types 3.4 Notes on the Boundary Conditions 4 Nano-Scaled Beams 4.1 Applying Boundary Conditions for Nano-Scaled Beams 4.2 Bending of Elastic Nano-Scaled Beams 4.3 Vibration of Nano-Scaled Beams 4.4 Buckling of Nano-Scaled Beams 5 Nano-Scaled Plates 5.1 Applying Boundary Conditions for Nano-Scaled Plates 5.2 Bending of Nano-Scaled Plates 5.3 Vibration of Nano-Scaled Plates 5.4 Buckling of Nano-Scaled Plates 6 Numerical Examples and Discussions 6.1 Elastic Beam and Plate Bending 6.2 Elastic Beam and Plate Vibration 6.3 Elastic Beam and Plate Buckling 6.4 Viscoelastic Free Vibration 7 Conclusions References `Explicit' and `Implicit' Non-local Continuum Descriptions: Plate with Circular Hole 1 Introduction 2 Materials and Methods 2.1 Micropolar Model 2.2 Eringen's Non-local Model 3 Differential Evolution Method 4 Numerical Simulations 5 Conclusion References Micromorphic Continuum Theory: Finite Element Analysis of 3D Elasticity with Applications in Beam- and Plate-Type Structures 1 Introduction 2 Micromorphic Elasticity Theory 2.1 Kinematics 2.2 Equation of Motion 2.3 Elasticity 2.4 Conditions of Elastic Parameters 3 Matrix-Vector Representation 4 Finite Element Formulation 5 Results and Discussion 6 Conclusions References Peridynamic Modeling of Laminated Composites 1 Introduction 2 Fundamentals of Peridynamics 3 Peridynamic Modeling of a Laminate 4 Numerical Results 4.1 Bond-Based PD 4.2 Ordinary State-Based PD 4.3 Non-ordinary State-Based PD 5 Conclusions References Nonlocal Approaches to the Dynamics of Metamaterials 1 Introduction 2 Phenomenological Models 3 High-Order Homogenization Methods 3.1 Asymptotic Expansions 3.2 Other Higher-Order Homogenization Techniques 4 Averaging Techniques 5 Concluding Remarks References Gradient Extension of Classical Material Models: From Nuclear & Condensed Matter Scales to Earth & Cosmological Scales 1 Introduction 2 State-of-the Art: Previous Literature & Current State of Affairs 2.1 Plastic Instabilities & Size Effects 2.2 Chemomechanical Instabilities in LiBs & Biomechanical Instabilities in Brain 3 ILG Formulation Through Continuum & Statistical Mechanics 4 ILG Applications: Mechanics, ChemoMechanics, and BioChemoMechanics 4.1 Mechanical Deformation Instabilities & Intermittent Plasticity 4.2 Chemomechanical Instabilities in LiB Anodes 4.3 Glioblastoma Instabilities in Brain 5 ILG and Rheology: Newtonian and Complex Fluids 6 ILG in Other Disciplines & Scales 7 ILG Modification of Newton's Gravitational Law 8 Gradient Interatomic Potentials 9 Fractional Considerations 10 Conclusions References