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دانلود کتاب Encyclopedia of Continuum Mechanics
دانلود کتاب دایره المعارف مکانیک پیوسته
مشخصات کتاب
Encyclopedia of Continuum Mechanics
ویرایش: 1st ed. 2020
نویسندگان: Holm Altenbach (editor). Andreas Öchsner (editor)
سری:
ISBN (شابک) : 3662557703, 9783662557709
ناشر: Springer
سال نشر: 2020
تعداد صفحات: 0
زبان: English
فرمت فایل : EPUB (درصورت درخواست کاربر به PDF، EPUB یا AZW3 تبدیل می شود)
حجم فایل: 341 مگابایت
قیمت کتاب (تومان) : 47,000
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این دایره المعارف کل علم مکانیک پیوسته از جمله مکانیک مواد و سیالات را پوشش می دهد. این دانشنامه شامل تعاریف ریاضی برای مدل‌سازی مکانیکی پیوسته، مفاهیم فیزیکی بنیادی، روش‌شناسی مدل‌سازی مکانیکی، رویکردهای عددی و بسیاری از کاربردهای اساسی است. تکنیک‌های مدل‌سازی و تحلیلی ابزارهای قدرتمندی در مهندسی عمران مکانیک و فضای فضایی، به علاوه در زمینه‌های مرتبط با پلاستیسیته، ویسکوالاستیسیته و رئولوژی هستند. مکانیک پیوسته مبتنی بر تانسور و مستقل از چارچوب مرجع اخیراً کاربردهایی در ژئوفیزیک و مواد پیدا کرده است.
توضیحاتی درمورد کتاب به خارجی
This Encyclopedia covers the entire science of continuum mechanics including the mechanics of materials and fluids. The encyclopedia comprises mathematical definitions for continuum mechanical modeling, fundamental physical concepts, mechanical modeling methodology, numerical approaches and many fundamental applications. The modelling and analytical techniques are powerful tools in mechanical civil and areospsace engineering, plus in related fields of plasticity, viscoelasticity and rheology. Tensor-based and reference-frame-independent, continuum mechanics has recently found applications in geophysics and materials.
فهرست مطالب
Encyclopedia of Continuum Mechanics Preface List of Topics Adaptive Structures Biographies of Scientists Biomechanics Composites Continuum Damage Mechanics Continuum Mechanics Basics Continuum Thermodynamics Continuum Waves Creep Mechanics Fractional Calculus in Continuum Mechanics Fracture Mechanics Generalized Continua Impact Mechanics Lightweight Structures Mechanical Behavior and Performance of Electronic and Photonic Materials, Packages and Systems Nanomechanics Numerical Continuum Mechanics Optimization Plane Problems Plasticity Plates Random and Fractal Media Ray Expansion in Continua Rods and Beams Shells Tensor Calculus Variational Principles About the Editors About the Section Editors List of Contributors A Active Control of Sound, Applications of Synonyms Definition Abbreviations Motivation Sound Control Transmission Loss Free-Field Radiation Cabin Noise Active Noise Control (ANC) Placement of Actuators and Sensors Active Windshield of a Passenger Car Active Fuselage Panels Active Aircraft Sidewall Panels Cross-References References Active Control of Vibration, Applications of Synonyms Abbreviations Definition Motivation Structural Damping Global Vibration Control Distributed Collocated Control Active Vibration Isolation Semi-active Vibration Control Nonlinear Mechanisms Cross-References References Active Deformation Active Noise Cancellation Active Noise Control Active Noise Reduction Active Soft Tissue Modelling Active Systems Adaptation Adaptation (Shakedown) Adaptive Structures Adaptive Structures, Principles of Synonyms Abbreviations Definition Classification of Adaptive Systems Application Target Frequency Considerations System Complexity and Robustness Components of Adaptive Structures Sensors Actuators Controllers Control Strategies Static Control Local Control Global Control Reference Signals Active Vibration Isolation Active Vibration Control (AVC) Active Structural Acoustic Control (ASAC) Active Noise Control (ANC) Design Principles Impedance Matching Cross-References References Advanced Composite Materials Aero, Eron Lyuttovich Education and Carrier Scientific Results References Airy, George Biddell Early Years and Education Professional Career and Honors Scientific Achievements Cross-References Ambartsumian, Sergey (Aleksandr) Education Professional Carrier Scientific Achievements Honors Other Activities References Analysis of Cracks in Piezoelectric Solids with Consideration of Electric Field and Strain Gradients Synonyms Definition Introduction Problem Statement Numerical Solution Algorithm J-Integral Numerical Examples A Square Plate with a Central Crack A Rectangular Plate with an Edge Crack Conclusion References Analysis of Non-linear Equilibria Analytical and Energy-Based Methods for Penetration Mechanics Synonyms Definition Introduction Ballistic Impact and Perforation Mechanics Perforation by Plugging Structural Models Energy-Based Models Monolithic Targets In-Contact Multilayered Targets Case Study: An In-Contact Metallic Three-Layered Target Perforation by Petalling References Analytical Methods Analytical Stability Considerations in Lightweight Design Synonyms Definition Stability Criteria and Post-buckling Behavior General Concept of Static Analytic Stability Considerations Beams Plates Cylindrical Shells Cross-References References Anisotropic and Refined Plate Theories Synonyms Definition Anisotropic Materials in Practice Voigt\'s Notation for Anisotropic, Linear Elastic Material Engineering Constants Special Kinds of Anisotropy Plate Problems Classical Theory of Monoclinic Plates Laminate Theories Refined Plate Theories Isotropic Refined Plate Theories Anisotropic Refined Plate Theories Cross-References References Anisotropic Bending Problems Anisotropic Damage Anisotropic Linear Viscoelasticity Anisotropic Out-of-Plane Problems Anisotropic Plasticity and Application to Plane Stress Synonyms Definition Introduction Tensors and Invariants Cauchy Stress Invariants Principal Values Principal Stresses Flow Theory of Plasticity Uniaxial Stress State Multiaxial Stress State Isotropic Yield Conditions Anisotropic Plasticity Material Axes Quadratic Formulations Linear Tensor Transformations Non-quadratic Approaches Specific Plane Stress Approach Yield Function Anisotropy Coefficients Anisotropy Coefficient Tensors Flow Rule Identification and Validation Conventional Mechanical Tests Uniaxial Tension Test Plane Strain Tension Test Hydraulic Bulge Test Advanced Mechanical Tests Biaxial Compressive Test Using an Adhesively Bonded Sheet Laminate Specimen Biaxial Tensile Test Using a Cruciform Specimen Combined Biaxial Tension-Compression Test In-Plane Uniaxial Stress Reversal Test Multiaxial Tube Expansion Test Combined Tension-Shear Test Applications Application to Steel Application to Aluminum Alloy Concluding Remarks Cross-References References Anisotropy Anisotropy of Linear Creep Synonyms Definition Elastic-Viscoelastic Correspondence Principle for the Case of Isotropic Materials Generalized Elastic-Viscoelastic Correspondence Principle for the Case of Anisotropic Materials The Generalized Correspondence Principle Applied to the Case of Fiber-Reinforced Composites References Anti-plane Motions Localized Near a Boundary of a Half-Space Anti-plane Surface Waves in Materials with Surface Energy Synonyms Definition Introduction Anti-plane Motions in the Bulk Boundary Conditions Within the Surface Elasticity Conclusions Cross-References References Anti-sound Application of Analytical Modeling in the Design for Reliability of Electronic Packages and Systems Definition Analytical Classical Mechanics and Applied Physics-Based Modeling, Its Role, and Significance Analytical vs. FEA Modeling Interfacial Thermal Stresses Concentrate at the Assembly Ends Incentive for Using Low-Modulus and/or Low-Yield-Stress Bonding Materials Transverse Grooves in Bonded Components Could Relieve Interfacial Stresses Conclusion References Approaches to the Description of Fractal Media Argyris, John Hadji Education Professional Career Scientific Achievements and Honors Cross-References References Armour Arutyunyan, Nagush Khachaturovich Education Professional Carrier Scientific Achievements References Asymptotic Analysis Asymptotic Solution Autofrettage Auxeticity in Two-Dimensional (2D) Elasticity Avalanches in Solids Synonyms Definition Avalanches in Solids Some Open Questions in the Field Comparison of Model Predictions for the Avalanche Statistics and Dynamics to Recent High-Resolution Experiments on Bulk Metallic Glasses On the Proper Determination of Power-Law Exponents for Slip Statistics Using Experimental Data from Bulk Metallic Glasses References Axiomatic/Asymptotic Method (AAM) Axiomatic/Asymptotic Method and Best Theory Diagram for Composite Plates and Shells Synonyms Definition Background Theory Carrera Unified Formulation (CUF) Axiomatic/Asymptotic Method (AAM) and Best Theory Diagram (BTD) Applications Influence of Primary Variables vs Characteristic Parameters Reduced Models Providing Quasi-3D Accuracy Best Theory Diagrams Conclusions Cross-References References B Balance of Energy Balance of Entropy Balance of Mass Balance of Moment of Momentum Balance of Momentum Ballistic Protection Bars Bauschinger, Johann Family and Education Professional Career Scientific Achievements and Honors Cross-References References Beams Beams and Plates Beams, Plates, and Shells Synonyms Definition Bernoulli-Euler Beams Kinematics Equilibrium Conditions Stress Distribution Kirchhoff Plates Kinematics Equilibrium Conditions Stress Distribution Kirchhoff-Love Shells Kinematics Constitutive Relations Equilibrium Conditions Stress Distribution Cross-References References Bearing Capacity Bernoulli, Daniel Early Years and Education Professional Carrier Scientific Achievements Cross-References Bernoulli, Jakob I Early Years and Education Professional Carrier and Achievments Cross-References Bernoulli, Johann Early Years and Education Professional Carrier Scientific Achievements Cross-References Bernoulli-Euler Beam Best Theory Diagram (BTD) Biaxial Experiments Bifurcation Buckling Bifurcation of Equilibrium Continuum Systems Bifurcation of Nanolayers with Direct Current Bingham, Eugene Cook Education Professional Carrier Scientific Achievements Honors Other Activities References Biological Systems Biological Tissues Biomechanical Rupture Risk Assessment of Abdominal Aortic Aneurysms Biomolecules Biosensor Bolotin, Vladimir Vasilyevich Early Life Bolotin\'s Contributions Bolotin\'s Books Cross-References References Bone Adaption Bone Remodeling Bone-Resorbable Graft Interaction Boundary Conditions for Generalized Continua Boundary Element Method for Composite Laminates Synonyms Definition Generalized Plane Strain Problem Governing Equations Boundary Integral Representation and Equations Fundamental Solutions Beam-Type Composite Laminates Boundary Integral Representation and Equations Numerical Solution Cross-References References Boundary Element Methods Synonyms Definition Introduction Derivation Discretization Modelling Crack Problems Other BEM Formulations Plates and Shells Elastoplasticity Transient Thermoelasticity Elastodynamics Summary References Boundary Integral Equation Methods Boundary Integral Equations Boundary Integral Equations and Solution for Composite Laminates Boundary Variational Functional Boundary-Value Dynamic Problems of Thin Bodies, Ray Expansion Approach Synonyms Definition Preliminary Remarks Problem Formulation and Governing Equations The Ray Method and Its Applications Pressing of a Rigid Washer into a Circular Opening of an Elastic Plate Pressing of a Rigid Circular Washer into an Opening in a Lateral Surface of an Elastic Shallow Cylindrical Shell Conclusion Cross-References References Bounding Box; Surrounding Box, Container Branch-Switching Breakage Bridgman, Percy Williams Early Years and Education Scientific Achievements and Professional Career Honors References Brittle Fracture B-Spline Curve Buckling and Postbuckling of Composite Plates and Shells Synonyms Definition Introduction Differential Equations Governing Buckling of Composite Plates and Shells Levels of Discretization Complexity and Stages of Nonlinearity Analysis Basic Shell Buckling and Postbuckling Analyses Design Aspects Conclusions Cross-References References Buckling and Post-buckling of Plates Synonyms Definition Overview The Buckling of a Discrete Elastica, Revisited An Elastic Spring Resting on an Elastic Hinge First and Second Gradients of Strain Non-linear Equilibrium of the Discrete Elastica Non-linear Solutions Fundamental Branch of Solution Linearized Equilibrium Analysis of the Small η Limit Comparison with the Predictions of a Föppl-von Kàrmàn-Type of Model Linear Buckling Analysis of Plates A Generic Plate Model Non-linear Equilibrium Fundamental Branch of Solutions Linear Bifurcation Condition Asymptotic Form of the Linearized Bifurcation Problem for Small Aspect Ratio Least-Effort Calculation of Buckling Loads Using the Föppl-von Kàrmàn Model Post-buckling Analysis of the Discrete Elastica A Fully Non-linear Discrete Elastica Set of Formal Rules for the Expansion Method Fundamental Branch Linear Bifurcation Analysis Weakly Non-linear Expansion Conclusion Cross-References References Budiansky, Bernard Early Life Scientific Contributions Recognition Cross-References References Burgers, Johannes Martinus Early Years and Education Professional Career Scientific Achievements and Honors Cross-References References Burzyński, Włodzimierz Stanisław Trzywdar Family and Education Professional Carrier Scientific Achievements References C Calculus of Variations Canonical Equations Caratheodory, Constantin Early Years and Education Professional Career and Honors Cross-References Carbon Carbon Nanotubes Synonyms Definition Introduction Geometry of Carbon Nanotubes Properties of Carbon Nanotubes Mechanical Properties Electrical Properties Thermal Properties Optical Properties Nanoobjects Based on Carbon Nanotubes Hetero-Nanotubes Hybrid-Nanotubes Functionalized-Nanotubes Defects in Carbon Nanotubes Application of Carbon Nanotubes Modelling of Carbon Nanotubes Atomic Models Continuum Models Multiscale Models Summary Cross-References References Carbon Nanotubules Carrera Unified Formulation (CUF) Carroll, Michael Mary Biographical Information, Education, and Academic Career Scientific Accomplishments Professional Service Honors and Awards Cross-References References Cauchy, Augustin Louis Education Scientific and Professional Career Scientific Achievements Cross-References Cauchy-Born Rule Cavitation Phenomenon Cellular Materials Center Manifold Characteristics Chemical Affinity Tensor Chemical Affinity Tensor in Coupled Problems of Mechanochemistry Synonyms Definition Introduction Diffusion-Controlled and Reaction Rate-Controlled Reaction Fronts Propagations On Tensorial Nature of Chemical Potential and Chemical Affinity Chemical Affinity Tensor and Stress-Affected Reaction Front Kinetics Kinematics Dissipation, Chemical Affinity Tensor, and Front Velocity Invariants of the Chemical Affinity Tensor and Reaction Intensity Boundary Value Problems of Mechanochemistry: Forbidden Zones Concluding Remarks Cross-References References Chemical Potential Chemical Potential Tensor Chemical Reaction Front Kinetics Chemical Reaction Rates Chemomechanics Christoffel, Elwin Bruno Early Years and Education Professional Carrier Scientific Achievments and Honors Cross-References Circular Plate Classical and Advanced Modeling of Rotating Composite Blades and Rotors Synonyms Definition Theory Preliminaries Classical and Advanced Kinematic Theories EoM in the Corotating Reference Frame EoM in the Inertial Reference Frame Applications Spinning Structures Rotating Blades Conclusion References Classical Beams and Plates in a Fractional Derivative Medium, Impact Response Synonyms Definition Backgrounds of Fractional Calculus in Mechanics and Tautochrone Problem Impact Response of a Bernoulli-Euler Beam Green Function for the Fractional Derivative Kelvin-Voigt Model Determination of the Contact Force Defining the Local Indentation. Approximate Solution Impact Response of a Kirchhoff-Love Plate Conclusion Cross-References References Classical Lamination Theory (CLT) Classical Plate Problems Synonyms Definition Introduction Bending of Thick Plates Bending of Thin Plates Axisymmetrically Loaded Circular Plates Simply Supported, Uniformly Loaded Circular Plates Simply Supported, Circular Plates Loaded at the Center Semi-infinite Plates: Hard Vs. Soft Simple Support Simply Supported Rectangular Plates: Navier Solution Summary Cross-References References Classical Shell Theory Cleavage Closed Cross Section CNTs Codazzi, Delfino Professional Carrier Scientific Achievements and Honors Cross-References Cold Flow Coleman, Bernard David Education Professional Carrier Scientific Achievements Honors Cross-References References Coleman-Noll Procedure Coleman-Noll Procedure for Classical and Generalized Continuum Theories Synonyms Definition The Coleman-Noll Procedure Balance Equations Entropy Inequality General Principles of the Theory of Materials Constitutive Equations Internal State Variables Viscous Materials Generalized Continua Strain Gradient Theory Micromorphic Theory Theory of Microforces Final Remark Cross-References References Collapse Load Collision Collision of Two Spherical Shells, Fractional Operator Models Synonyms Definition Background Collision of Two Elastic Spherical Shells The Main Kinematic and Dynamic Characteristics of the Wave Surface Construction of One-Term Ray Expansions Central Impact of Two Elastic Spherical Shells Solution in the Case of the Collision of Two Identical Spherical Shells Central Impact of Two Identical Viscoelastic Spherical Shells Conclusion Cross-References References Composite = Hybrid, Compound, Complex, Combined, Blended, Mixed Composite Materials Composite Media Composite Shell Structures and Materials Composites Composites with Fibers Computational Dynamics Synonyms Definition Introduction Nonlinear Elastodynamics Discretization in Space Discretization in Time Newmark\'s Method Schemes for First-Order ODEs Energy-Momentum Schemes Balance of Angular Momentum Balance of Energy Midpoint-Based EM Scheme Conclusions Cross-References References Computational Mechanics Computational Mechanics of Generalized Continua Synonyms Definition Overview Kinematics Variational Principles Gradient Continuum Dirichlet Principle Mixed Variational Principle Regularized Mixed Variational Principle Micromorphic Formulation Constitutive Modelling Computational Aspects Higher-Order Continuous Discretization Continuous Discretization Discretization of the Variational Principles Gradient Continuum Micromorphic Continuum References Computational Methods for Coupled Problems Synonyms Definition The Mathematical Model The Numerical Model Relevant Applications References Computational Methods for Dynamic Fracture Computational Methods for Impact Computational Modeling and Simulation Computational Models for Hemodynamics Synonyms Definition Background The Vascular Circulation Mathematical Modeling Numerical Approximation The Heart Mathematical Modeling Numerical Approximation Conclusions Cross-References References Conductivity Configurational Force Connections of Shell Elements Constitutive Equations Constitutive Equations of Elastic Shells Constitutive Framework Constitutive Modeling Constitutive Modeling of Beams Using Mechanics of Structure Genome Synonyms Definition Beam Modeling as Taught in Traditional Textbooks Kinematics Kinetics Constitutive Relations Constitutive Modeling of Beams Using MSG Mechanics of Structure Genome Conclusion References Constrained Mechanical Systems Constrained Mixture Contact Detection; Contact Search Contact Interaction; Contact Calculation, Resolution of Contact Contact Problem Continua with Microstructure Continuous Media Continuous-Molecular Approach to Modeling of Nanostructures Synonyms Definition Introduction Continuum Rod Network of the Atomic Bond Structure Contact Problem of a Graphene Monolayer Cauchy-Borne Rules Concluding Remarks Cross-References References Continuum Biomechanics Continuum Damage Mechanics Continuum Damage Model for Ductile Materials Based on Stress-State-Dependent Damage Functions Synonyms Definition Introduction Continuum Damage Model Damage Variables Constitutive Equations Undamaged Configurations Damaged Configurations Numerical Simulations on the Microscale Biaxial Experiments Conclusions References Continuum Mechanics Basics, Introduction and Notations Synonyms Definition Overview Notation List of Symbols Cross-References References Continuum Mechanics of Fractal Media Synonyms Definition Introduction Approaches to Theory of Fractal Media I.1 Approach Based on the Analysis on Fractal (AF-Approach) I.2. Fractional-Differential Approach (FD-Approach) II.1. Fractional-Integral Approach (FI-Approach) II.2. Fractional Space Approach (FS-Approach) II.3. Non-integer Dimensional Space Approach (NIDS-Approach) Cross-References References Continuum Mechanics of Soft Tissue Growth and Remodeling Synonyms Definition Introduction Arterial Mechanics and Mechanobiology Approaches to Model Growth and Remodeling A Constrained Mixture Framework Illustrative Constitutive Relations for Arterial Growth and Remodeling Conclusions Cross-References References Continuum Mechanics with Spontaneous Violations of the Second Law of Thermodynamics Synonyms Definition Introduction Background: Fluctuation Theorem Basics Entropy as a Submartingale Continuum Mechanics with Violations of the Second Law Stochastic Thermomechanics Basic Framework Axioms of Continuum Thermomechanics Random Fields Conclusion References Continuum Micromechanics Continuum Model Continuum Models of Fractal Media Contitutive Equations Coordinate-Independent Quantities Cosserat Media Synonyms Definition The Physical Picture of the Cosserat Continuum Continuum Mechanics of Cosserat Media Kinematics Stress Concept Balance Relations Cosserat Material Theory Thermoelastic Cosserat Solid Isotropic, Geometrically Linear, Isothermal Cosserat Solid Cosserat Elastoplasticity Applications Material Modeling Computational Regularization Size Effect Modeling Conclusions Cross-References References Cosserat, Eugene and Francois Education of the Cosserat Brothers Eugène Cosserat François Cosserat Translation Activities Scientific Joint Works of Eugène and François Cosserat Euclidean Action Group Theory Impact of Cosserat\'s Contribution in the Modern Period References de Coulomb, Charles-Augustin Education Professional Carrier Scientific Achievements Honors Coupled Problems Coupled Problems in Biological Systems Synonyms Definition Introduction Continuum Biomechanics of Multicomponent Materials Basic Quantities Balance Relations Selected Examples of Coupled Effects in Biomaterials The Intervertebral Disc as a Binary Tissue of a Solid Skeleton and a Single Pore Fluid Swelling Tissue as a Biphasic, Four-Component Aggregate Ternary, Four-Component Aggregate for Vascularized Tissue Subjected to Drug Delivery Conclusions Cross-References References Covalent Bonds Crack Crack Analysis Crack Growth Crack Propagation Cracks Crandall, Stephen Harry Education, Professional, and Scientific Career Awards Books by S.H. Crandall References Crash Impact Protection Crashworthiness Synonyms Definition Crashworthiness Assessment Crash Tests and Regulations Crash Energy Metrics Absorbed Energy Specific Energy Absorption Peak Force Crush Force Efficiency Stroke Efficiency Energy-Absorbing Effectiveness Factor Biomechanical Injury Criteria Head Injury Criterion Viscous Criterion Neck Injury Criteria Tibia Index Crashworthiness of Structural Systems Behavior of Metallic Impact-Absorbing Members Crashworthiness of Composites and Structural Foams Computational Simulation of Structural Crashworthiness Good Practice Guidelines for the Finite Element Modelling and Simulation of Structures Under Impact Geometrical Modelling Material Modelling Element Choice Analysis Configuration and Post-processing References Creep Creep at Extremely Low Temperatures Synonyms Definition Overview Basic Methodology Near 0 K Thermodynamic Instability Physical Background of Low Temperature Plasticity Smooth Plastic Flow (SPF) Discontinuous Plastic Flow (DPF) Physical Background of Low Temperature Creep Experimental Evidence of Low Temperature Creep Experimental Setup LT Creep in Copper LT Creep in Stainless Steels Modeling of Low Temperature Creep Key Research Findings Cross-References References Creep Deformation Synonyms Definition Introduction Theoretical Analysis of Creep Damage Experimental Creep Investigations Under Complex Stress State Creep Results of Pure Copper Creep Results of 2017 Aluminum Alloy New Trends in Creep Damage Analysis: Interdisciplinary Approach References Creep Fatigue Synonyms Definition Thermal Stress and Damages Creep-Fatigue Life Evaluation and Failure Criteria The Damage Summation Method Frequency-Modified Method Strain Range Partitioning Others Background of Creep-Fatigue Interaction: Interpretation Based on Exploration of Crack Growth Process References Creep in Modern Materials Synonyms Definition Introduction Fundamentals of Designing Properties of Modern Materials Creep in Modern Metallic Materials Examples of Possibilities to Control Properties of Advanced Metallic Materials Consequences of a Material Creep During Operation: Example References Creep in Structures Synonyms Definition Introduction Large Creep Deformations of Rotating Bar Large Creep Deformations of Rotating Disk Large Creep Deformations of Axially Symmetric Thin-Walled Shells in Membrane State Large Creep Deformations of Axially Symmetric Thin Membranes References Creep in the Proximity of Absolute Zero Creep Plasticity Creep Rupture Creep-Fatigue Failures Creep-Fatigue Interaction Critical Loads Cryogenic Temperatures Crystal Plasticity Curved Rod Curved Shear Webs Cyber-Physical Systems D Damage Damage and Failure in Layered Composite Structures Synonyms Definition Introduction and Motivation Models for Intra-ply Damage and Failure Mechanisms in Composite Laminates Models for Delamination Models for Intra-ply Damage Continuum Damage Mechanics for Composites Damage Onset Stiffness Degradation and Damage Evolution for MDF Stiffness Degradation and Damage Evolution for FF Numerical Aspects Further Developments Extensions of UD-Ply Models Non-UD Topologies Combined Ply Damage and Delamination Cross-References References Damage and Failure of the Vascular Wall Synonyms Definition Introduction Histology of the Vessel Wall The Extracellular Matrix (ECM) Collagen and Its Organization Determine High-Strain Vessel Wall Properties Proteoglycans Interlink Collagen Fibrils Histomechanical Modeling of Vascular Tissue Modeling Irreversible Properties of Collagen Fibers Failure Represented by Cohesive Interface Models Traction-Separation Law (TSL) Defines Failure Properties Failure Propagation Constraint Along Finite Element Faces Discontinuities Embedded in the Finite Element Discontinuous Kinematics Variational Formulation Conclusions Cross-References References Damage Assessment Damage in Soft Biological Tissues Synonyms Definition Introduction Continuum Damage Mechanics Approach Classical Continuum Damage Mechanics at Large Strains Continuum Damage Mechanics Modeling of Soft Tissues Pseudo-Elastic Approach Mesh-Independent Numerical Calculations Integral Formulations Gradient Damage Relaxed Incremental Variational Formulation Cross-References References Damped Vibrations of One-Mass Systems Modelled by Fractional Operators Damping Features of Which Are Modelled by Fractional Operators Deactivation of Damage Effects Synonyms Definition and Literature Survey Projection Tensors of Discontinuous Damage Deactivation Effect of Continuous Damage Deactivation Uniaxial State of Stress and Scalar Damage Triaxial State of Stress and Isotropic Damage Triaxial State of Stress and Orthotropic Damage References Decline and Collapse of the Vessel Wall Defects Defects in Nanostructures Synonyms Definition Introduction Carbon Nanotubes and Their Derivatives Structural Defects Vacant Sites Stone-Wales Defect Chemical Doping Structurally Modified Carbon Nanotubes Summary and Concluding Remarks References Deformable Body Deformation Map Deformation Waves Del Operations Delamination Derivatives of Vectors and Tensors Synonyms Definition Derivatives of Vector Fields in Curvilinear Coordinates Curvilinear Coordinates The First Differential dr Derivatives of a Vector Function Derivative Operations for a Tensor Field Equation Recap Cross-References References Descartes, René Education Professional Career Scientific Achievements Cross-References Design for Reliability of Electronic Materials and Systems Definition Analytical Modeling in the Design-for-Reliability of Electronic Systems Bow-Free Assembly Thermally and Lattice Mismatch-Induced Stresses Could Shock Tests Adequately Mimic Drop Test Conditions? Is the Maximum Acceleration an Adequate Criterion of the Dynamic Strength of a Product? Could a Static Load Be More Damaging Than a Dynamic One? Closed-Form Solution for Highly Nonlinear Vibrations When Could a Short Impact Load Be Substituted with an Instantaneous Impulse? Larger Standoff Heights Could Relieve Stress in Solder Joint Interconnections Stretchable Electronics Incentive for Pre-stressing Accelerated Test Specimens Thinner and Longer Legs Relieve Stress in a Thermoelectric Module Conclusion References Deterioration and Breakdown of the Vessel Wall Determination of Effective Properties of Random Media Determinism Differential Equation Solver Differential Geometry Diffusive-Ballistic Transition Dimensional Reduction Dimensionless Numbers Direct an Inverse Problem in Electromagnetic Casting Direct Derivation of Plate Theories Synonyms Definition Introduction Equilibrium of Plates Contact Forces and Moments: Stress-Resultants and Stress-Couples Assigned Forces and Moments Equilibrium Equations The Thick Plate Model Stress–Strain and Strain–Displacement Relations Physical Meaning of the Lagrange Multipliers Boundary Conditions The Thin Plate Model Stress–Strain and Strain–Displacement Relations Physical Meaning of the Lagrange Multipliers Boundary Conditions Summary Cross-References References Direct Method of Calculus of Variations in Elasticity Definition Introduction The Method and Its Applications to Elasticity Outline of the Method Role in Relaxation Theory Direct Method as an Approximation Method A Model-Case Application in 1D Nonlinear Elasticity Cross-References References Directed Quantities Director Theory Discontinuity Surfaces in Elasto-Visco-Plastic Media Synonyms Definition Preliminary Remarks Problem Formulation Attenuation Equations for the Acceleration and Shock Waves Tresca Yield Condition Maximal Reduced Stress Yield Condition Numerical Examples An Irrotational Spherical Wave An Irrotational Plane Wave Conclusion Cross-References References Discontinuum-Based; Particle-Based, Distinct Entity-Based Discrete and Continuum Thermoelasticity Discrete and Continuum Thermomechanics Synonyms Definition Introduction Basic Designations and Assumptions One-Dimensional Crystal General Equations The First Thermal Approximation The Second Thermal Approximation Kinetic Temperature Quasi-One-Dimensional Crystal General Equations The First Thermal Approximation The Second Thermal Approximation Two- and Three-Dimensional Crystals General Equations The First Thermal Approximation Heat Transfer in One-Dimensional Crystal Nonlocal Temperature Heat Impact Heat Flux Summary Cross-References References Discrete Element and Particle Methods Synonyms Definition Introduction Molecular Dynamics Discontinuum Deformation Analysis (DDA) Discrete Element Method (DEM) The Combined Finite-Discrete Element Method (FDEM) Contact Detection Contact Interaction Fracture Fluid-Particle Interaction Toward Physics of Discontinua References Discrete Element Method; Distinct Element Method Discrete Element; Distinct Element, Discrete Particle, Distinct Entity, Solid Particle Discretization Methods Dislocation Core Dislocations Dislocations and Cracks in Generalized Continua Synonyms Definition Background Strain Gradient Elasticity Screw Dislocation Classical Solution Gradient Solution Edge Dislocation Classical Solution Gradient Solution Toward Singularity-Free Cracks Singularity-Free Dislocation Dynamics Conclusions Cross-References References Dislocations and Disclinations in Elastic Shells Dispersion Dispersion of Waves Dispersion Relations for the Coupled Hyperbolic Thermoelasticity Synonyms Definition Introduction A Statement of the Coupled Problem of Hyperbolic Thermoelasticity The Dispersion Relations of Coupled Thermoelasticity in the Form of an Exponential Function Decreasing with Time The Dispersion Relations of Coupled Thermoelasticity in the Form of an Exponential Function Decreasing with the Space Coordinate Cross-References References Dispersion Relations for the Thermoelasticity of the Lord–Shulmann Type Dissipation in Concrete Double Forces Double-Shearing Model Drilling Rotation Driving Force Drucker, Daniel Charles Education Professional Career Scientific Achievements Honors Other Activities Cross-References References Ductile Damage Ductile Metals Dyadics Dynamic Equations, Verification of Hyperbolicity via the Theory of Discontinuities Synonyms Definition Backgrounds and Some Historical Remarks Procedure for Defining the Correctness of Engineering Approaches Velocities of Transient Waves in Thin-Walled Beams of Open Section Due to the Korbut-Lazarev Theory and Its Generalizations Velocities of Transient Waves in Thin-Walled Beams of Open Section Due to the Vlasov Theory and Its Modifications Conclusion Cross-References References Dynamic Loading Response Dynamic Response Dynamic Simulation of the Behavior of Biomechanical Systems Dynamic Stiffness Method (DSM) Dynamic Variational Principles with Application for Contact Problems with Friction Synonyms Definition Introduction Classical Formulation of the Problem Variational Formulation of the Problem Without Contact Conditions Nonsmooth Functionals for Unilateral Contact Conditions with Friction Variational Formulation of the Problem with Contact and Friction Dual Variational Formulation and Uzawa\'s Optimization Algorithm Cross-References References Dynamical Contact Problems of Fracture Mechanics Synonyms Definition Introduction Statement of the Problem Interaction of the Finite I-mode Crack with P-wave Statement of the Problem Integral Equations and Fundamental Solutions Algorithm for the Problem Solution Numerical Results Cross-References References Dynamically and Thermally Exact Shell Theory Dynamics Dynamics and Thermodynamics of Fracture Mechanics Synonyms Definition Cross-References References Dynamics in Low Natural Frequency Range E Effect of Pressure on Material Properties of Polymers Effect of Pressure on Mechanical Properties of Polymers Synonyms Definition Introduction Effect of Pressure on Molecular Structure of Polymers Effect of Pressure on Polymer Phase Transitions and Its Volume Effect of Pressure on Mechanical Properties of Polymers Time-Pressure Superposition Modeling the Effect of Pressure on Polymers Measurements of the Effect of Pressure and Temperature on the Mechanical Time-Dependent Properties of Solid Polymers Cross-References References Effect of Temperature on Mechanical Properties of Polymers Definition Introduction Structure and Classification of Polymers Mechanical Properties of Polymers Theory of Linear Viscoelasticity Creep and Relaxation Process: Example of Responses to a Simple Step Function Input Creep Process Relaxation Process Material Functions (General Constitutive Description of Viscoelastic Materials) Effect of Temperature on Behavior of Polymers Effect of Temperature on Volume Effect of Temperature on Coefficient of Thermal Expansion Physical Aging Effect of Temperature on Material Functions Time–Temperature Superposition Closed Form Shifting (CFS) Algorithm Modeling Effect of Temperature Cross-References References Effective Modulus Effective Variables and Damage Effect Functions Synonyms Definition State Variables Microstructural Variables Hardening Variables Effective State Variables Damageable Material Mechanical Equivalence Hypotheses Damage Effect Functions General Multi-dissipative Material Total Energy Equivalence Hypothesis Extended to Other Dissipative Phenomena References Eigenstrain Elastic and Inelastic Stress Waves Synonyms Definition Continuum Mechanics (Necessary Facts) General Theory of Waves (Necessary Facts) Theory of Elasticity (Necessary Facts) Continuous Elastic Waves Discontinuous Elastic Waves References Elastic Failure of Composites Elastic Sensitivity Analysis and Structural Optimization for Discrete Parameter Variables Elastic Shells, Dislocations, and Disclinations Synonyms Definition Introduction Theory of Nonlinear Shells with Dislocations and Disclinations Disclinations in a Spherical Shell Buckling of a Plate with Distributed Dislocations Cross-References References Elastic Shells, Linear Shear-Deformable Theory Synonyms Definition Introduction Geometry of a Thick Shell Kinematics of a Thick Shell Variational Statement of Elastodynamics for a Thick Shell Construction of Two-Dimensional Shell Models Higher-Order Theory of Shear-Deformable Shells Extended Higher-Order Theory of Shear-Deformable Shells First-Order Shear Deformation Shell Theory Third-Order Shear Deformation Shell Theories Conclusion Cross-References References Elastic Shells, Material Symmetry Group Synonyms Definition Introduction Basic Shell Relations Change of Reference Placement Material Symmetry Group Solid Elastic Shells Physically Linear Elastic Shells Cross-References References Elastic Shells, Resultant Nonlinear Theory Synonyms Definition Introduction Notation Resultant Equilibrium Conditions Shell Kinematics Constitutive Equations Isotropic Elastic Shell Undergoing Small Strains Cross-References References Elastic Waves in Microstructured Solids Synonyms Definition Background One-Dimensional Dispersive Wave Equations Classical Wave Equation Strain Gradient Model Linear Version of the Boussinesq Equation Microstructure Model Accounting for Microinertia Refined Models Models with Higher-Order Time Derivatives Notes from History Cross-References References Elastic Waves of Stresses Elasticity Elastic-Plastic Fracture Mechanics Elastodynamics Electric Field and Strain Gradient Electrochemistry Electro-elastic and Magneto-elastic Systems Electromagnetic Shaping of Molten Metals Elements of Linear Spaces Elements of Tensor Spaces Emergent Property; Property Obtained as a Result of the Interaction Among Many Constitutive Parts of the System Energy Conserving Algorithms in Shell Dynamics Energy Methods Energy Methods for Granular Materials Energy-Based Methods Engesser, Friedrich Education Professional Career Scientific Achievements and Honors Cross-References Entropy Inequality Entropy Introduction Entropy Principle Entropy Principle Exploited by Lagrange Multipliers Synonyms Definition Introduction Thermodynamic Processes in Mixtures of Inviscid Fluids Objective Equations of Balance in Regular Points of the Mixture Equations of Balance on Singular Surfaces Stoichiometry and Interaction Force Constitutive Equations Thermodynamic Processes Entropy Principle Statement Lagrange Multipliers Consequences of the Entropy Principle Exploitation of the Entropy Inequality ʌ ε and Absolute Temperature ʌ ρα I and Chemical Potentials Summary of Results Equilibrium Properties Strategy of Thermodynamics Simple Mixtures Diffusion: Comparison with Linear Irreversible Thermodynamics The Laws of Fick and Fourier The Laws of Fick and Fourier in the Present Theory Symmetry Relations for Diffusion and Thermal Diffusion First and Second Sound References Entropy Production Epoxy Resin Equilibrium Forms Bifurcation of the Nonlinear NEMS/MEMS Synonyms Definition Introduction Branching of Equilibrium Forms of Ferromagnetic Membranes Branching of Equilibriums of a System of Nanolayers Conclusions References Equivalent Single Layer (ESL) Eringen, Ahmed Cemal Beginnings and Family Education Professional Career Academic Collaboration and Dissent Scientific Achievements and Awards Cross-References References Eshelby Problem Eshelby Tensor Estimating Optic Aberration Euclid Scientific Achievements Euler, Leonhard Education St. Petersburg: First Term Berlin Eyesight Deterioration St. Petersburg: Second Term and Death Achievements in Mathematics and Physics Mathematical Notation Analysis Graph Theory Solid Mechanics Fluid Mechanics Further Readings Cross-References References Excitation-Contraction Coupling Expansions Methods Near a Bifurcation Point Explicit Geometry Description Explicit Time Integration; Explicit Time Marching, Time Marching, Temporal Discretization Extended Finite Element Methods (XFEM) Synonyms Definition Introduction Extended Approximations The Level-Set Method General Formulation of the XFEM Variants of the XFEM Enrichments in Applications Implementation of the XFEM Numerical Integration Linear Dependencies and Ill-Conditioning Interface and Boundary Conditions Conclusions Cross-References References Eyes Optics F Failure Assessment Failure Assessment Diagram (FAD) Failure Criteria Fatigue FEM Fiber Failures Fiber-Reinforced Plastics (FRP) Fiber-Reinforced Polymers Fibers Finite Crack Finite Deformation Fracture Mechanics Finite Dimensional Lagrangian Systems Synonyms Definition Introduction Basic Terminology Least Action Principle Kinetic and Potential Energy Stable and Unstable Equilibrium Configurations Modal Analysis of Euler-Lagrange Equations Linearization of Lagrange Equations in a Neighborhood of a Stable Equilibrium Configuration Cross-References References Finite Element (FE) Method Finite Element Method (FEM) Finite Element Methods Synonyms Definition Introduction The Finite Element Method Meshfree Discretizations The Partition-of-Unity Approach Isogeometric Analysis as a New Paradigm Concluding Remarks References Finite Element Modelling Finite Elements Finite Elements for Plates and Shells Synonyms Definition Introduction Kirchhoff-Love Type Plate and Shell Models Kirchhoff Plate Model Governing Equations and Weak Form Finite Element Formulation Remarks on Modeling and Discretization Benchmarking Kirchhoff-Love Shell Model Kinematic Equations Finite Element Formulation Membrane Locking and Methods to Avoid It Benchmarking Reissner-Mindlin-Type Plate and Shell Models Mindlin Plate Model Governing Equations and Weak Form Finite Element Formulation Remarks on Modeling and Discretization Transverse Shear Locking and Methods to Avoid It Benchmarking Reissner-Mindlin Shell Model Kinematic Equations Finite Element Formulation Remarks on Modeling and Discretization Benchmarking Three-Dimensional and Solid Shell Finite Elements Cross-References References First-Order Shear Deformation Theory First-Ply Failure Flows in Nanochannels Synonyms Definition Nanochannels Flows: Occurrence and Application Nanochannel Fluid Flows Features Effect of Characteristic Length Scale Fluid-Wall Material Interaction Fluid Microrotation Electrical Phenomena Associated with the Double Electrical Layer Modeling Flows in Nanochannels Continuum Model Breakdown Micropolar Fluid Theory Molecular Modeling Characteristics of Fluid Flows in Nanochannels Inhomogeneous Fluid Structure Velocity Profiles and Boundary Slip Microrotation On the Validity of Navier-Stokes Equation for Nanochannels Flow Prediction Limit Applicability of Continuum Theory The Role of Nanochannel Size Cross-References References Fluctuation Theorem Foppl, August Otto Family and Education Professional Carrier Academic Pupils Scientific Achievements and Honors Cross-References References Foppl, Ludwig Family and Education Professional Career Academic Pupils Scientific Achievements and Honors Cross-References References Föppl-von Kármán Plate Theory Form Invariance Formal Asymptotic Expansions Formulation and Analysis of Plane Elasticity and Thermoelasticity Problems for Inhomogeneous Materials Fractal Fractal Continuum Mechanics Fractal Derivative Fractal Viscoelastic Models Synonyms Definition Introduction Viscoelastic Behaviors of Materials Mechanical Constitutive Relationship Models Methodology: Hausdorff Derivative and the Corresponding Statistical Interpretations Hausdorff Derivative Hausdorff Derivative and Fractal Metric The Hausdorff Derivative Diffusion Models and its Statistical Implications Statistical Interpretations Fractal Viscoelastic Models Fractal Dashpot and Viscoelastic Models Scaling Transformation Experimental Validation Comparison between Hausdorff and Fractional Viscoelastic Models Perspectives and Problems Physical Interpretation of Hausdorff Derivative Models Hausdorff Derivative Vs. Fractional Derivative Models Anomalous Relaxation Equation Anomalous Diffusion Equation Viscoelastic Models by Structural Derivative Open Issues Cross-References References Fractional Calculus in Biomechanics Synonyms Definition Introduction Model of Tooth Root in the PDL Geometrical Model Equations of Tooth Root Motions Translational Displacements of Tooth Root Effect of the Fractional Kernel Parameters Strains in PDL During Translational Displacement of Tooth Root Effect of Series Truncation Effect of Inertia Conclusion Cross-References References Fractional Calculus in Thermoelasticity Synonyms Definition Essentials of Fractional Calculus Time Fractional Operators Space Fractional Operators Balance Equations Equations of State Generalizations of the Fourier Law Example Conclusions References Fractional Derivative Models Fractional Derivative Standard Linear Solid Model Fractional Derivative Timoshenko Beams and Uflyand-Mindlin Plates, Impact Response Synonyms Definition Preliminary Remarks Impact Response of a Timoshenko-Type Beam Method of Solution Impact Response of a Viscoelastic Uflyand-Mindlin-Type Plate Analysis of the System\'s Critical State Conclusion Cross-References References Fractional Derivative Viscoelastic Medium Fractional Heat Conduction Fractional Operator Models of Viscoelasticity Synonyms Definition Backgrounds The Simplest Fractional Derivative Viscoelastic Models Fractional Derivative Kelvin-Voigt Model Fractional Derivative Maxwell Model Fractional Derivative Standard Linear Solid Model Application of the Fractional Derivative Viscoelastic Models for the Description of Dynamic Response of Current Materials Modelling the Young Operator E Using the Fractional Derivative Kelvin-Voigt Model Without Volumetric Relaxation Modelling the Shear Operator μ Using the Fractional Derivative Kelvin-Voigt Model Without Volumetric Relaxation Modelling the Shear Operator μ Using the Fractional Derivative Maxwell Model Modelling the Shear Operator μ Using the Fractional Derivative Standard Linear Solid Model Scott Blair Model for Shear Relaxation Models Considering Volumetric Relaxation Concluding Remarks Cross-References References Fractional-Analytical Methods in Mechanics of Delayed Fracture Fractionally Damped Medium Fractional-Order Operators in Fracture Mechanics Synonyms Definition Introduction Methods of Solving Boundary Value Problems of Linear Viscoelasticity for Bodies with Growing Cracks Some Approaches and Methods in the Theory of Linear Viscoelasticity for Bodies with Growing Cracks Method of Operator-Valued Continued Fractions (for Resolvent Operators) Mechanics of Slow Crack Growth in Viscoelastic Bodies Modelling Crack in Viscoelastic Media Basic Equations Describing Subcritical Crack Growth in Viscoelastic Bodies Solving of Kinetic Equations by Example Cross-References References Fracture Mechanics in Lightweight Design Synonyms Definition A Few Fundamental Aspects Monotonic, i.e., Static Loading Limitation of the Applicability of the LEFM The Two-Criteria Method: Failure Assessment Diagram (FAD) Crack Growth Under Cyclic Loading Cross-References References Fracture Mechanics of Composites Synonyms Definition Introduction Formulation of the Problem Solution of the Considered Problem Solution of the Integral Equations Conclusion Appendix References Fracture Mechanics of Elastic Composites Synonyms Definition Introduction One Crack in Modes I, II, and III of Classical Fracture Representations of Incremental Fields Asymptotical Representation of Complex Potentials: Crack Propagation Interaction of Two Unequal, Collinear Cracks in a Prestressed Elastic Composite in Mode I of Classical Fracture Conclusions References Fracture; Crack Frame Invariance Frameworks for Material Modeling Synonyms Definition Classification of Material Models Internal Kinematic Constraints Determinism in Case of Internal Constraints Elasticity Conservation of Energy Material Symmetries in Elasticity Well-Posedness of the Elasto-Static Boundary Value Problem Viscosity Eulerian and Lagrangian Description Volume Change and Shape Change Elastoplacticity The Multiplicative Decomposition Elastic Ranges Flow Rule The Plastic Strain Rate λ The Flow Direction N Isochoric Plasticity Maximum Plastic Dissipation Normality Summary Numerics of Elastoplasticity Cross-References References Friction Functional Derivative G Gauß, Johann Carl Friedrich Early Years and Education Scientific Achievements Honors Cross-References Geiringer, Hilda Early Years and Education Professional Career Scientific Achievements and Honors Cross-References References Generalized Contact Actions Synonyms Definition Introduction Generalized Contact Action Deduction of General Euler-Lagrange Equations Forces Concentrated on Lines and Points Explicit Expression for Higher-Order Contact Actions Further Considerations References Generalized Continua Generalized Corresponding Principle Generalized Finite Element Method (GFEM) Geometrical Nonlinearities Geometrically Exact Equations for Beams Synonyms Definition Introduction History of Geometrically Exact Equations History of Fully Intrinsic Formulation Intrinsic Formulation for Beams Equations of Motion Constitutive Equations Closing the Formulation Derivation of Intrinsic Kinematical Equations Conservation Laws for the Intrinsic Formulation Conclusion References Geometrically Nonlinear Equations Geometry of Curves and Surfaces Geometry of Motion Germain, Marie-Sophie Early Years and Education Scientific Achievements and Honors Cross-References Germain, Paul Family and Education Professional Career Scientific Achievements Academic Students Service to the Community Cross-References References Gol\'denveizer, Alexei L\'vovich Family and Education Professional Carrier Academic Pupils Scientific Achievements References Gradient Elasticity Green, George Early Years and Education Professional Carrier Scientific Achievements Honors Green-Naghdi Theory Grioli, Giuseppe Early Life and Education Professional Career Scientific Work Cross-References References Growth and Decay of Acceleration Waves Guided Waves Guyer-Krumhansl Equation H Hashing Algorithms; Mapping Algorithms Hausdorff Derivative Hayes, Michael Alphonsus Early Life and Education Professional Career Scientific Work Cross-References References Heart Tissue Activation Hencky, Heinrich Early Years and Education Professional Career Scientific Achievements Cross-References References Heterogeneity Heterogeneous Plate Heterogeneous Plate Theories Hierarchical Modeling of Functionally Graded Beams Synonyms Definition Functionally Graded Materials and Beam Structures Hierarchical Beam Modeling Numerical Results Conclusions Cross-References References Hierarchical Shell Models High Rate and Short Pulse Fracture Mechanics High Strain Rate Loading High Strain Rate Metal Plasticity Synonyms Definition Introduction Metal Plasticity Dislocations and Slip Remobilization Thermal Barriers Peierls-Nabarro Barrier Dislocation Interactions and the Taylor Model Path Dependence Impurities Long-Range Barriers Summary of Low Rate Response Behavior Above 102 s-1 Wave Dynamics Adiabatic Deformation Behavior Above 104 s-1 Phonon Drag Enhanced Work Hardening Shock Waves Deformation Twinning Failure Cross-References References High Strain Rate Plasticity Higher Gradient Continua Higher Gradient Theories and Their Foundations Synonyms Definition Introduction Interactions Modeled as Work Distributions The Foundation of Higher Gradient Theories à la D\'Alembert and à la Cauchy Some Remarks on Wave Propagation in Second Gradient Continua References Higher Order Continua Higher Order Theory Higher Order Theory of Functionally Graded Shells Synonyms Definition Introduction 3-D Formulation of the Problem Higher Order FG Shell Theory First Order FG Shell Theory High Order Theory of the Axisymmetric Shells of Revolution Higher Order Theory of the Axisymmetric Spherical Shells Shear Deformation Theory of the Axisymmetric Spherical Shells Kirchhoff-Love Theory of the Axisymmetric Spherical Shells Comparison of the Higher Order and Classical Shell Theories Results and Discussions Cross-References References Higher Order Theory of Micropolar Curved Rods Synonyms Definition Introduction 2-D Equations of Micropolar Elasticity 2-D Micropolar Elasticity in Coordinates Related to the Middle Line 2-D Formulation of the Problem First-Order Approximation Cross-References References Higher-Order Beam Theories Higher-Order Continua Higher-Order Rod Theory Higher-Order Shell Theories Higher-Order Theories Hill Muscle Modelling Hill, Rodney Education Professional Career Scientific Achievements and Honors Cross-References References Historical Introduction into the Theory of Plasticity History of Plasticity Synonyms Definition Introduction The Origins and Basics: Development Until 1945 Some Basic Relations Slip-Line Fields and the Origin of Metal Forming Constitutive Relations I: First Attempts Contributions of Material Science The Exodus and World War II The Expansion of These Basics Until the 1980s Constitutive Relations II: Small Deformation Theories Some Amendments Influence of Temperature and Rate Dependence Variational and Extremum Principles Limit Analysis Finite Element Method Step Toward Finite Deformations Initial Proposals and Concepts Different Objective Rates and Hypoelasticity The Logarithmic Rate and Related Properties Large Deformations One-Dimensional Considerations Basic Facts for a Deforming Continuous Body Eulerian Formulations with the Logarithmic Rate Lagrangean Formulations with Plastic Strain Formulations with Unstressed Configurations Director Triads and Isoclinic Configurations Cross-References References History of the Principle of Virtual Displacements History of the Principle of Virtual Powers History of the Principle of Virtual Velocities History of the Principle of Virtual Work Synonyms Definition Introduction Greek Science Renaissance and Early Modernity The Principle of Virtual Work After the Birth of Modern Calculus Cross-References References Homogenization Homogenization Methods and Generalized Continua in Linear Elasticity Synonyms Definition Introduction Overview of the Homogenization Method of Periodic Media Homogenization Method for Continuous Periodic Media On Some Subtleties of the Method Deriving Generalized Continua Through Homogenization Generalized Continua Derived by High-Order Homogenization Elastic Composites in Statics Asymptotic Formulation Resolution Process Expression of the Fields Physical Interpretation of the Hierarchical Description Link with Phenomenological Approaches of Generalized Continua Comments Elastic Composites in Dynamics Conventional Elasto-dynamics at the Leading Order Nonconventional Correctors at Higher Orders Physical Meaning: Wave Propagation and Rayleigh Scattering Generalized Continua at the Leading Order Reinforced Media Strain Gradient Model of Highly Contrasted Reinforced Media Multifaceted Nature of the Behavior Intrinsic Length, Elastic Energy, and Boundary Conditions Reticulated Media Preliminary Analysis Homogenization Method for Discrete Periodic Media An Archetypical Case: Orthogonal Grid of Rigidly Connected Beams Current Advances and Perspectives Inner Resonance Media Derivation of the Inner Resonance Model Inner Modal Amplitudes as Additional Kinematic Descriptors Comments on ``Generalized Inertia\'\' Conclusion Cross-References References Homogenization of Random Media Homogenization of Thin Periodic Plates Synonyms Definition Introduction The Homogenized Plate Problem Determination of the Homogenized Plate Elastic Stiffness Tensors Effect of Symmetries Symmetric Periodic Plate Material Symmetry of the Homogenized Plate Important Special Cases The Case of Symmetric Rectangular Parallelepipedic Unit Cell Cross-Reference References Hooke, Robert Early Years and Education Scientific and Professional Career Scientific Achievements in Mechanics Cross-References Hopf Bifurcation Huber, Maksymilian Tytus Family and Education Professional Career at Lwów Polytechnic Professorship at Warsaw Polytechnic Aviation: One of Huber\'s Passions The Hardship of World War II Professor Huber Activity in the Postwar Time Recognition After the Death Scientific Achievements A Great Personality References Human Brain Hurst Hybrid Approaches; Combined Finite-Discrete Approaches, Combined Continuum-Discontinuum Approaches Hygrothermoelastic Stress in Organic Matrix Composite Materials Synonyms Definition Introduction: Importance of Calculating Hygrothermoelastic Stress in OMC Materials and Short Review of the State of the Art Models for Calculating Hygrothermoelastic Stress in OMC Materials Hygrothermal Field Hygrothermoelastic Stress Field Hygrothermoelastic Stress in Laminated OMC Plates Calculation of Hygrothermoelastic Stress in OMC Plates Subjected to Complex Environmental Conditions Measurement of Hygrothermoelastic Stress in OMC Plates Hygrothermoelastic Stress in OMC Materials with Complex Microstructure Cross-References References Hyperbolic Elasticity Hyperbolic Theories Hyperstress Hypotheses of Mechanical Equivalence I Ilyushin, Alexei Antonovich Family and Education Professional Career Academic Pupils Scientific Achievements Cross-References References Impact Impact Dynamics Impact Energy Absorption Impact of Cellular Materials Synonyms Definition Introduction to Cellular Materials Experimental Work on Cellular Materials Quasi-static Compression: Features of the Stress-Strain Curve Impact Tests Blast Experiments Blast-Induced Compression Blast Loading of Sandwich Structures Models for Uniaxial Compression Due to Impact Loading Simple Shock Wave Propagation Approach to Dynamic Compaction Advanced Dynamic Compaction Models Stress-Wave Propagation Phenomena in Cellular Rod Primary Stress Wave Due to an Impact with Initial Velocity Reflected Wave from a Stationary Boundary Compaction of Layered Cellular Materials: Positive Density Gradient Cellular Material Topologies Which Do Not Tolerate the Shock Wave Propagation Cross-References References Impact of One Shell Against the Other Impact of Soft Projectiles Impact of Textile Structures Synonyms Definition Introduction Fibers for Bullet-Resistant Vests Ballistic Response of Textile Armor Transverse Ballistic Impact of Single Fibers Ballistic Response of Fabrics Fibers and Panels Ballistic Impact on Fabric Layers Vest Design Layup Stitching and Lamination References Impact on Ceramic Materials Synonyms Definition An Introduction to Ceramic Materials Ceramic Armour Development: The Early Days The Drive to Understand Mechanisms Ballistic Testing Ballistic Impact of Ceramics Hardness Impedance Geometry The “dwell-ish-ness” High Velocity Impact of Ceramics Plate-Impact Studies in Ceramics Improving Resilience to Impact and Penetration Epilogue: Future Challenges Cross-References References Impact on Reinforced Concrete Structures Synonyms Definition Synopsis Overview Definition of Hard and Soft Impact on Concrete Structures; Impact Loading Soft Impact Hard Impact Definition of “High-” and “Low”-Impact Velocities for RC Structures Punching Shear and Plugging Failure of Reinforced Concrete Structures Penetration and Perforation of Concrete Structures Subject to Hard Projectile Impact Strain-Rate Effects on Reinforced Concrete Materials Strain-Rate Effect on Concrete A Note of Caution on the Use of a DIF for Concrete in Numerical Modelling Strain-Rate Effect on Reinforcing Steel Bars Strain-Rate Effect on Bond Between Concrete and Reinforcing Steel Bars Numerical Modelling and Simulation of Reinforced Concrete Structures Subject to Impact Loading Concrete Material Modelling for Impact Loading Computational Framework References Impact Response Impact Response of Viscoelastic Timoshenko-Type Beams and Uflyand-Mindlin-Type Plates Described via Fractional Calculus Models Imperfections Infinite Strain Rate Infinite-Dimensional Constrained Mechanical Systems Inhomogeneous Waves Integrals and Derivatives of Non-integer Order Integral-Type Nonlocal Models Integration Formulas for Vectors and Tensors Synonyms Definition Basic Theorems Integral Formulas Involving the ∇Operator Further Reading Equation Recap Cross-References References Intelligent Structures Interaction Force; Force Arising from the Interference Between Two or More Particle Force Fields Interface Fracture Mechanics Interfacial Waves Internal State Variables Internal Stress in Organic Matrix Composites Intervertebral Disc Introduction to Creep Mechanics Synonyms Definition Introduction and Historical Remarks Creep of Metallic Materials Creep Stages Types of Creep Materials Science-Based Creep Equations General Creep Equation Dislocation Creep Nabarro-Herring Creep Coble Creep Creep Equations for Structural Analysis Basic Idea: Separation Ansatz Extension to Three-Dimensional Behavior Anisotropic Creep Cross-References References Invariance Under Rigid Body Motions Invariant Manifolds in Beam Dynamics: Free Vibrations and Nonlinear Normal Modes Synonyms Definition Overview Intrinsic Form of the Geometrically Exact Beam Equations Linear Normal Modes General Case Single Beams of Constant Cross Section Nonlinear Equations of Motion in Modal Coordinates Residualization and Quadratic Modes Nonlinear Normal Modes Cross-References References Isogeometric Analysis Isogeometric Finite Element Analysis Synonyms Definition Overview Integrating Computer-Aided Design and Finite Element Analysis Leveraging Higher-Order Smoothness of Splines in Analysis Objectives and Structure of This Article B-Splines and NURBS as a Basis for Finite Element Analysis Univariate B-Splines Knot Insertion, Degree Elevation, and k-Refinement Multivariate B-Splines Nonuniform Rational B-Splines Finite Element Implementation Through Spline Extraction Spline Extraction Spline Decomposition The Extraction Operator Using Extraction Operators in a Finite Element Code Adaptivity and Local Refinement in Isogeometric Analysis Refinability of B-Spline Functions by Subdivision Construction of Adaptive Hierarchical Approximation Spaces Adaptive Isogeometric Analysis Based on Hierarchical Refinement Implementation of Hierarchical Refinement Summary and Conclusions Cross-References References Isotropy Group of Hyperelastic Shells J J-integral Johnson, William Early Years and Education Professional Career Scientific Achievements and Honors Further Reading Cross-References References Junctions in Irregular Shell Structures Synonyms Definition Introduction Notation and Shell Equilibrium Conditions Principle of Virtual Displacements Mechanical Description of Junction The Stiff Junction The Hinge Junction The Deformable Junction The Elastic Junction Cross-References References K Kachanov, Lazar Markovich Biographical Data Contributions to Science Key Publications Books Selected Articles Abbreviations Karman, Theodore von Education Professional Career Scientific Achievements Honors and Legacy Cross-References References Kinematics Synonyms Definition The Deformation Map The Deformation Gradient Transformation Rules Jump Conditions on F/Local Kinematic Compatibility Polar Decomposition The Role of F When Changing Between the Material and the Spatial Description Local Balance of Momentum with the Material Nabla Operator The Velocity Gradient L Strain Tensors Small Strain Theory Large Strain Theory References Kinetic Collective Model for Phonons Kinetic Theory and Thermodynamics, Non-equilibrium Reacting Gas Flows Synonyms Definition Introduction General Procedure of Non-equilibrium Flow Modeling One-Temperature Approach Internal Energy Relaxation Conclusion Cross References References Kirchhoff Plate Kirchhoff, Gustav Robert Education Professional Carrier Scientific Achievements Further Readings References Kirchhoff–Love Model Kirchhoff-Love Nonlinear Shell Model Kirchhoff-Love Shell Kirchhoff-Love Shell Model Knowles, James Kenyon Family Professional Career Awards and Recognitions Scientific Results Teaching and Mentoring Cross-References Publications Koiter, Warner Tjardus Early Life Famous PhD Thesis Cross-References References Korn\'s Inequality Synonyms Definition A Version of Korn\'s Inequality The Friedrich Inequality Demonstrating (1) for Homogeneous Boundary Conditions Korn\'s Inequality for a Free Elastic Body Further Reading Cross-References References Kudo, Hideaki Education Professional Career Further Reading Cross-References References Kupradze, Victor Education Professional Career Scientific Achievements Honors References L L2 Estimate for Components of Linearly Elastic Displacement Lagrange Expansion (LE) Lagrange Multipliers Lagrange Multipliers in Infinite Dimensional Spaces, Examples of Application Synonyms Definition Introduction Incompressible Materials Application of Lagrange Multiplier Theorem Approach Through Penalization Second Gradient as Constrained Generalized Model Application of Lagrange Multiplier Theorem Approach Through Penalization Cross-References References Lagrange Multipliers in Infinite-Dimensional Systems, Methods of Synonyms Definition Introduction Lagrange Multipliers Theorem Finite-Dimensional Case Constraints in Banach Spaces Submersions Lagrange Multipliers Theorem in Banach Spaces Example References Lagrange, Joseph-Louis Education Scientific and Professional Career Scientific Achievements Cross-References Lagrangian Systems with a Finite Number of Degrees of Freedom Laminate Solution by BEM Laminate Theories Laminated Composite Elastic Shells, Nonlinear Theory Synonyms Definition Introduction Equations of Motion Parametrization of the Shell Displacement Fields 7-Parameter Formulation 12-Parameter Formulation Mechanical Strains Stresses and Constitutive Equations Equations of Motion Finite Element Model Numerical Examples A Simply Supported Isotropic Cylindrical Shell Subjected to Uniformly Distributed Radial Forces Post-buckling Response of Laminated Composite Plate A Pinched Composite Hyperboloidal Shell Transient Response of an Isotropic Plate Transient Response of a Laminated Composite Clamped Cylindrical Shell Under Internal Pressure Conclusions Cross-References References Laminated Composites Lamination Theories Large Deformation Analysis Large Deformations Large Strains; Large Deformations Layer-wise (LW) Lee, Erastus Henry Professional Career Scientific Contributions Awards and Recognitions Further Reeding Cross-References References Legendre Polynomial Legendres Polynomial Lehmann, Johannes Theodor Education Professional Career Scientific Achievements Cross-References References Levi-Civita, Tullio Introduction Biography Scientific Activity The Mathematical Genealogy of Levi-Civita An Appreciation of the Impact of Levi-Civita on Mathematical Physics and of His Human Personality References Levy, Maurice Education Professional Carrier Scientific Achievements Honors Cross-References Life Estimation Lifetime Light Scattering Limit Analysis of Plates Synonyms Definition Introduction Lower Bound Static Approach The Case of Thin Plates Upper Bound Kinematic Approach The Case of Thin Plates Usual Strength Criteria for Thin Plates von Mises Criterion Tresca Criterion Johansen Square Criterion Illustrative Examples Circular Plate Under Uniform Loading Lower Bound Static Approach Upper Bound Kinematic Approach Von Mises Criterion Square Plate Obeying Johansen\'s Criterion A First Lower Bound Approach A Second Lower Bound Approach Upper Bound Approach The Case of Clamped Supports Conclusions Cross-References References Linear and Nonlinear Vibrations, Fractional Oscillators Synonyms Definition Backgrounds Dynamics of Linear Fractionally Damped Oscillators Solution via the Laplace Transform Free Damped Vibrations of a Linear Oscillator with Fractional Derivative in the Inertial Force Description in a Medium with Fractional Viscosity Linear and Nonlinear Oscillators with Small Viscosity Free Damped Vibrations of a Linear Oscillator with Small Viscosity Damped Vibrations of Nonlinear Oscillators Vibrations of a Nonlinear Oscillator in a Medium with Viscosity of the Order of Vibrations of a Nonlinear Oscillator in a Medium with Viscosity of the Order of 2 Concluding Remarks Cross-References References Linear Bifurcation and Weakly Non-linear Analyses in Thin Elastic Bodies Linear Complexity; Scaling Linearly with the Number of Elements Linear Elastic Fracture Mechanics (LEFM) Lippmann, Horst Education Professional Career Scientific Achievements and Honors Cross-References References Local Action Logarithmic Strains Long Fiber Composite Long Rods Lord-Shulman Thermoelasticity Love, Augustus Edward Hough Early Years and Education Professional Carrier Scientific Achievements and Honors Cross-References References Low Thermal Energy Low-Dimensional Structures = 0D-, 1D-, 2D-Structures = Low-Dimensional Solids = Low-Dimensional Bodies Lower-Dimensional Models Low-Strength Projectile Impact Ludwik, Paul Education Professional Career Scientific Achievements Honors and Legacy Cross-References Bibliography Lurie, Anatoli Isakowitsch Education Professional Carrier Academic Pupils Scientific Achievements References M Macromechanics of Laminates Synonyms Definition Ply Mechanics Classical Lamination Theory (CLT) Special Laminates Inclusion of Thermal and Moisture Stresses Failure Modes and Mechanisms in Composite Laminates Failure Criteria for First-Ply Failure Cross-References References Mainardi, Gaspare Professional Carrier Scientific Achievements and Honors Cross-References Marguerre, Karl Early Years and Education Scientific Career Scientific Achievements Honors Activities as a Musician References Material Laws; Material Constitutive Laws Material Nodes; Gauss Point, Material Integration Points Material Nonlinearities Material with Continuous Space Variation of Properties Material-Independent Balances Synonyms Definition Overview General Balances and the Principle of Section Material (Lagrangian) and Spatial (Eulerian) Description Mass Balance Localization of the Mass Balance Global Balance of Momentum Localization of the Balance of Momentum Cauchy\'s Theorem Jump Conditions on T/Local Dynamic Compatibility Localization of the Balance of Momentum Global Balance of Moment of Momentum Localization of the Balance of Moment of Momentum Comments on the Balances of Mass and Moment of Momentum Limitations of the Classical Balances Cross-References References Mathematical Justifications of Plate Models Synonyms Definition Introduction The Three-Dimensional Model Principle of a Formal Asymptotic Approach Implementation The Limit Kirchhoff-Love Model: Results and Comments Some Extensions References Mathematical Modeling of the Cardiovascular System Mathematical Models for Coupled Problems Matrix Cracks Maugin, Gerard A Education Professional Carrier Scientific Achievements Honors Other Activities Cross-References References Maximum Friction Law Maximum Friction Surface Maxwell-Cattaneo Mean Weighted Residual Methods Mechanical Behavior of Optical Fibers and Interconnects: Application of Analytical Modelling Definition Thermostatic Compensation in Temperature-Sensitive Devices The Curvature of a Coated Fiber Could Be Different from the Curvature of Its Coating Threshold of the Added Transmission Losses What Could Possibly Be Done to Reduce the Bending Stress in an Optical Fiber? Analytical Modeling Could Be of Help in Optical Fiber Proof Tests Conclusion References Mechanical Field Theory Mechanical Properties Mechanics of 3D Fibre Reinforced Polymer Composites Synonyms Definition Introduction In-Plane Mechanical Properties of 3D Fibre Composites Elastic Properties of 3D Fibre Composites Failure Stress Properties of 3D Fibre Composites Fatigue Mechanics of 3D Fibre Composites Through-Thickness Mechanical Properties of 3D Fibre Composites Interlaminar Fracture Mechanics of 3D Fibre Composites Impact Damage Mechanics of 3D Fibre Composites Concluding Remarks References Mechanics of Discrete Media: Thermoelasticity Mechanics of Time-Dependent Material Behavior Mechanobiology Mechanochemistry Members of Vector Spaces MEMS Mesh; Domain Discretization Meshfree Method Meshless Discretization Methods Synonyms Definition Introduction Meshless Shape Functions Smoothed Particle Hydrodynamics Element-Free Galerkin Method Local Maximum Entropy Approximations Imposing Essential Boundary Conditions Advantageous Properties of Meshless Methods Applications of Meshless Methods to Material Failure Continuum-Based Methods to Material Failure Discrete Crack Methods Future Challenges Determination of Nodal Support Boundary Representation Computational Efficiency References Meshless Methods Mesomechanics Mesoscale Metal Forming Metal Forming Design Metal Forming Simulation Based on Advanced Mechanical Model Strongly Coupled with Ductile Damage Synonyms Definition Introduction Formulation of Thermodynamically Consistent Constitutive Equations Fully Coupled with Ductile Damage The Main Physical Phenomena Inherent to Large Plastic Strains with Ductile Damage On Some Mathematical Representation of the Ductile Damage Variables Formulation of Thermodynamically Consistent Constitutive Equations with Ductile Isotropic Damage On the Kinematics of Large Transformation Stress Measures and Stress Rates Choice of the State Variables Choice of State Potential and Derivation of State Relations Choice of the Dissipation Potentials and Derivation of the Evolution Equations Thermodynamical Admissibility Numerical Aspects in the Framework of the Finite Element Method Dynamic Explicit Scheme Global Resolution Schemes Strong and Weak Forms of the IBVP Space Discretization of the IBVP Time Discretization of the IBVP Global Resolution Scheme Local Integration Scheme: State Variables Computation Application to Various Metal Forming Processes Material Constants Studies Microcracks Closure Effects on Damage Distortional Parameters Effects on the Yield Surfaces Distortional Parameters Effect on Damage Evolution Application to Sheet Metal-Forming Processes Discussion Concluding Remarks and Perspectives References Methods of Plasticity Theory in Lightweight Design Synonyms Definition Principal Features of the Stress-Strain Diagram Multiaxial Stress States: Different Yield Conditions Residual Stresses: Autofrettage Melan\'s Shakedown Theorems Melan\'s Theorem I Melan\'s Theorem II Elastic-Plastic Buckling Cross-References References Methods of Weighted Residuals Methods to Determine the Mechanical Properties of Composite Materials with Continuous Fibers Synonyms Definition Overview Mechanical Models Historical Consideration Model That Use as Parameters the Phase Concentration Bounding Methods General Principles Upper and Lower Bounds for Elastic Constants Determining the Exact Displacement Solution for Some Particular Types of Networks Variational Methods Semiempirical Models: Halpin-Tsai\'s Formula Finite Element Method Conclusions Cross-References References Microcrack Opening/Closure Effect Microfluidic Devices Micromechanical Modeling of Viscoplasticity Micromechanics Synonyms Definition Aims and Basic Concepts of Micromechanics Mean-Field and Bounding Approaches Effective Field Schemes Effective Medium Schemes Bounding Methods Full-Field Approaches Periodic Homogenization Windowing Embedding Models Multiscale Models Cross-References References Micromechanics of Creep Synonyms Definition Overview Micromechanical Modeling of Viscoplastic Materials Basic Relations Micro-Macro Transition Schemes Creep of Polcyrystals Creep of Composites Micromechanical Modeling of Viscoelastic and Elastic Viscoplastic Materials Summary and Outlook Cross-References References Micromorphic Approach to Gradient Elasticity Micromorphic Approach to Materials with Internal Length Synonyms Definition Overview Kinematics General Micromorphic Media Balance Laws for Momentum and Generalized Moment of Momentum Thermomechanical Setting and Constitutive Laws Linearized Constitutive Laws The Micromorphic Approach to Gradient Elasticity, Viscoplasticity, and Damage Micromorphic Media and Heterogeneous Materials Application to Strain and Damage Localization Phenomena Cross-References References Micromorphic Media Micromorphic Theory Micropolar Curved Rods Micropolar Elasticity Micropolar Media/Continuum Microscale Microscopic Damage Mindlin, Raymond David Early Years and Education Professional Carrier Scientific Achievements and Honors/Awards Cross-References References Minimum Principles Minimum Work Path von Mises, Richard Early Years and Education Professional Career Scientific Achievements and Honors Cross-References References Further Reading Mixed Formulations in Continuum Mechanics Mixed Variational Methods: Considerations on Numerical Applications Synonyms Definition Introduction The Structural BV Problem Variational Formulation Finite Element Approximation Matrix Form of the Mixed Problem A Sample Problem Shear Locking Unlocking Procedures Assumed Strain Method Selective\\Reduced Integration Strain Gap Method Extension to Isogeometric Analysis Conclusions Cross-References References Model of Microstructure Modeling Corneal Refractive Surgery Modeling Damage Initiation and Propagation in Composite Structures Synonyms Definition Introduction Criteria for Isotropic Materials Maximum Normal Stress Criterion (Rankine) Maximum Shear Stress Criterion (Tresca) Maximum Distortional Energy Criterion Pressure-Dependent Criteria Criteria for Composite Materials Noninteracting Criteria Partially Interacting Criteria Interacting Failure Criteria Inter-ply Failure Failure Initiation Criteria for Crack Extension Virtual Crack Closure Technique Cohesive Elements Strain Rate Effects Progressive Damage Analysis Ply Discount Method Continuum Damage Mechanics Approach Discrete Damage Modeling Conclusion Cross-References References Modeling Framework for Computational Physiology Definition Introduction Characteristics of Physiological Systems and Biological Materials Imaging at Multiple Scales Units Associated with Physical Processes Physical Processes Defined with Bond Graphs Multiscale Modeling of the Heart Cell-Level Modeling The CellML Standard Modeling Organ Anatomy and Tissue Structure Modeling Whole Heart Function Discussion Cross-References References Modeling Individual Abdominal Aortic Aneurysms Modeling of Bone Adaption Processes Synonyms Definition Introduction Continuum Mechanics Framework The Overall Modeling Approach Model Generation from Radiological Data Model Parameter Identification Numerical Studies on Implant Effects Model Refinement Critical Discussion Summary and Outlook Cross-References References Modeling of Heart Muscles Synonyms Definition Introduction Morphology, Structure, and Physiological Function of the Heart Introductory Notions of Nonlinear Solid Mechanics Passive Deformations of the Heart Tissue Active Deformations of the Heart Tissue Active-Stress Contractile Response Myocardia Active-Strain Contractile Response Myocardia Modeling of the Heart Tissue Activation Cross-References References Modeling of the Human Cornea Synonyms Definition Structure and Microstructure Patient-Specific Models Geometry Material Model Unstressed Configuration Boundary Conditions Loading Examples of Simulations Conclusion Cross-References References Models of Viscoelasticity Based on Utilization of Fractional Operators Moisture Stresses Molecular Potentials Morphing Structures, Applications of Definition Abbreviations Introduction Aircraft and Rotorcraft Applications Space Applications Marine Applications Road Applications Energy Applications Buildings and Furniture Concluding Remarks References Morphological Models Synonyms Definition Introduction Probabilistic Characterization of Random Media Choquet Capacity Principle of Random Sets and of Random Function Modeling Random Sets An Archetype Model: The Boolean Random Set Multiscale Random Sets Cox Boolean Model Percolation of the Cox Boolean Model Long-Range Random Sets Random Functions Construction of Boolean Random Functions Random Tessellations and Polycrystals Boolean Random Functions and Fracture Statistics Weakest Link Model Crack Arrest Criterion Models of Random Damage Prediction of Effective Properties by Numerical Simulations Conclusion Cross-References References Moving Morphable Component (MMC) Moving Morphable Void (MMV) Murakami, Sumio Early Years and Education Professional Career Scientific Achievements and Honors References Muscle Fibers Muscle Mechanics Mushtari, Khamid Muzafarovich Family and Education Professional Carrier Scientific Achievements References Muskhelishvili, Nikoloz Education Scientific Achievements Professional Career Honors References Myocardium N Nabla Operations Nádai, Arpád Education Professional Career Scientific Achievements and Honors Cross-References References Naghdi, Paul Mansour Biographical Information, Education, and Academic Career Scientific Accomplishments Professional Service Honors and Awards Cross-References References Nanocomposites Nanoflows Nanomaterials Nanomechanics = Mechanics of Nanoscale Events and Processes Nanomechanics of Stress Relaxation in Composite Low-Dimensional Structures Synonyms Definition Misfit Stresses and Their Relaxation in Composite Low-Dimensional Nanostructures Composite 0D Nanostructures Core-Shell Nanoparticles Quantum Dots and Satellite Dislocation Loops Relative Stability of Composite Low-Dimensional Nanostructures to Stress Relaxation References Nanostructures Natural Modes of Vibration Negative Poissons Ratio in Plane Elasticity Synonyms Definition Introduction Classical Limits of Poisson\'s Ratio Effective Poisson\'s Ratio of 2D Materials with Microstructure Effects of Anisotropy Summary and Outlook Cross-References References Neighboring; Close to Each Other, Side by Side NEMS Neuber, Heinz August Paul Family and Education Professional Career Academic Pupils Scientific Achievements and Honors Cross-References References Neuromuscular Modelling Newton, Isaac Early Life and Education Academic Career Scientific Achievements Honors Other Activities Cross-References References Newton-Raphson Method Nikuradze, Johann (Ivane) Education Professional Career Scientific Achievements Node-Dependent Kinematic (NDK) Node-Dependent Kinematics, Multilayered Beam, Plate, and Shell Elements Synonyms Definition Background of Multilayered Structures Analysis Theory Carrera Unified Formulation (CUF) One-Dimensional Model Two-Dimensional Model Node-Dependent Kinematic Models A Two-Node Element with Node-Dependent Kinematics General Formulation of a Node-Dependent Kinematics Element Governing Equation Applications Convergence Analysis A Three-Layer Plate Analysis Cantilevered Beam with Piezoelectric Patches Conclusions Cross-References References Noll, Walter Education Professional Carrier Scientific Achievements Cross-References References Non-equilibrium Gas Dynamics Non-Gaussian Random Fields in Multiscale Mechanics of Heterogeneous Materials Synonyms Definition Introduction Notations Background in Linear Elasticity Construction of Information-Theoretic Models Preliminaries and Algebraic Decomposition Construction of the Probabilistic Representation Remarks About Model Identification Application in Computational Multiscale Analysis Conclusion Cross-References References Non-isotropic; Anisotropic Nonlinear Behavior Nonlinear Computations Synonyms Definition Introduction Weak Formulation Discretization and Numerical Integration Linearization Load and Displacement Control Path-Following Methods Concluding Remarks References Nonlinear Dynamic Problems Enabling Self-Similar Solutions Nonlinear Dynamics of Shells Nonlinear Evolution Equations Nonlinear Fracture Mechanics Synonyms Definition Linear Versus Nonlinear Fracture LEFM Geometric Nonlinearity Nonlinear Elasticity Plasticity Time-History Dependence The J-Integral Original Formulation Generalizations Cohesive Models Diffuse Interface Models Conclusions Cross-References References Nonlinear Models of Materials Failure Nonlinear Rays and Fronts Dynamics in Stochastically Inhomogeneous Media and in Media with Deterministic Structure Synonyms Definition Preliminary Remarks Rays\' Propagation in Stochastically Inhomogeneous Media Deterministic Chaos of Ray Trajectories and Fronts in Inhomogeneous Media with Deterministic Structure Cross-References References Nonlinear Response Nonlinear Solution Procedures Nonlinear Vibrations of Fractionally Damped Suspension Bridges Nonlinear Wave Equations Nonlinear Waves in Continuous Media Synonyms Definition Derivation of Governing Equations Nonlinear Effects Historical Notes References Nonlocal Continuum Nonlocal Damage Mechanics Nonlocal Elasticity Nonlocal Plasticity Nonlocal Theories Synonyms Definition Background Nonlocal Elasticity Nonlocal Plasticity Nonlocal Damage Mechanics Thick Level Set Approach Peridynamic Models Cross-References References Nonlocal Theory of a Thermoelastic Beams and Its Application to the MEMS/NEMS Analysis and Simulations Synonyms Definition Introduction 2-D Formulation 1-D Formulation First Order Approximation Thermoelastic Analysis of Microbeam Cross-References References Nonlocal Thermoelasticity Nonlocal Transport Equations for Small Systems and Fast Processes Synonyms Definition Introduction Some Physical Consequences of the Nonlocal Transport Equation Relaxation Effects and Thermal Wave Propagation Phonon Hydrodynamics and Heat Transport in Nanowires Heat Transport in Superfluid Liquid Helium Microscopic Interpretation of the Nonlocal Terms: Phonon Normal Collisions Thermodynamic Consistency of Nonlocal Transport Equation Concluding Remarks References Nonlocality Non-singular Dislocation Continuum Theory Nonsmooth Optimization Algorithm Normal and Shear Loadings Normal Form Novozhilov, Valentin Family and Education Professional Carrier Academic Pupils Scientific Achievements Cross-References References Nth Order Stresses Numerical Approximation Method Numerical Methods Numerical Methods and Modeling in Impulsive Dynamics Synonyms Definition Introduction Constitutive Modeling of Materials Under Impulsive Loading Models Which Modify the Yield Surface Viscous Constitutive Models Viscoelastic Models Viscoplastic Models Dynamic Damage Models Computational Methods for Structures Under Impulsive Loading Finite Element Methods Meshless Methods Peridynamics Discrete Element Methods Cross-References References Numerical Methods for the Fractional Differential Equations of Viscoelasticity Synonyms Definition Background Basic Definitions and Properties Numerical Methods for Fractional Ordinary Differential Equations Grünwald-Letnikov Methods Fractional Backward Differentiation Formulas Adams Methods Collocation Methods Piecewise Polynomial Collocation Nonpolynomial Collocation Spectral Methods Numerical Methods for Time-Fractional Partial Differential Equations Cross-References References Numerical Solution Algorithm Numerical Treatment of Stability Problems in Lightweight Design Synonyms Definition Introduction Nonlinear Finite Element Equations and Stiffness Matrices Detecting Critical Points in Conservative Systems Predicting Critical Points in Conservative Systems Detecting Critical Points in Nonconservative Systems Traversing Critical Points Example Summary Cross-References References O Objectivity Observer Invariance On Microfluidics Devices for Clinical Biosensor Synonyms Definition Introduction Materials for Fabrication of Microfluidic Channels: Polydimethylsiloxane Microfluidic Channels Fabrication Technique: UV Lithography Types of Microfluidic Systems Applications Biosensors Electrochemical Biosensors Amperometric Biosensors Potentiometric Biosensors Conductometric biosensors Thermometric biosensors Optical biosensors Acoustic biosensors Classification by Bio-receptor Integration of Microfluidic into Clinical Biosensors References On Treatment of Finite Rotations in FEM Analyses of Irregular Shell Structures Synonyms Definition Introduction Sixth Degree of Freedom Problem Finite Rotations Conclusions Cross-References References One-Dimensional Continua Oniashvili, Otar Education and Scientific Achievements Political Activities Honors References Open Cross Section Optimization of Structures Orowan, Egon Education Professional Carrier Scientific Achievements and Honors Further Reading Cross-References References P Palmov, Vladimir Alexandrovich Education Professional Carrier Scientific Achievements Honors Cross-References References Papkovich, Petr Fedorovich Education Professional Carrier Scientific Achievements Honors Cross-References References Parametric Sensitivity Analysis and Optimization of Elastic Problems Synonyms Definition Introduction Linear Elasticity Problem Sensitivity Analysis Discrete Formulation Direct Method Adjoint Method Continuous Formulation Direct Method Adjoint Method Plane Stress State Functionals of Structural Performance and Their Gradients Mass Functional Stress Functional Displacement Functional Conrod Thickness Optimization Conclusion Cross-References References Partition of Unity Method (PUM) Partition of Unity Property Passive Deformation Path-Following Techniques Patient-Specific Eye Models Patient-Specific Simulation of Abdominal Aortic Aneurysms Synonyms Definition Introduction Biomechanical Modeling Organ-Level Model Constitutive Modeling of AAA Tissues Models Based on Hyperelasticity Models Based on the General Theory of Fibrous Connective Tissue Models of Irreversible Deformation Wall Strength Modeling AAA Wall Adaptation Models Clinical Validation Summary and Conclusion Cross-References References PEM Vibration Damping Penalty Function Method; Penalty Method Penetration and Perforation Mechanics Penetration Mechanics of Rigid Projectiles in Metallic Targets Synonyms Definition Overview The Deceleration of a Rigid Projectile Analyzing Experimental Results Numerical Simulations A Simple Penetration Model A Numerically Based Model for Rt Impact at High Velocities: The Cavitation Phenomenon A Numerically Based Model for the Entrance Phase Effect Cross-References References Perfectly Plastic Material Personnel Armor Petalling Phenomenological Model of Orthotropic Damage in Polymer Matrix Composites Synonyms Definition Introduction Materials and Methods Results and Discussion Mathematical Problem Formulation Description of the Orthotropic PCM Response in 2-D Case Illustrative Example Conclusions References Phonon Hydrodynamics Piezoelectric Piola, Gabrio Introduction Piola\'s Life and Contribution to Mathematics and Mechanics Piola\'s Life Piola\'s Scientific Contributions Short Assessment of Piola\'s Role and Originality in Mechanical Sciences References Pipkin, Allen Comporte Biographical Information, Education, and Academic Career Scientific Accomplishments Professional Service References Planar Ideal Plastic Flows Synonyms Definition Introduction Basic Equations Existence of Ideal Flows Stationary Flows Nonstationary Flows Mapping Between the Principal Lines and Cartesian Coordinate Systems Applications Conclusions References Planar Problems in Rigid-Plasticity for Granular Materials and Soils Synonyms Definition Introduction Mathematical Formulation Classical Continuum Cosserat Continuum Models for a Classical Continuum Balance Laws Coulomb and Other Yield Criteria Flow Rules and Kinematic Equations Plastic Potential Model Double Shearing Model Linear Ill-Posedness Double-Slip and Double-Spin Model Cosserat Models Balance Law Yield Criteria Flow Rules and Kinematic Equations Problems and Applications Conclusions Cross-References References Plane Stress Description of Plastic Anisotropy Plane Surfaces of Strong and Weak Discontinuities in a Thin Elastic Anisotropic Layer Plane Thermoelasticity of Inhomogeneous Solids Synonyms Definition Introduction Formulation and Basic Assumptions General Three-Dimensional Formulation of Elasticity and Thermoelasticity Problems Plane Strain Plane Stress Unified Formulation of Plane Problems in the Cartesian and Polar Coordinate Systems Material Inhomogeneity Governing Equations of Plane Elasticity and Thermoelasticity Problems for Inhomogeneous Materials Compatibility Equation in Terms of Stresses Lamé Equations for Inhomogeneous Material Dominant Analytical Methods Cross-References References Plane Transient Waves in Anisotropic Layer, Ray Expansion Approach Synonyms Definition Introduction The Ray Method Numerical Example Conclusion Cross-References References Plane-Strain and Plane-Stress Problems of the Elasticity and Thermoelasticity Theories for Inhomogeneous Solids Plastic Buckling Plasticity Plasticity and Damage Plasticity Theory at Small Scales Plate Model or Theory Plate Theory Plate Theory After von Karman Synonyms Definition Introduction Föppl-von Kármán Plate Theory Assumptions Strain-Displacement Equations Equilibrium Equations Constitutive Equations Equations in Displacements Governing Equations Boundary Conditions Equations in Mixed Form: Application of Airy\'s Functions Governing Equations Boundary Conditions Special Cases Decoupled In-Plane and Plate Equations (First-Order Theory) Coupled In-Plane and Plate Equations (Second-Order Theory) In-Plane Equations for Large Deflections Buckling Equation Some Comments Further Readings Cross-References References Plate Theory of Second Order Plates Plugging PMC Composite Pobedrya, Boris Efimovich Family and Education Professional Career Academic Pupils Scientific Achievements Cross-References References Poisson, Simeon Denis Education Scientific Carrier Scientific Achievements Cross-References Polanyi, Mihaly (Michael) Education Professional Career Scientific Achievements and Honors Further Reading Cross-References References Polarization Tensor Porous Materials Post-Buckling Post-Buckling Behavior Prager, William Early Years and Education Professional Career Scientific Achievements and Honors Cross-References References Prandtl, Ludwig Early Years and Education Professional Career Scientific Achievements and Honors Cross-References References Further Reading Predicting the Risk of Aortic Rupture Principal Lines Coordinate System Principles of Material Modeling Synonyms Definition Principles of Material Modeling The Principle of Determinism Limitations The Principle of Local Action Limitations Principle of a Finite-Dimensional State Space Limitations The Principle of Invariance Under Rigid Body Motions Limitations Frame Invariance, Observer Invariance, Form Invariance, and Objectivity Balances Constitutive Laws Remarks Energetic Considerations: The Stress Power and Pullbacks of the Cauchy Stress Tensor Remarks Cross-References References Probabilistic Modeling Propagation of Discontinuity Surfaces of Strong and Weak Discontinuity in Elasto-Visco-Plastic Media Propagation of Thermal Elastic Waves in a Medium with Heat-Flux Relaxation Constant Proper Rotation Group Pseudo-Continuum Cosserat Pure Saint-Venant Torsion Pure Torsion of Thin-Walled Cross Sections Synonyms Definition Pure Torsion of Prismatic Rods Thin-Walled Profiles with Open Cross Section Thin-Walled Profiles with Closed Cross Section Cross-References References Q Quasi-3D Vibration Analysis of Laminated Composite Shells Synonyms Definition Introduction Laminated Composite Shell Geometry Hierarchical Shell Models and Ritz Method Numerical Results and Discussion Conclusions Cross-References References R Rabotnov, Yuri Nikolaevich Family and Education Professional Career Academic Pupils Scientific Achievements Honors Cross-References References Radial Basis Functions (RBFs) Random Random Fields Random Fields with Fractal and Hurst Effects in Mechanics Synonyms Definition Why Random Fields with Fractal and Hurst Effects? Scalar Random Fields with Fractal and Hurst Effects Covariance Functions, Fractal Dimension, and Hurst Effect Parametric Classes of Covariance Functions Responses of Dynamical Systems to Fractal-Hurst Forcings Elastic Rods, Shear Beams, and Euler-Bernoulli Beams with Random Field Properties Under Random Field Forcings Waves on Random Mass Density Fields with Fractal and Hurst Effects Lamb\'s Problem Acceleration Waves Cross-References References Random Media Random Set Rate-Dependent Plasticity Rate-Sensitive Plasticity Rational Thermomechanics of Shells Ray Expansion Theory Synonyms Definition Preliminary Remarks Basic Insights into the Differential Geometry of Moving Surfaces Conditions of Compatibility Conditions of Compatibility in Arbitrary Coordinates The Case When the Spatial Coordinates Coincide with the Ray Coordinates Conditions of Compatibility for the Physical Components Conditions of Compatibility for Rods and Beams Ray Series and Their Application for Solving Boundary-Value Dynamic Problems Concluding Remarks Cross-References References Ray Expansions in Dynamic Contact Problems Synonyms Definition Preliminary Remarks Dynamic Contact Interaction of an Infinite Cylindrical Shell with an Unbounded Elastic Medium Solution for the Space with a Cylindrical Cavity Solution for a Circular Cylindrical Shell in the Case of Welded Contact with a Surrounding Medium Solution for a Circular Cylindrical Shell in the Case of Smooth Contact with a Surrounding Medium Solutions of Simplified Equations for Circular Cylindrical Shells Numerical Investigations Conclusion Cross-References References Ray Expansions in Impact Interaction Problems Synonyms Definition Preliminary Remarks Impact Response of a Spherical Shell of the Timoshenko Type Geometry of the Wave Surface The Main Kinematic and Dynamic Characteristics of the Wave Surface Governing Equations Normal Impact of an Elastic Sphere Upon an Elastic Spherical Shell Normal Impact of an Elastic Long Hemisphere-Nose Bar Against an Elastic Spherical Shell Conclusion Cross-References References Ray Series Application for the Description of Transient Wave Propagation Ray Series Method for the Analysis of Boundary-Value Dynamic Problems of Thin Bodies Ray Series Method for the Analysis of Dynamic Problems of Contact Interaction Ray Series Method for the Analysis of Dynamic Problems of Impact Interaction Rays Propagation in Inhomogeneous Media Synonyms Definition Introduction Expansion of Rays and Fronts in Inhomogeneous Media Conclusion Cross-References References Rectangular Plate Rectangular Shear Webs Reduced Equation of Energy Balance Reduced Forms Regularization Reinforced Concrete Structures Subjected to Impact Loads Reissner or Hencky or Reissner-Mindlin Thick Plate Theory Reissner, Eric Early Years and Education Professional Carrier Scientific Achievements and Honors Cross-References References Reissner–Mindlin Model Relation Between Eshelbyan Mechanics and Topological Derivative Concept Synonyms Definition Introduction The Topological Derivative Concept Problem Statement: Elasticity System into Three Spatial Dimensions Topological Sensitivity Analysis Existence of the Topological Derivative Topological Derivative Evaluation Direct Approach Alternative Approach A Numerical Example Conclusion References Relaxation Times Residual Stress in Composites Residual Stresses Resistive Stress Resonance Modes in Beams Reuss, Andras Education Professional Career Scientific Achievements and Honors Cross-References References Rheological Activity of Materials Rheology Rigid Penetrations Rigid Perfectly Plastic Model Rigid Plastic Material Rigid Plastic Models Ritz–Galerkin Methods Rod Theory Rods Rotordynamics Rupture Rychlewski, Jan Family and Education Professional Carrier Academic Pupils Scientific Achievements Cross-References References S de Saint-Venant, Adhémar-Jean-Claude Barré Biographical Information, Education, and Career Scientific Accomplishments Honors and Awards Cross-References References SAW (Surface Acoustic Waves) Scaling Function in Mechanics of Random Materials Synonyms Definition Introduction Notion of a Scaling Function Effect of Phase Contrast, k, on Scaling Function Effect of Volume Fraction, vf, on Scaling Function Effect of an Anisotropy Measure, A, on Scaling Function Effect of Mesoscale, δ, on Scaling Function Application Problems Conductivity in 2D Two-Phase Microstructures Conductivity in 3D Polycrystals Elasticity in 2D Two-Phase Checkerboards Elasticity in 3D Polycrystals Viscoelasticity in 2D Two-Phase Checkerboards Conclusion References Search Algorithms; Contact Search Algorithms Second Gradient Continua Second Law of Thermodynamics Second Sound Self-Similarity Problems in Nonlinear Elastic Media Synonyms Definition Introduction The System of Model Relationships Plane One-Dimensional Shock Waves in an Incompressible Medium Shock Wave Velocities The Problem of the Shear of the Half-Pace Boundary Plane Self-Similar Problems Problem Formulation The Solution of the Boundary Problem Cross-References References Sensitivity Sensitivity Analysis Sensitivity Analysis in Structural and Multidisciplinary Problems Synonyms Definition Introduction Variational Sensitivity Analysis Direct Method for Sensitivity Analysis Adjoint System Method for Sensitivity Analysis Discrete Sensitivity Analysis Direct Method for Discrete Sensitivity Analysis Adjoint Method for Discrete Sensitivity Analysis Nonlinear Sensitivity Analysis Example: A Vibrating Beam with an Elastic Foundation Second-Order Sensitivity Analysis Concluding Remarks Cross-References References Series Expansion Serrations Seven- and Twelve-Parameter Shell Finite Element Models SH Surface Waves in Solids with Surface Stresses Shakedown and Plastic Collapse in Plane Stress Problems Synonyms Definition Limited Kinematic Hardening Plasticity Theory Maximal Dissipation Hypothesis Strictly Stable Hardening Hypothesis (Pham 2017) Shakedown Theorems and Plastic Collapse Modes Examples of Application Initial Yield Stress Cross-References References Shape Optimization for Direct and Inverse Problems in Electromagnetic Casting Synonyms Definition Introduction The Model Problem The Direct Problem The Inverse Problem Shape Derivatives Numerical Method Shape Discretization Projection of the Equilibrium Equation First Inverse Problem Formulation Second Inverse Problem Formulation Numerical Results The Direct Problem, Three-Dimensional Case Numerical Results in the Case of the Two Dimensional Inverse Problem Conclusion Cross-References References Shape Sensitivity Analysis Sharp Interface Shear Center Shear Deformation Shell Theories Shear Deformation Theories Shear Web Assemblies Shear Web Theory Synonyms Definition Introduction Plane Rectangular Shear Webs Curved Shear Webs Combined Shear Web Assemblies From Shear Web Theory to Tension Field Theory Cross-References References Shear Webs Shell Branchings and Intersections Shell Thermomechanics, Resultant Nonlinear Theory Synonyms Definition Introduction Basic Principles Modified Resultant Energy Balance Modified Resultant Entropy Inequality Remarks on Constitutive Equations Cross-References References Shells Shock Shock Response Shock Structures Shock Waves Via Ray Expansions Synonyms Definition Introduction Plane Shock Waves in an Incompressible Elastic Medium Plane Shock Waves in a Compressible Elastic Medium Ray Method in the Dynamics of Shock Waves Ray Method in the Case of One-Dimensional Cylindrical Waves Conclusion Cross-References References Siebel, Erich Education Professional Career Scientific Achievements and Honors Cross-References References Simple Mixtures Simulation Singular Solutions in Plane Strain Plasticity Synonyms Definition Introduction Coordinate Systems Fundamental Equations of Continuum Mechanics Constitutive Equations Pressure-Independent Plasticity Pressure-Dependent Plasticity Maximum Friction Law Singularity in Solutions Near a Maximum Friction Surface Rigid Perfectly Plastic Solids Viscoplastic Solids Double-Shearing Model Conclusions Cross-References References Singularity Six-Field Theory of Shells Six-Parameter Shell Theory Sixth Degrees of Freedom Size Dependence at the Nanoscale Size Effect in Nanomaterials Synonyms Definition Experimental Observation Scaling Law Concluding Remarks Cross-References References Size Effect Phenomenon Size-Dependent Plasticity Theory Skeletal Muscle Modelling Synonyms Definition Background Kinematics and Governing Equations Geometry Constitutive Modelling Additive Split of the Stress Tensor Multiplicative Split Software Tools Applications Cross-References References Slip Avalanches Slips Small-Strain Thin Shell Theory Smart Structures Snap-Through Buckling Sneddon, Ian Naismith Early Life and Education Professional Life Scientific Contributions Family and the Arts Cross-References References SO(3) Soft Body Impacts Soft Impact Synonyms Definition Introduction Bird Strike Bird Strike Experimental Tests Numerical Modelling of Bird Strike Ice Impact Ice Impact Experimental Tests Numerical Modelling of Ice Impact Tire Fragment Impact Tire Impact Experimental Tests Numerical Modelling of Tire Impact Other Soft Impacts Cross-References References Softening Softening and Fracture of the Arterial Wall Solid Particles; Discrete Elements, Distinct Elements Solitons Solution Techniques Spline Finite Element Methods Stability Stability Behavior of Composite Plates and Shells Stability Limit Stability of Structures State Space Formulation Static and Dynamic Bifurcations Synonyms Definition Introduction Lyapunov Stability Stability of Equilibria of Conservative Systems Center Manifolds for Equilibria Example: Calculation of the Leading Center Manifold Expansion for a Hopf Bifurcation Center Manifolds for Perturbed Parameter Values Normal Form Reduction of the Bifurcation Equations Normal Form Calculation for the Hopf Bifurcation Unfolding of the Linear System Using Normal Forms Short Discussion of the Hopf Bifurcation Cross-References References Stiffness and Strength of Anisotropic Heterogeneous Slender Structures Stochastic Continuum Stochastics Stokes, George Gabriel Early Years and Education Professional Career and Scientific Achievements Honors Cross-References Strain Gradient Plasticity Synonyms Definition Background Notation Theory Kinematics A Quite General Higher-Order Theoretical Framework Particularization to a Nye\'s Tensor-Based SGP and Introduction of Energetic and Dissipative HO Contributions Helmholtz Free Energy Density and Dissipation Minimum Principles Example of Application: The Torsion of Thin Metal Wires Open Problems Predictions Under Nonproportional Loading in the Presence of HO Dissipative Contributions Cyclic Behavior Utilizing a Defect Energy that Predicts Strengthening Cross-References References Strain Gradient Theory Strain Rate Intensity Factor Strains Stress Stress Constraint Stress Intensity Factor Stress Power Stress Relaxation = Stress Release = Total or Partial Restoration of Unstressed State Stress Shielding Stress Tensor Stress Tensor in Compressible Flows Stress-Constrained Topology Optimization for Lattice Materials Synonyms Definition Introduction Methodology Mechanical Properties of Lattice Materials RVE Elastic Properties Topology Optimization of Lattice Materials with Stress Constraints Problem Formulation Filtering of Design Variables Stress Constraints Sensitivity Analysis Numerical Examples and Discussion Conclusions Appendix: Sensitivity Calculation of the Stiffness Tensor of the Lattice References Stress-State Dependence Structural Assessment of the Aneurysmatic Aorta Structural Control Structural Materials Structural Safety Structure-Born Noise Reduction Structured Materials Structure-Preserving Methods Super-Quadrics; Shapes Resembling Ellipsoids or Other Quadrics Supraphysiological Loading Surface Energy and Its Effects on Nanomaterials Synonyms Definition Introduction Surface Free Energy and Surface Stress Effective Modulus of a Particle Thin Films Thin Wire of Square Cross Section Spherical Particles Numerical Examples and Discussions Summary Cross-References Appendix A: Coordinate Transformation Bulk and Surface Elasticity Tensors Special Cases Films Wires Spherical Particles References Surface Geometry, Elements Synonyms Definition Introduction Geometry of a Surface Covariant Differentiation Surface Curves Surface Divergence Theorem Cross-References References Surface Tension Surface Wave Propagation in 3D Medium Synonyms Definition Preliminary Remarks Problem Formulation Horizontally Polarized Surface Wave Vertically Polarized Surface Wave Numerical Example Closing Remarks References Surface Waves Synonyms Definition General Remarks Separation of the Equation of Motion for Surface Waves Surface Waves with SH-Polarization (Love Waves) Homogeneous Half-Space Love Waves in a Model ``Layer Over Half-Space\'\' (LOH) Rayleigh Waves in a Homogeneous Half-Space Stoneley and Scholte Waves at a Solid-Solid and Solid-Fluid Interface, Resp. Surface Waves, Eigenvalue Problems, and Some Extensions Historical Remarks and Applications Cross-References References Surface/Interfacial Energy Theory of Solids Definition Deformation and Kinematics of Two-Dimensional Curved Surface Strain Measures of the Interface Constitutive Relations of Elastic Interface Residual Stress Field Caused by the Interfacial Energy Equilibrium Equation of the Interface The Effect of Curvature-Dependence of Interfacial Energy Stress and Energy of Isotropic Interface Under Small Deformation Condition Summary Cross-References References Surfaces of Strong Discontinuity in One-Dimensional Nonlinearly Elastic Media Symmetric Hyperbolicity System of Symmetric Hyperbolic Equations, Extended Thermodynamics of Gases Synonyms Definition Introduction Formal Structure of Extended Thermodynamics Field Equations, Equilibrium, and Entropy Principle Exploitation of the Entropy Principle: Symmetric Hyperbolic Systems Wave Speeds: Growth and Decay of Acceleration Waves Extended Thermodynamics of Moments Reminder of the Kinetic Theory of Gases Analogies Between Extended Thermodynamics and Kinetic Theory The Distribution Function and the Constitutive Relations for a Gas Extended Thermodynamics of 20 Moments: Subsystems Results Characteristic Speeds Shock Waves Light Scattering Stationary Heat Conduction Between Coaxial Cylinders References T Taylor Expansion (TE) Taylor, Geoffrey Ingram Early Life and Education Professional Career Scientific Achievements Honors Cross-References Temperature and Entropy Introduction Temporal Discretization Methods; Temporal Integration Methods; Time Marching Methods Tension Field Theory Tensor Integral Theorems Tensor Notation Tensor Random Fields in Continuum Mechanics Synonyms Definition Why Tensor Random Fields in Continuum Mechanics? Representations of Rank 1 and Rank 2 TRFs Rank 1 TRF Rank 2 TRF Spectral Expansions of Homogeneous and Isotropic TRFs The Spectral Expansion of the Elasticity Random Field TRFs Dependent Fields Cross-References References Tensors Synonyms Definition Second-Order Tensors Dyads Components of a Second-Order Tensor Dot Product of a Second-Order Tensor by a Vector The Unit or Metric Tensor Dot Product of Tensors Trace of a Second-Order Tensor Determinant of a Second-Order Tensor Tensor Transposition Conjugate Vector Scalar Product of Second-Order Tensors Orthogonal Tensors Eigenproblem for a Second-Order Tensor Cross Product of Vectors and Tensors A Few Words About Higher-Order Tensors; Isotropic Tensors Equation Recap Cross-References References Terminal Ballistics Textiles Theoretical Damage Evolution Modeling Theory of Discontinuities Theory of Materials Theory of Microforces Theory of Porous Media Theory of structures Thermal Conduction and Thermal Radiation by Means of Generalized Continuum Thermal Effects by Means of Two-Component Cosserat Continuum Synonyms Definition Introduction Different Views on the Nature of Heat The Physical Meaning of the Model The Basic Ideas of the Approach A Model of the Ether The Interaction of the Ether with Material Particles The One-Rotor Gyrostat Continuum The Main Hypotheses Thermodynamic Analogies A Model of Heat-Conductive Elastic Material An Isotropic Chiral Medium The Hyperbolic Thermoelasticity and the Classical One Nonlinear Models of Heat Transfer: State of the Art Cross-References References Thermal Phenomena by Means of Continuum with Microstructure Thermal Processes by Means of Micropolar Continuum Thermal Stress and Low-Cycle Fatigue Thermal Stresses Thermal Waves Thermodynamic Processes by Means of Continuum with Rotational Degrees of Freedom Thermoelastic Diffusion Thermoelastic Diffusion Theory for Piezoelectric Materials Synonyms Definition Overview Nonlinear Theory Linear Theory Conclusion References Thermoelastic Waves Synonyms Definition Introduction Thermoelastic Waves Propagating in the L-S Model Thermoelastic Waves Propagating in the G-L Model Thermoelastic Waves Propagating in the H-I Model Thermoelastic Waves Propagating in the G-N Model Thermoelastic Waves Propagating in the C-T Model Closure References Thermoelastic Waves in a Medium with Heat-Flux Relaxation Synonyms Definition Introduction The Equations of the Hyperbolic Thermal Conductivity and Thermoelasticity Thermoelastic Waves in a Layer with Internal Heat Sources Problem Statement A Constant Temperature Kept at the Unconstrained Boundaries of the Layer A Constant Temperature Kept at the Constrained Boundaries of the Layer The Insulated and Constrained Boundaries of the Layer The Insulated and Unconstrained Boundaries of the Layer The Stress Distribution Conclusion Cross-References References Thermoelasticity Thermoelasticity: From Particle Dynamics to Continuum Mechanics Thermomechanical Processing of Steels and Alloys: Multilevel Modeling Synonyms Definition Introduction Multilevel Constitutive Models of Materials Main Physical Mechanisms Occurring in the Processes of Thermomechanical Treatment About the Multilevel Models of Steels and Alloys Aiming to Describe Thermomechanical Processing Cross-References References Thermomechanical Treatment of Polycrystals Thermomechanics Thick Plate Thick Shells Thin Bodies Embedded in Fractional Derivative Viscoelastic Medium, Dynamic Response Synonyms Definition Backgrounds Damped Vibrations of N-Degree-of-Freedom Mechanical Systems Free Vibrations of an Elastic Plate in a Viscous Medium Linear Vibrations of a Circular Cylindrical Shell Being in a Viscous Medium Conclusion Cross-References References Thin Elastic Shells, Lagrangian Geometrically Nonlinear Theory Synonyms Definition Introduction Geometry and Deformation of the Shell Base Surface Lagrangian Equilibrium Conditions Small Elastic Strains Restricted Rotations Cross-References References Thin Elastic Shells, Linear Theory Synonyms Definition Introduction Geometry and Small Deformation of a Thin Shell Equilibrium Conditions Compatibility Conditions Static-Geometric Analogy and Complex Shell Relations Constitutive Equations Conclusions Cross-References References Thin Plate Thin Plates and Shells Thin-Walled Beams of Open Profile Thin-Walled Profiles Thin-Walled Structures Three-Dimensional (3D) Composites: Through-Thickness Reinforcement Three-Dimensional Derivations of Static Plate Theories Synonyms Definition Introduction The 3D Problem Kinematic Derivation of Hencky\'s Plate Model Plate Kinematics Formulation of Hencky\'s Plate Model Minimum Potential Energy Plate Generalized Stresses Plate Equilibrium Natural Scaling of Stresses in Plates Constitutive Equations Static Derivation of Reissner Plate Model Derivation of a Statically Compatible Stress Field Formulation of the Reissner Plate Model Minimum of the Complementary Energy Plate Kinematics Constitutive Equations Static Boundary Conditions Conclusion References Time to Creep Rupture Time-Dependent Behavior of Solids Time-Dependent Permanent Strain Time-Stepping Algorithms in Nonlinear Resultant Shell Dynamics Synonyms Definition Introduction Initial-Boundary Value Problem Extended Newmark Algorithm Energy Conserving Algorithm (ECA) Example Conclusions Cross-References References Time-Stepping Schemes Timoshenko, Stepan Prokofievitch Russian Empire Period European and American Period Timoshenko\'s Books Cross-References References Topological Derivative Topology Optimization Topology Optimization Based on Explicit Geometry Description Synonyms Definition Introduction Theory Geometry Description Optimization Formulation Sensitivity Analysis Examples Some Extensions Conclusions References Transient Surface Waves of Strong and Weak Discontinuity Transient Wave Propagation Transient Waves in Cosserat Beams: Ray Expansion Approach Synonyms Definition Backgrounds and Some Historical Remarks Problem Formulation Determination of Transient Wave Velocities Propagating in Thin-Walled Beams with Cosserat-Type Microstructure Determination of Transient Wave Velocities Propagating in a Cosserat Continuum Conclusion Cross-References References Transport Coefficients Trescas Solids Truesdell, Clifford Ambrose III Education Professional Career Scientific Achievements and Honors Further Reading Cross-References References Truesdell\'s and Zhilin\'s Approaches: Derivation of Constitutive Equations Synonyms Definition Introduction Balance Equations for the Micropolar Continuum Temperature and Entropy Introduction Truesdell\'s Approach Zhilin\'s Approach Second Law of Thermodynamics Constitutive Equations for Inelastic Components of the Stress Tensors Summary and Comparison Cross-References References Two Dimensional Structural Models Two Dimensional Elastic Materials with Microstructure U Ultimate State Uncertainty Quantification Unilateral Contact Unit Cell Analysis V Variability Variable-Kinematics, Meshless Analysis of Composite Beams Synonyms Definition Equations of Motion in Strong Form Simplified Closed-Form Solution Approximate Solution via Collocation Scheme Use of Exact, Frequency-Dependent Shape Functions Exact Wave Solution in the Frequency Domain The Dynamic Stiffness Matrix Numerical Results Static Response and Stress Analysis of Laminated Composite Beams Free Vibration Analysis of Angle-Ply Laminates Thin-Walled Structure Conclusions References Variational Approach to Damage and Fracture Mechanics Variational Derivative Variational Formulation of Damage and Fracture Mechanics Variational Formulations and Galerkin Methods for Strain Gradient Elasticity Synonyms Definition Introduction First Strain Gradient Elasticity Theory of Mindlin and Toupin Variational Formulations for First Strain Gradient Elasticity Theory Galerkin Methods and Error Estimates Second Strain Gradient Elasticity Theory of Mindlin Variational Formulations for Second Strain Gradient Theory Isogeometric Galerkin Approximations Cross-References References Variational Methods for Continuum Models of Granular Materials Synonyms Definition Continuum Models of Granular Materials Historical Context and Experimental Motivation Granular Micromechanics Micro-Macro Identification First Gradient Approximation Perspectives Concluding Remarks Cross-References References Variational Methods for Discrete Models of Granular Materials Synonyms Definition Granular Solids Grain Interactions Phenomenology and Fundamental Kinematical Quantities Action and Variational Inequality Discrete Models Classical Models Models with Finite Grain Spin and Generalized Grain Interactions Concluding Remarks Cross-References References Variational Methods in Continuum Damage and Fracture Mechanics Synonyms Definition Principle of Least Action for Dissipative Systems Reduction to the Standard Variational Principle The Derivations of KKT Conditions The Definition of the Total Deformation Energy Functional in Nonlocal Continuum Mechanics Damage Strain Gradient Formulation for the 1D Case Cross-References References Variational Methods in Granular Micromechanics Variational Methods in Structural Optimization Synonyms Definition Introduction Basic Formulation Analysis Problem Objective Functional and the Global and Local Constraints Necessary Conditions for the Extremum of the Objective Functional Mechanical Interpretation of the Adjoint System Sensitivity Analysis Conclusions Appendix Cross-References References Variational Methods in the Theory of Beams and Lattices Synonyms Definition Introduction Kinematics and Constitutive Hypothesis for Beams Total Strain Energy and Its First Variation Equilibrium Conditions for Beams Further Necessary Conditions for Beams (Steigmann and Faulkner 1993) Equilibrium Conditions for Lattices Further Necessary Conditions for Lattices (Steigmann 1996) Cross-References References Variational Principles Variational Principles in Numerical Practice Synonyms Definition Introduction Interaction Between Bone Tissue and Bio-resorbable Material Effects of the Internal Friction in Concrete Smart PEM Structures with Distributed Control References Vector Differential Operators Vector Integral Theorems Vectors Synonyms Definition Vectors of Linear Algebra and of Mechanics Vectors of Rn (n=3) Reciprocal Basis Transformation of Basis Metric Coefficients More on Vectors: Cross Product in Three-Dimensional Space Cross Product Equation Recap Cross-References References Vectors in Finite Dimensional Spaces Vekua, Ilia Education Professional Career Scientific Achievements Honors References Vibration Isolation Vibration Suppression Vibrations of Rods and Beams Synonyms Definition Introduction Geometry and Kinematics One-Dimensional Theory Asymptotic Analysis References Vibrations of Small Amplitude of Thin Bodies Regarded as One-Dimensional Continua Vibrations of Suspension Bridges: Fractional Derivative Model Synonyms Definition Backgrounds Problem Formulation Method of Solution The Case k=1 The Case k=2 The Case of Free Damped Vibrations at γ1=γ2= γ The Case of Force-Driven Vibrations at γ1=γ2=γ The Case of Free Damped Vibrations at γ1=γ2 Conclusion Cross-References References Vibrations of Viscoelastic Rods Vibrations of Viscoelastic Rods, Fractional Operator Models Synonyms Definition Backgrounds The Models Involving More Than One Fractional Operators of Different Orders Thermodynamic Constraints Longitudinal Vibrations of Viscoelastic Rods Concluding Remarks Cross-References References Violations of the Second Law of Thermodynamics Virtual Experimentation; Virtual Laboratory, Numerically Simulated Experiments Virtual Microstructure Viscoelastic Composite Materials Viscoelasticity Synonyms Definition Introduction: Temperature and Time-Dependent Behavior of Solids Viscoelastic Transition Creep and Stress Relaxation Functions Invariant and Non-invariant Viscoelastic Materials Linear Heredity Theory of Viscoelasticity: Boltzmann Superposition Principle Linear Mechanical Models First- and Multi-order Viscoelastic Material Models Generalized Linear Viscoelasticity Time-Temperature Equivalence Law Dynamic Mechanical Analysis: Frequency Domain Analysis of Viscoelasticity of Solids Cross-References References Viscoplastic Flow Viscoplastic Model Viscoplasticity Synonyms Definition General Remarks One-Dimensional Viscoplasticity: Small Strains Three-Dimensional Viscoplasticity: Small Strains Other Formulations Generalizations References Viscosity Voigt, Woldemar Education Professional Carrier Scientific Achievements Honors References Vorovich, Iosif Izrailevich Education Professional Life Scientific Results References W Wagners Tension Field Wave Wave Equations Wave in Microstructured Solids Wave Motion Wave Propagation Wave Propagation in Inhomogeneous Media Wave Propagation in Stochastically Inhomogeneous Media Wave Propagation in Viscoelastic Rods Wave Propagation in Viscoelastic Rods, Fractional Operator Models Synonyms Definition Backgrounds The Models Involving Fractional Derivatives of Two Different Orders Rheological Characteristics Harmonic Waves Stress Waves Concluding Remarks Cross-References References Wave Types Wave Velocities Waves Waves and Generalized Continua Synonyms Definition Introduction Basic Toolbox for Studying Linear Plane Wave Propagation Generalized Continua Notations Uniaxial Propagation and Secular Equation Example: Dispersion Curves in Strain Gradient Elasticity The Generalized Acoustic Tensor Example: Generalized Acoustic Tensor in Strain Gradient Elasticity for a Centrosymmetric Material Energy Conservation, Poynting Vector, and Reflection/Transmission at Interfaces Example: Poynting Vector in Strain Gradient Elasticity Cross-References References Waves in Continuous Media: Classical Theory Synonyms Definition Modeling of Waves Types of Waves Notes from History Cross-References References Weak Form Weakly Excited Lattice Web Buckling Weighted Average Methods Weighted Residual Methods for Finite Elements Synonyms Definition Basic Definitions and Classification of Approximate Methods Galerkin Techniques: Derivation of Principal Finite Element Equations Rod Elements Euler-Bernoulli Beam Elements Timoshenko Beam Elements Plane Elasticity Elements Classical Plate Elements Shear Deformable Plate Elements Three-Dimensional Elements Concluding Remarks Cross-References References Westergaard, Harold Malcolm Education Professional Carrier Scientific Achievements Honors Cross-References References Y Yield and Kinematic Equations for Planar Deformations Yield Design Young, Thomas Education Scientific and Professional Career Scientific Achievements Cross-References Z Z-Binder Reinforcement Zerna, Wolfgang Early Years and Education Professional Career Scientific Achievements and Honors Reference Zhilin, Pavel Andreevich Family and Education Professional Carrier Academic Pupils Scientific Achievements Cross-References References Zhilins Approach Zhilin\'s Method and Its Modifications Synonyms Definition Introduction Balance Equations for a Continuum with Microstructure Transformation of the Energy Balance Equation Different Ways of Entropy and Chemical Potential Introduction Variant 1 Variant 2 Variant 3 Closing Remarks Cross-References References Zienkiewicz, Olgierd Cecil Education Professional Career Scientific Achievements and Honors Further Reading Cross-References References
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