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دانلود کتاب Encyclopedia of Continuum Mechanics

دانلود کتاب دایره المعارف مکانیک پیوسته

Encyclopedia of Continuum Mechanics

مشخصات کتاب

Encyclopedia of Continuum Mechanics

ویرایش: 1st ed. 2020 
نویسندگان:   
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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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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