Encyclopedia of Continuum Mechanics

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