Theory of Elasticity and Plasticity: A Textbook of Solid Body Mechanics

Dedication
Preface to the English-Language Edition
Preface
Abstract
Contents
Notation Conventions
	Loads and Stresses
	Deformations and Movements
	Physical and Mechanical Characteristics of Materials
Part I Basis of Elasticity Theory
	1 Summary of Elasticity Theory: Basic Concepts
		1.1 From the History of Elasticity Theory
		1.2 Elasticity of Solid Bodies
		1.3 Homogeneous Strain
		1.4 Internal Forces: Method of Sections
		1.5 Homogeneous Body
		1.6 Stress Vector
		1.7 Elongation of Steel Specimens
		1.8 Permanent Deformations
		1.9 Elastic Limit
		1.10 Elastic Shear Deformation
		1.11 Law of Twoness of Tangential Stresses
		1.12 Homogeneous Stressed State
		1.13 Generalized Hooke's Law
		1.14 Another Form of Hooke's Law
		1.15 Plane Stress-Strain State
		1.16 Homogeneous Model of a Solid Body
		1.17 Axisymmetric Plane Strain
		1.18 Lame Task
		1.19 Phenomenon of Stress Concentration
		1.20 Saint-Venant Principle
		References
	2 The First Basic Problem of Elasticity Theory
		2.1 Equilibrium Equations
		2.2 Expression of Strains Through Movements
		2.3 Definition of Movements
		2.4 Saint-Venant Identities
		2.5 Compatibility Conditions
		2.6 Boundary Conditions
		2.7 The First Basic Problem of Elasticity Theory
		References
	3 The Second Primary Problem of Elasticity Theory
		3.1 Definition of Stresses Through Deformations
		3.2 Equations of Elastic Body Strain
		3.3 Application of Harmonic Functions
		3.4 Trefftz Integral
		3.5 Grodsky–Neyber–Papkovich Integral
		References
	4 Three-Dimensional Harmonic Function
		4.1 Simplest Examples of Harmonic Functions
		4.2 Green Function
		4.3 Green's Spatial Functions
		4.4 Boundary Problems for Half-Space
		4.5 Other Properties of Harmonic Functions
		References
	5 Elastic Half-Space
		5.1 Volumetric Expansion on Surface
		5.2 Stress on Surface
		5.3 Strain of Elastic Half-Space
			5.3.1 Integral Operator of Formulas (5.18)–(5.20)
		5.4 Examples
		References
	6 Herz's Task
		6.1 Deformation of Adjoining Bodies
		6.2 Primary Assumptions
		6.3 Axisymmetric Hertz Problem
		6.4 Compression of Orthogonal Cylinders
			6.4.1 Simplest Case
			6.4.2 Primary Case
		6.5 Compression of Barrel-Shaped Bodies
			6.5.1 Rotation Bodies with Parallel Axes
			6.5.2 Case of Intersecting Axes
		6.6 Elongated Contact Area
		6.7 Compression of Parallel Cylinders
		References
	7 Stressed State in a Body Point
		7.1 Principal Stresses
		7.2 Maximum Stresses
		7.3 Intensity of Stresses
		7.4 Some Properties of Tangential Stresses
		References
	8 Linear Elastic Systems
		8.1 General Comments
		8.2 Linear System
		8.3 Potential Energy of a Helical Spring
		8.4 Principle of Mutuality of Works
		8.5 Castigliano's Theorem
		8.6 Specific Potential Energy of Elastic Deformation
		References
	9 Plane Problem of Elasticity Theory
		9.1 Functions of Stresses
			9.1.1 Example 1: Concentrated Force in the Wedge Apex
			9.1.2 Example 2: Wedge Bending by Uniform Pressure
		9.2 Complex Representation of a Bi-Harmonic Function
		9.3 Kolosov Displacement Integral
		9.4 Action of Concentrated Force
		9.5 Solution of the First Principal Problem for a Circle
		9.6 Annex to the Brazilian Test
		References
	10 Mathematical Structural Imperfections
		10.1 Mathematical and Physical Theories of Structural Imperfections
		10.2 Edge Dislocation in an Infinite Body
		10.3 Mathematical Wedge-Shaped Dislocation
		10.4 Mathematical Biclination
		10.5 Flat Dislocation of Somigliana
		10.6 Somigliana Dislocation in Half-Plane
			10.6.1 Functions ,  for the Plane with Dislocation
			10.6.2 Functions ,  for a Half-Plane with Dislocation
			10.6.3 Calculation of Galin Functions
			10.6.4 Completion of Problem Solution
			10.6.5 Addition to Geomechanics
		10.7 Pair of Fislocations in a Plane
		10.8 Edge Dislocation in a Half-Plane
		10.9 Half-Plane with a System of Dislocations
		References
	11 The Beginning of the Theory of Stability of Equilibrium
		11.1 Stability and Instability
		11.2 Work and Classification of Forces
		11.3 Stability with Conservative and Dissipative Forces
		11.4 Lyapunov–Chetaev Theorem
		11.5 Instability in the First Approximation
		11.6 Critical Load
		11.7 The Theorem on Stability by the First Approximation
		11.8 The Raus–Hurwitz criterion
		11.9 Main Types of Stability Loss
		11.10 Methods for Determining Critical Load
		11.11 The Perturbed Motion of the Compressed Rod
		11.12 Stability Under Non-conservative Load (Example)
			11.12.1 Equations of Perturbed Motion
			11.12.2 Area of Valid Stability
			11.12.3 Investigation of the Value μ, (Formula (11.31))
			11.12.4 Investigation of the Effect of Friction
			11.12.5 The influence of the spacing of the End Masses
		References
Part II Principal Variants of Mathematical Plasticity Theory
	12 Origin and Development of Plasticity Theory
		12.1 Primary Definitions
		12.2 The Subject and Tasks of the Theory of Plasticity
		12.3 Early Development Stages of Plasticity Theory
		12.4 Development of Plasticity Theory in the Twentieth Century
		12.5 Soviet Period of Plasticity Theory Development
		12.6 Russian Mechanics in the Post-Soviet Period
			12.6.1 General Situation and Dangerous Trends
			12.6.2 Plasticity Theory in Russia in the Post-Soviet Period
		12.7 Abstract
		References
	13 Initial Concepts of Plasticity Theory
		13.1 Second-Rank Tensor in Euclidean Space
		13.2 Tensors in Plasticity Theory
		13.3 Decomposition of Stress and Strain Tensors
		13.4 Other Invariants in Plasticity Theory
		13.5 On the Criterion of Similarity of Stress and Strain Deviators
		13.6 Stress Diagrams and Their Idealization
		References
	14 On the Plasticity Conditions of an Isotropic Body
		14.1 General Considerations
		14.2 General Notes
		14.3 Tresca Plasticity Condition
		14.4 Huber–Mises Plasticity Condition
		14.5 Experimental Study of Elastic–Plastic Materials
		14.6 Volumetric Elasticity of Materials
		14.7 Invariant Form of Hooke's Law
		References
	15 Plasticity Theory of Henky–Nadai–Ilyushin
		15.1 Laws of Active Elastic–Plastic Deformation
		15.2 Defining the Universal Hardening Function
		15.3 Some Properties of the Hardening Function
		15.4 Another Form of Strain Ratios
		15.5 Unloading Laws
		15.6 Work of Stresses, Potential Energy, and Potentials
			15.6.1 Stress Potential
			15.6.2 Potential of Strains
		15.7 Theorem of the Minimal Work of Inner Forces
		15.8 Lagrange Equilibrium Variation Equation
		15.9 Setting Boundary Problems of Plasticity Theory
		15.10 Theorem of Simple Loading
		15.11 Theorem of Unloading
		References
	16 Solution of the Simplest Problems for the Strain Theory of Plasticity
		16.1 Pure Bending of a Straight Beam
		16.2 Torsion of a Round-Section Beam
		16.3 Elastic–Plastic Inflation of a Spherical Vessel
		16.4 Symmetric Strain of a Cylindrical Tube
		16.5 Torsion of a Beam of Ideally Plastic Material
			16.5.1 Elastic Torsion: Prandtl Analogy
			16.5.2 Elastic–Plastic Beam Torsion
		16.6 Rod of a Variable Section: Method of Elastic Solutions
			16.6.1 Preparation of Initial Ratios
			16.6.2 Specification of Problem Setting
			16.6.3 Algorithm of the Elastic Solutions Method
		References
	17 Additions and Generalizations to the Strain Theory of Plasticity
		17.1 Generalizations of Goldenblatt and Prager
		17.2 Tensor–Linear Ratios in Plasticity Theories
		17.3 Vector Representation of Tensors
		17.4 Transformations of Rotation and Reflection
		17.5 Ilyushin's Isotropy Postulate
		17.6 Delay Law
		17.7 Loading Surface
		17.8 Drucker Postulate
		17.9 On the Applicability Limits of the Strain Theory of Plasticity
		References
	18 Theories of Plastic Yield
		18.1 General Ratios
		18.2 Prandtl–Reuss Yield
		18.3 Saint-Venant–Mises Yield Theory
		18.4 Plastic Yield in Isotropic Hardening
		18.5 Handelman–Lin–Prager Plasticity Theory
		18.6 Yield for Plane Loading Surfaces
		18.7 Yield for Some Loading Surfaces
		18.8 Kadashevich–Novozhilov Plasticity Theory
		18.9 Singular Loading Surfaces
		References
	19 Other Variants of Plasticity Theories
		19.1 Batdorf–Budiansky Slip Theory
		19.2 Two-Dimensional Klyushnikov Model
		19.3 Endochronic Plasticity Theory
		19.4 On the Methods of Physical Mesomechanics and Synergetics
		References
Part III Development of the Slip Concept in Plasticity Theory
	20 Problem Setting
		20.1 Initial Concepts and Definitions
		20.2 Shift Resistance
		20.3 Slip Synthesis
		20.4 Definition of Principal Strains
		References
	21 Strain Specifics of Plastic Bodies
		21.1 Elongation Diagram of a Plastic Material Specimen
		21.2 Delay of Yield
		21.3 Yield Stress and Loading Rate
		References
	22 Axioms of the Inelastic Body Model
		22.1 Deformational Softening
		22.2 Initial Shear Resistance
		22.3 Function of Elastic Softening
		References
	23 The Fluidity at the Finite Speed of Loading
		23.1 Yield Strength at the Final Loading Speed
		23.2 Defining the Aging Function
			23.2.1 Example
		23.3 Components of Deformational Softening
		23.4 Almost Simple Strain
		References
	24 Specimen Elongation with Yield Drop
		24.1 Original Assumption
		24.2 Occurrence of Non-elastic Strain
		24.3 Origins of Boundary Layer Theory
		24.4 Simplified Model of Non-elastic Strain Growth
		24.5 Definition of the Plastic Zone Growth Rate
		24.6 Steady-State Yield
		24.7 Building an Elongation Diagram
		References
	25 Building a Shear Resistance Operator
		25.1 General Form of the Shear Resistance Operator
		25.2 Boundary Condition
		25.3 Special Cases
		References
	26 Full Bauschinger Effect
		26.1 Secondary Yield Stress
		26.2 Proportional Primary Loading
		26.3 Proportional Loading of an Opposite Sign
		26.4 Function  in Almost Simple Strain
		References
	27 Non-elastic Uniaxial Elongation–Compression
		27.1 Calculating Slip Intensity
		27.2 Calculation of the Integral (27.5)
		27.3 Solving the Integral Equation
		27.4 Study of the Tensor Intensity of Slips
		27.5 Determinant Equations in Uniaxial Elongation
		27.6 Plastic Strain in Loading and Compression
			27.6.1 Increment of Non-elastic Strain in Loading
			27.6.2 Strain in Compression
		27.7 Strain Creep and Stress Relaxation
		27.8 Examples of Building Diagrams in an Uniaxial Stressed State
		References
	28 Module of Additional Orthogonal Load
		28.1 Problem Statement
		28.2 Determining the Intensity of Additional Slips
		28.3 Calculation of the Strain Increments and Additional Loading Modulus
		28.4 Analysis of Results and Conclusions
		References
	29 Plane-Plastic Strain
		29.1 Theorem of Strain in Pure Shear
		29.2 General Dependencies in Pure Shear
		29.3 Monotonous Plane-Plastic Strain
			29.3.1 Preparation of Initial Dependencies
			29.3.2 Determinant Ratios
			29.3.3 Continuity Condition
			29.3.4 Monotony Conditions
		References
Part IV Non-elastic Strain of Geomaterials
	30 Complex Strain of Soils
		30.1 Real State of the Mechanics of Non-elastic Strains
		30.2 Simple Strain Model of Hardening Dense Soils
		30.3 Defining the Form of the Function G
			30.3.1 Building the G Function for a Material with High Hardening
			30.3.2 Universal G Function for Hardening Soils
		References
	31 Simple Loadings of Geomaterials
		31.1 Uniaxial Compression
		31.2 Creep in Uniaxial Compression
		31.3 Uniaxial Elongation
		31.4 Pure Shift
		31.5 Determination of Model Parameters
		31.6 Comparison of Experimental and Calculation Results
		References
	32 On Boundary Value Problems of Inelastic Body Mechanics
		32.1 General Formulation of the Problem of Inelastic Solid Mechanics
		32.2 More About the Method of Elastic Solutions
		32.3 An Example of Using the Birger Method
			32.3.1 The Initial Stage of the Process with Linear Hardening
			32.3.2 Case of Semi-Infinite Plastic Zone
			32.3.3 Auxiliary Task
			32.3.4 Final Length of the Plastic Zone
			32.3.5 The Dependence of the Tensile Force and Pressure p on the Length of the Plastic Zone
		32.4 Perfectly Plastic Body Case
		32.5 Using the Kröner Theory of Residual Stresses
		32.6 Kröner Method for Plane Deformation
		32.7 More About Incompatible Deformations
			32.7.1 Distributed Wedge Dislocations
			32.7.2 Strain Incompatibility Tensor
		32.8 The Application of Kröner's Method to the Brazilian Test
			32.8.1 Zero Approximation
			32.8.2 Green's Tensor Function for a Circle
			32.8.3 Definition of Deformation in a First Approximation
		References
Index