Theory of Elasticity and Plasticity: A Textbook of Solid Body Mechanics [1 ed.] 9783030666217, 9783030666224

Table of contents :
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