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Mechanics of Solid Deformable Body

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This textbook contains sections with fundamental, classical knowledge in solid mechanics, as well as original modern mathematical models to describe the state and behavior of solid deformable bodies. It has original sections with the basics of mathematical modeling in the solid mechanics, material on the basic principles, and features of mathematical formulation of model problems of solid mechanics.

For successful mastering of the material, it is necessary to have basic knowledge of the relevant sections of the courses of mathematical analysis, linear algebra and tensor analysis, differential equations, and equations of mathematical physics. Each section contains a list of test questions and exercises to check the level of assimilation of the material.

The textbook is intended for senior university students, postgraduates, and research fellows. It can be used in the study of general and special disciplines in various sections of solid mechanics, applied mechanics for students and undergraduates of various specializations and specialties, such as mechanics and mathematical modeling, applied mathematics, solid physics, and engineering mechanics.

Author(s): Michael Zhuravkov, Yongtao Lyu, Eduard Starovoitov
Publisher: Springer
Year: 2023

Language: English
Pages: 316
City: Singapore

Preface
Introduction
Contents
1 Statements of Boundary Problems of Solid Mechanics
2 Basic Concepts of Stress–Strain State Theory
2.1 Defining Concepts of Continuum Mechanics
2.2 Stress Tensor
2.3 Stress Tensor Properties
2.4 Equilibrium Equations of Deformed Solid
2.5 Equilibrium Conditions at the Boundary
2.6 Principal Axes and Principal Values of Stress Tensor
2.7 Maximal Tangent Stresses
2.8 Stress Deviator and Spherical Part of Stress Tensor
2.9 Strains and Displacements
2.10 Principal Axes and Principal Values of Strain Tensor
2.11 Strain Deviator and Spherical Part of Strain Tensor
2.12 Strain Compatibility Equations
3 Material and Solid Mechanical Characteristics (Properties)
3.1 Main Mechanical Characteristics of Materials and Solids
3.2 Classification of Materials by the Nature of Deformation
3.3 Complete Diagrams of Deformation and Fracture
4 Construction of Mathematical Model Problems
4.1 The System of Equations to Describe the Medium Stress–Strain State
4.2 Physical Relationships Determining the Solid Behavior
5 Mathematical Models of the Theory of Elasticity
5.1 Basic Concepts and Definitions
5.2 Hooke’s Law
5.3 Clapeyron’s Formula and Clapeyron’s Theorem
5.4 Thermoelasticity
5.5 The Boundary Problems of Theory of Elasticity
5.6 The Theory of Elastic Boundary Problems in Displacements
5.7 Boundary Problems of the Theory of Elasticity in Stresses
5.8 Homogeneous Problem of the Theory of Elasticity in Stresses
5.9 The Problem of the Setting of Theory of Elasticity
5.10 2D-Problems of the Theory of Elasticity
6 Mathematical Models of Solid with Rheological Properties
6.1 Creep and Relaxation
6.2 Solid Mathematical Models
6.2.1 Structural Rheological Models
6.2.2 Creep Damage
6.2.3 General Comments
6.3 Rheological Equations of Linear Viscoelasticity
6.3.1 General Concepts
6.3.2 Examples of Creep and Relaxation Kernels
6.3.3 Volterra Principle
6.4 Equations of Mechanics of Linear-Hereditary Media
7 Mathematical Models of Plasticity Theory
7.1 Solid-Plastic Behavior Models
7.2 Material Plasticity During Tension and Compression
7.3 Elastic–Plastic Models of Material Behavior
7.4 Viscoplasticity Models
7.5 Plasticity Conditions
7.6 Simple and Complex Loading
7.7 Hypotheses of the Small Elastoplastic Deformation Theory
7.8 Theory of Problems of Small Elastic–Plastic Deformations
7.9 The Method of Elasticity Solutions
7.10 Plastic Flow Theory
7.11 Relationship Between Plastic Flow and Theory of Plasticity
7.12 Formula and General Methods
7.12.1 Formula of the Problems of Plasticity Theory
7.12.2 General Methods of Solving the Problems of Plasticity Theory
8 Fundamental Solutions
8.1 General Remarks
8.2 Boussinesq’s Problem
8.3 Hertz’s Formulas
8.4 Flamant’s Problem
8.4.1 Flamant’s Problem for a Half-Space
8.4.2 Flamant’s Problem for a Half-Plane
8.5 Kelvin’s Problem
8.6 Cerrutti’s Problem
8.7 General Case of Point Load Action on Surface of Falf-Space
8.8 Mindlin’s Problem
8.9 Some Generalizations
9 Dynamic Problems of Solid Mechanics
9.1 General Concepts and Definitions
9.2 Wave Equation
9.3 Model Problems of Dynamics of Elastic Body
9.3.1 General Comments
9.3.2 Cauchy Problem and Boundary Problems
9.4 Stokes Problem
9.5 Natural and Forced Harmonic Oscillations
9.5.1 Natural Oscillations of Elastic Bodies
9.5.2 Forced Oscillations of Elastic Bodies
9.6 Propagation of Shock Waves in Unlimited Elastic Bodies
9.7 Progressive Waves
9.8 Rayleigh Waves
9.9 Progressive Waves in a Plane Layer
9.10 Semi-plane Under Action of Moving Surface Force
9.11 Model Problems on Perturbation Propagation in Elastic Space
9.12 Model Problems on Non-stationary Perturbation Propagation
9.13 Model Problems on Volume Perturbation Propagation
10 Mathematical Models of Special Classes of Solid Mechanics Problems
10.1 Solving Problems for Heterogeneous Nonlinear Elastic Media
10.1.1 Solving Problems for Physically Nonlinear Elastic Medium
10.1.2 Solution of Problems for Orthotropic Solids
10.2 Solving the Problems of Linear Theory of Viscoelasticity for Inhomogeneous Solids
10.3 The Theory of Creep of Isotropic Hardening Media
10.3.1 Consider the Situation of a Complex Stress State
10.4 Bilinear Theory of Elasticity
10.5 Nonlinear Viscoelastic Media
10.6 Method of Successive Approximations in Viscoplastic Problems