Mechanics of Continuous Media

The following sections are included:

• Foreword to the first edition

• Foreword to the third edition

• Foreword to the fourth edition

• Contents

The following sections are included:

• Continuum mechanics and its methods

• Fundamental hypotheses
  • The structure of real bodies and the hypothesis of continuity.
  • On space and time

The following sections are included:

• Lagrangian approach to the study of the motion of a continuous medium

• Eulerian approach to the study of motion of continua

• Scalar and vector fields and their characteristics

• Elements of tensor calculus

• Theory of deformation

• Strain Rate Tensor

• Distribution of velocities in an infinitesimal particle of a continuum

• Theorems of Stokes and Gauss–Ostrogradskii and certain related properties of vector fields

The following sections are included:

• Equation of continuity

• The equations of motion of a continuous medium

• The equation of angular momentum

• Principal axes and principal components of a symmetric stress tensor

The following sections are included:

• Ideal fluid and gas

• Linear elastic body and linear viscous fluid

• Examples of equations in curvilinear coordinates and additional results from tensor analysis

The following sections are included:

• Kinetic energy theorem and work of internal surface forces

• The first law of thermodynamics (principal of conservation of energy) and the equation of heat flow

• Thermodynamic equilibrium; reversible and irreversible processes

• Two-parameter media. Perfect gas. Carnot cycle

• The second law of thermodynamics and the concept of entropy

• Thermodynamic potentials of two-parameter media

• Examples of ideal and viscous fluids and their thermodynamic properties. Heat conduction
  • Model of the ideal incompressible fluid
  • Model of an ideal gas, i.e., of an ideal compressible fluid
  • Model of a viscous fluid

• First and second laws of thermodynamics for finite volumes of continuous media. Generation of entropy in certain irreversible processes

• The introduction to the theory of models of gas and fluid mixtures with chemical interaction and the diffusion of the components

• The modelling of a mixture for reversible processes

• The modelling of a mixture for irreversible processes

The following sections are included:

• Basic concepts of electrodynamics. Electromagnetic field. Maxwell’s equations in vacuum

• Maxwell’s equations in Minkowski space

• Lorentz transformations and inertial reference frames

• Interaction of electromagnetic field with conductors

• Interaction of an electromagnetic field with bodies in the presence of polarization and magnetization

• Hydrodynamics of the conducting fluid

• The laws of freezing in of magnetic and vortex flux lines

The following sections are included:

• General foundations of the formulation of concrete problems

• Typical simplifications in the formulations of certain problems, leading to a reduction of the number of independent variables

• Linearization of equations and of problems of continuum mechanics

• Conditions on surfaces of strong discontinuity

• Strong discontinuities in an electromagnetic field

• Surfaces of discontinuity inside ideal compressible media

• Dimensionality of physical quantities and Π-theorem

• Parameters determining a class of phenomena and typical examples of the application of the methods of dimensional analysis

• Similarity and modelling of phenomena

The following sections are included:

• Appendix I Nonlinear tensor functions of several tensor arguments
  • Fundamental concepts
  • Symmetry groups of tensors
  • Tensors which specify the geometric symmetry of oriented media and crystals [29]
  • Tensor functions of tensors characterizing the geometric properties of oriented media and crystals
  • Tensor functions for oriented media and crystals with additional tensor arguments
  • On the Riemann curvature tensor and a generalization of Schur’s theorem
  • BIBLIOGRAPHY TO APPENDIX I

• Appendix II Models of Continuous Media with Internal Degrees of Freedom
  • BIBLIOGRAPHY TO APPENDIX II

• Subject Index

The following sections are included:

• Contents

The following sections are included:

• Hydrostatics

• General theory of steady motions of an ideal fluid and gas. Bernoulli’s integral

• Bernoulli’s integral for an incompressible heavy fluid

• The Phenomenon of Cavitation

• Bernoulli’s integral for adiabatic flow of a perfect gas

• Influence of compressibility on the shape of flow tubes. Elementary theory of laval nozzle

• Application of integral relations to finite volumes of material media under steady motion

• Interactions of fluids and gases with bodies in steady stream line flow

• Basic components on hydrodynamic and gas engines

• Foundations of the theory of reactive power

• Potential flows of an ideal fluid. The integral of Cauchy–Lagrange

• Potential motions of an incompressible fluid. Properties of harmonic functions

• The problem of the motion of a sphere in an unbounded volume of an ideal incompressible fluid

• The kinematic problem of the motion of a rigid body in an unbounded volume of an ideal incompressible fluid

• Energy momentum, moment of momentum of fluid motion when a rigid body moves inside it, and the foundations of the theory of virtual masses

• The forces exerted by an ideal fluid on a body which moves in an unbounded mass of fluid

• Motion of a gas with small disturbances

• Propagation of plane waves of finite amplitude (Riemann Waves)

• Oscillations of a bubble in a fluid

• Motion of a sphere inside a viscous incompressible fluid

• Motion of an incompressible viscous fluid in cylindrical tubes

• Turbulent fluid motion

• Equations of the laminar boundary layer

• Boundary layer for the flow of an incompressible fluid around a flat plate. Blasius’ problem

• Certain important effects of the motion of a viscous fluid in a boundary layer

• Determination of the velocity fields for given vortices and sources

• Important examples of vortex fields

• Dynamic theory of cylindrical vortices

• Motion of systems of continuously distributed vortices in an ideal fluid

• Diffusion of vortices in a viscous incompressible fluid

The following sections are included:

• Introductory remarks

• Model of the elastic body

• Problems of uni-axial extension of an elastic bar

• Strains and stresses in a circular elastic tube under the influence of internal and external pressures (Lamé’s problem).

• Formulation of problems of the theory of elasticity. Clapeyron’s equation. Uniqueness theorem for the solution of the problems of the theory of elasticity. Saint–Venant’s principle

• The problem of bending of beams

• Torsion of cylindrical rods

• The methods of strength of materials in problems of bending of beams

• Variational methods in the theory of elasticity

• Elastic waves in isotropic media

The following sections are included:

• Certain effects which arise during the deformation of solids and are not described by the model of the elastic body

• Residual strains. Load surface

• The basic determining relations in the theory of plastic bodies

• Examples of Models of Plastic Bodies

• The problem of torsion of a cylindrical rod made of elastic-plastic material without hardening

The following sections are included:

• The plane problems of the theory of elasticity

• Stress concentrations

• A theory of cracks

The following sections are included:

• References

• Subject Index