Continuum mechanics

The continuum mechanics is a section of mechanics , continuum physics and condensed matter physics devoted to the motion of gaseous, liquid and deformable solids, as well as force interactions in such bodies.

A.Ilyushin, Corresponding Member of the USSR Academy of Sciences , described continuum mechanics as “an extensive and very extensive science, including the theory of elasticity, viscoelasticity, plasticity and creep, hydrodynamics, aerodynamics and gas dynamics with the theory of plasma, the dynamics of media with non-equilibrium processes of structure change and phase transitions " ^[1] .

In addition to the usual material bodies like water, air, or iron, special media are considered in continuum mechanics - fields : the electromagnetic field , the gravitational field, and others.

Continuum mechanics is divided into the following main sections: deformable solid body mechanics , hydromechanics , gas dynamics . Each of these disciplines is also divided into sections (narrower now); so, the mechanics of a deformable solid body is divided into the theory of elasticity , the theory of plasticity , the theory of cracks , etc.

Content

Continuous Media Mechanics Methods

In continuum mechanics based on the methods developed in theoretical mechanics , motions of such material bodies are considered that fill the space continuously, neglecting their molecular structure. At the same time, the characteristics of bodies are also considered to be continuous - such as density , stresses, velocities , etc. An applied explanation for this is that the linear dimensions with which we deal in continuum mechanics are much longer than intermolecular distances. The minimum possible volume of the body, which allows you to explore some of its specified properties, is called a representative volume or a physically small volume. This simplification makes it possible to use in mechanics of continuous media well developed for continuous functions of the apparatus of higher mathematics . In addition to the continuity hypothesis, the hypothesis of space and time is accepted - all processes are considered in space , in which the distances between points are determined, and develop in time , and in classical mechanics of continuous media, time flows the same for all observers, and in relativistic space and time are connected in single space-time .

Continuum mechanics is the extension of the Newtonian mechanics of a material point to the case of a continuous material medium ; the systems of differential equations that are compiled to solve various problems of continuum mechanics reflect the classical laws of Newton , but in the form specific to this section of mechanics. In particular, such fundamental physical quantities of Newtonian mechanics as mass and force are presented in the equations of continuum mechanics in specific forms: mass — as density , and force — as voltage (or — in statics of gases and liquids — as pressure ).

In continuum mechanics, methods are developed to reduce mechanical problems to mathematical problems, that is, to find some numbers or numerical functions using various mathematical operations. In addition, an important goal of continuum mechanics is to establish the general properties and laws of motion of deformable bodies and force interactions in these bodies.

Under the influence of continuum mechanics, a number of branches of mathematics have been greatly developed - for example, certain sections of the theory of functions of complex variable , boundary problems for partial differential equations , integral equations, and others.

Axiomatics of continuum mechanics

Academician A. Yu. Ishlinsky , describing the state of affairs in the field of axiomatization of mechanics, noted: “The mechanics of Galileo - Newton are still not adequately axiomatized, unlike geometry , the axiomatization of which was completed by the great mathematician D. Hilbert ... Nevertheless, it is possible and it is necessary (the time has come) to construct classical mechanics , as well as geometry, based on a number of independent postulates and axioms established as a result of generalizing practice ” ^[2] .

However, a number of attempts to axiomatize mechanics (and, in particular, continuum mechanics ) have been made. Below are the basic concepts of continuum mechanics, which play (in various axiomatic constructions) the role of either axioms or the most important theorems .

Euclidean space . The space in which the movement of the body is considered is a three-dimensional Euclidean point space (denoted by ^[3] ${\mathcal {E}}$ ${\ displaystyle {\ mathcal {E}}}$ , and $E_{3}$ ${\ displaystyle E_ {3}}$ ).
The absoluteness of time $t$ ${\ displaystyle t}$ . The passage of time does not depend on the choice of reference system.
Hypothesis of continuity . The material body is a continuous medium (the continuum in space $E_{3}$ ${\ displaystyle E_ {3}}$ ).
The law of conservation of mass . Every material body $V$ ${\ displaystyle V}$ has a scalar non-negative characteristic - mass $M$ ${\ displaystyle M}$ which: a) does not change with any movements of the body, if the body consists of the same material points, b) is an additive quantity: $M(V)=M(V_{1})+M(V_{2})$ ${\ displaystyle M (V) = M (V_ {1}) + M (V_ {2})}$ where $V=V_{1}+V_{2}$ ${\ displaystyle V = V_ {1} + V_ {2}}$ .
The law of conservation of momentum (change in the amount of motion).
The law of conservation of angular momentum (changes in angular momentum).
The law of conservation of energy (the first law of thermodynamics).
The existence of absolute temperature (the third law of thermodynamics).
The law of entropy balance (second law of thermodynamics).

In non-classical models of continuum mechanics, these axioms can be replaced by others. For example, instead of the first two axioms, the relevant provisions of the theory of relativity ^[4] can be used.

Notes

↑ Ilyushin, 1978 , p. five.
↑ Ishlinsky, 1985 , p. 473.
↑ Truesdell, 1975 , p. 33.
↑ Gorshkov A.G. , Rabinsky L.N., Tarlakovsky D.V. Fundamentals of tensor analysis and continuum mechanics. - M .: Science, 2000. - 214 p. - ISBN 5-02-002494-5 .

Literature

Baranov A. A., Kolpashchikov V. L. Relativistic thermomechanics of continuous media . - Minsk: Science and technology, 1974. - 152 p.
Dimitrienko Yu. I. Nonlinear continuum mechanics . - M .: Fizmatlit, 2009. - 624 p. - ISBN 978-5-9221-1110-2 .
Ilyushin A. A. Mechanics of continuous medium. - M .: Publishing House of Moscow. Univ., 1978. - 287 p.
Ishlinsky A. Yu. Mechanics: Ideas, tasks, applications. - M .: Science, 1985. - 624 p.
Kolarov D., Baltov A., Boncheva N. Mechanics of plastic media. - M .: Science, 1979. - 302 p.
Loitsyansky L. G. Mechanics of fluid and gas. - M .: Drofa, 2003. - 840 p. - ISBN 5-7107-6327-6 .
Sedov L.I. Mechanics of continuum. Volume 1 . - M .: Science, 1970. - 492 p.
Sedov L.I. Mechanics of continuum. Volume 2 . - M .: Science, 1970. - 568 p.
Trusdell K. The initial course of rational continuum mechanics . - M .: Science, 1975. - 592 p.
Black L. T. Relativistic models of continuous media. - M .: Science, 1983. - 288 p.

[_91363cd9528b802c-1] Ilyushin, 1978 , p. five.

[_bba41dce21898574-2] Ishlinsky, 1985 , p. 473.

[_a7529c28a74b9fd3-3] Truesdell, 1975 , p. 33.

[4] Gorshkov A.G. , Rabinsky L.N., Tarlakovsky D.V. Fundamentals of tensor analysis and continuum mechanics. - M .: Science, 2000. - 214 p. - ISBN 5-02-002494-5 .