Elements of continuum mechanics and conservation laws. Elements of continuum mechanics
Plan
1. Elements of mechanics continuous media... Stationary motion of an ideal fluid. Bernoulli's equation.
2. Elastic stresses. Hooke's Law.
Abstracts
1. The volume of the gas is determined by the volume of the vessel that the gas occupies. In liquids, unlike gases, the average distance between molecules remains practically constant, therefore the liquid has a practically unchanged volume. In mechanics, with a high degree of accuracy, liquids and gases are considered as continuous, continuously distributed in the part of space they occupy. The density of the liquid depends little on pressure. The density of gases depends on pressure significantly. It is known from experience that the compressibility of a liquid and a gas in many problems can be neglected and the unified concept of an incompressible liquid can be used, the density of which is the same everywhere and does not change with time. Ideal liquid - physical abstraction, that is, an imaginary fluid in which there are no internal friction forces. An ideal fluid is an imaginary fluid in which there are no internal friction forces, and a viscous fluid contradicts it. Physical quantity, determined by the normal force acting from the side of the liquid per unit area, is called the pressure R liquids... The unit of pressure is pascal (Pa): 1 Pa is equal to the pressure created by a force of 1 N, evenly distributed over a surface normal to it with an area of 1 m 2 (1 Pa = 1 N / m 2). The pressure in any place of the fluid at rest is the same in all directions, and the pressure is equally transmitted over the entire volume occupied by the fluid at rest.
Pressure changes linearly with height... Pressure P = rgh called hydrostatic. The force of pressure on the lower layers of the liquid is greater than on the upper ones; therefore, a buoyant force acts on the body immersed in the liquid, determined Archimedes' law: an upward buoyancy force acts on a body immersed in a liquid (gas), equal to weight the liquid (gas) displaced by the body, where r is the density of the liquid, V- the volume of the body immersed in the liquid.
The movement of fluids is called a flow, and the collection of particles of a moving fluid is called a flow. Graphically, the movement of fluids is depicted using streamlines, which are drawn so that the tangents to them coincide in direction with the velocity vector of the fluid at the corresponding points in space (Fig. 45). By the pattern of streamlines, one can judge the direction and modulus of velocity at different points in space, i.e., the state of fluid motion can be determined. The part of the fluid bounded by the streamlines is called a stream tube. A fluid flow is called steady (or stationary) if the shape and location of the streamlines, as well as the values of the velocities at each of its points, do not change over time.
Consider a current tube. Let's choose two of its sections S 1 and S 2 ,
perpendicular to the direction of speed (Fig. 46). If the fluid is incompressible (r = const), then through the section S 2 will pass in 1 s the same volume of liquid as through the section S 1, i.e. 
The product of the flow velocity of an incompressible fluid and the cross-section of the flow tube is a constant value for a given flow tube. The ratio is called the continuity equation for an incompressible fluid. 
- Bernoulli's equation - expression of the law of conservation of energy as applied to the steady flow of an ideal fluid (here p - static pressure (pressure of the fluid on the surface of the body flown around it), value - dynamic pressure, - hydrostatic pressure). For a horizontal stream tube, the Bernoulli equation is written as 
, where left side
called total pressure. Toricelli's formula is written:
Viscosity is the property of real fluids to resist the movement of one part of the fluid relative to another. When some layers of a real fluid move relative to others, internal friction forces arise, directed tangentially to the surface of the layers. The force of internal friction F is the greater, the larger the considered surface area of the layer S, and depends on how quickly the fluid flow rate changes when passing from layer to layer. The Dv / Dx value shows how quickly the velocity changes when passing from layer to layer in the direction NS, perpendicular to the direction of movement of the layers, and is called the velocity gradient. Thus, internal friction force modulus is equal, where the proportionality coefficient h , depending on the nature of the liquid is called dynamic viscosity(or just viscosity). Viscosity unit- pascal second (Pa s) (1 Pa s = 1 N s / m 2). The higher the viscosity, the more the liquid differs from the ideal, the greater the forces of internal friction arise in it. The viscosity depends on temperature, and the nature of this dependence for liquids and gases is different (for liquids it decreases with increasing temperature, for gases, on the contrary, it increases), which indicates the difference in the mechanisms of internal friction in them. The viscosity of oils depends especially strongly on temperature. Methods for determining viscosity:
1) Stokes formula 
; 2) Poiseuille's formula
2. Deformation is called elastic if, after the cessation of the action of external forces, the body assumes its original size and shape. Deformations that persist in the body after the cessation of external forces are called plastic. The force per unit of cross-sectional area is called stress and is measured in pascals. A quantitative measure characterizing the degree of deformation experienced by a body is its relative deformation. The relative change in the length of the bar (longitudinal deformation), the relative transverse tension (compression), where d - rod diameter. Deformations e and e " always have different signs, where m is a positive factor depending on material properties, called Poisson's ratio.
Robert Hooke experimentally found that for small deformations, the elongation e and the stress s are directly proportional to each other:, where the proportionality coefficient E- Young's modulus.
Young's modulus is determined by the stress causing elongation, equal to one... Then Hooke's law can be written like this 
, where k- coefficient of elasticity: the elongation of the rod under elastic deformation is proportional to the force acting on the rod. Potential energy of an elastically stretched (compressed) rod 
Deformations solids obey Hooke's law only for elastic deformations. The relationship between strain and stress is represented as stress diagrams(fig. 35). It can be seen from the figure that the linear dependence s (e), established by Hooke, is fulfilled only within very narrow limits up to the so-called proportionality limit (s p). With a further increase in stress, the deformation is still elastic (although the dependence s (e) is no longer linear) and no residual deformations arise up to the elastic limit (s y). Residual deformations occur in the body beyond the elastic limit, and the graph describing the return of the body to its original state after the cessation of the action of the force is not displayed as a curve IN, and parallel to it - CF. The stress at which a noticeable permanent deformation appears (~ = 0.2%) is called the yield point (s t) - point WITH on the curve. In the area of CD deformation increases without increasing stress, that is, the body "flows", as it were. This area is called the yield area (or plastic deformation area). Materials for which the yield area is significant are called viscous, for which it is practically absent - brittle. With further stretching (per point D) the body is destroyed. The maximum stress that occurs in the body before fracture is the ultimate strength (s p).
Under the action of the applied forces, the bodies change their shape and volume, that is, they are deformed.
For solids, deformations are distinguished: elastic and plastic.
Elastic deformations are called deformations that disappear after the cessation of the action of the forces, and the bodies restore their shape and volume.
Plastic deformations are called deformations that persist after the cessation of the action of the forces, and the bodies do not restore their original shape and volume.
Plastic deformation occurs during cold working of metals: stamping, forging, etc.
The deformation will be elastic or plastic depends not only on the properties of the material of the body, but also on the magnitude of the applied forces.
Bodies that under the action of any forces experience only elastic deformations are called perfectly elastic.
For such bodies, there is an unambiguous relationship between the acting forces and the elastic deformations they cause.
We will restrict ourselves to elastic deformations that obey the law Hooke.
All solids can be divided into isotropic and anisotropic.
Isotropic bodies are called bodies, the physical properties of which are the same in all directions.
Anisotropic bodies are called bodies, the physical properties of which are different in different directions.
The above definitions are relative, since real bodies can behave as isotropic with respect to some properties and as anisotropic with respect to others.
For example, crystals of a cubic system behave as isotropic if light propagates through them, but they are anisotropic if we consider their elastic properties.
In what follows, we restrict ourselves to the study of isotropic bodies.
The most widespread in nature are metals with a polycrystalline structure.
Such metals are composed of many tiny, randomly oriented crystals.
As a result of plastic deformation, the randomness in crystal orientation can be violated.
After the cessation of the action of the forces, the substance will be anisotropic, which is observed, for example, when the wire is pulled and twisted.
The force per unit area of the surface on which they act is called mechanical stress n .
If the stress does not exceed the elastic limit, then the deformation will be elastic.
The limiting stresses applied to the body, after the action, which it still retains its elastic properties, is called the elastic limit.
There are stresses of compression, tension, bending, torsion, etc.
If, under the action of forces applied to the body (rod), it stretches, then the resulting stresses are called tension
If the rod is compressed, then the resulting stresses are called pressure:
.
(7.2)
Hence,
T = P. (7.3)
If 
Is the length of an undeformed bar, then after the application of force it gets elongation 
.
Then the length of the rod
. (7.4)
Attitude 
To 
, is called the relative lengthening, i.e.
. (7.5)
Based on experiments, Hooke established a law: within the limits of elasticity, the stress (pressure) is proportional to the relative elongation (compression), i.e.
(7.6)
,
(7.7)
where E is Young's modulus.
Relations (7.6) and (7.7) are valid for any rigid body, but up to a certain limit.
Up to point A (elastic limit) after the cessation of the action of the force, the length of the bar returns to the initial one (the area of elastic deformation).
Outside of elasticity, deformation becomes partially or completely irreversible (plastic deformation). For most solids, linearity is maintained almost up to the elastic limit. If the body continues to stretch, then it will collapse.
The maximum force that must be applied to the body without destroying it is called ultimate strength(point B, fig. 7.1).
Consider an arbitrary continuous medium. Let it be divided into parts 1 and 2 along the surface A – a – B – b (Fig. 7.2).
If the body is deformed, then its parts interact with each other along the interface along which they border.
To determine the resulting stresses, in addition to the forces acting in the section A – a – B – b, you need to know how these forces are distributed over the cross section.
,
(7.8)
where 
Is the unit normal vector to the site dS.
.
(7.9)
To determine the mechanical stress in a medium, on an oppositely oriented site, at any point, it is enough to set stresses on three mutually perpendicular sites: S x, S y, S– passing through this point, for example, point 0 (Fig. 7.3 ).
This position is valid for a medium at rest or moving with arbitrary acceleration.
In this case
,
(7.10)
where 
(8.11)
S - face area ABC; n is the outer normal to it.
Consequently, the stress at each point of the elastically deformed body can be characterized by three vectors 
or by their nine projections on the X, Y, Z axes:
(7.12)
who call tensor of elastic stresses.
General properties of liquids and gases. Equilibrium equation and fluid motion. Incompressible fluid hydrostatics. Stationary motion of an ideal fluid. Bernoulli's equation. Ideally elastic body. Elastic stresses and deformations. Hooke's Law. Young's modulus.
Relativistic mechanics.
Galileo's principle of relativity and transformation. Experimental justification special theory relativity (SRT). Postulates of Einstein's special theory of relativity. Lorentz transformations. The concept of simultaneity. Relativity of lengths and time intervals. The relativistic law of addition of velocities. Relativistic impulse. Equation of motion of a relativistic particle. Relativistic expression for kinetic energy. The relationship of mass and energy. The ratio between the total energy and momentum of a particle. The limits of applicability of classical (Newtonian) mechanics.
The basics molecular physics and thermodynamics
Thermodynamic Systems - Ideal Gas.
Dynamic and statistical laws in physics. Statistical and thermodynamic methods for studying macroscopic phenomena.
Thermal motion of molecules. Interaction between molecules. Perfect gas. State of the system. Thermodynamic parameters of state. Equilibrium states and processes, their representation on thermodynamic diagrams. Ideal gas equation of state.
Foundations of molecular kinetic theory.
The basic equation of the molecular-kinetic theory of ideal gases and its comparison with the Clapeyron-Mendeleev equation. Average kinetic energy of molecules. Molecular kinetic interpretation of thermodynamic temperature. The number of degrees of freedom of the molecule. The law of uniform distribution of energy over the degrees of freedom of molecules. Internal energy and heat capacity of an ideal gas.
Maxwell's law for the distribution of molecules in terms of velocities and energies of thermal motion. Ideal gas in a force field. Boltzmann distribution of molecules in a force field. Barometric formula.
Effective molecular diameter. The number of collisions and the mean free path of molecules. Transfer phenomena.
Fundamentals of Thermodynamics.
Gas work when its volume changes. Quantity of heat. The first law of thermodynamics. Application of the first law of thermodynamics to isoprocesses and the adiabatic process of an ideal gas. Dependence of the heat capacity of an ideal gas on the type of process. The second law of thermodynamics. Heat engine. Circular processes. Carnot cycle, efficiency of the Carnot cycle.
3 .Electrostatics
Electric field in a vacuum.
Electric charge conservation law. Electric field. The main characteristics of the electric field: strength and potential. Tension as a gradient of potential. Calculation of electrostatic fields by the superposition method. Tension vector flow. Ostrogradsky-Gauss theorem for an electrostatic field in a vacuum. Application of the Ostrogradsky-Gauss theorem to the calculation of the field.
Electric field in dielectrics.
Free and bound charges. Types of dielectrics. Electronic and orientation polarization. Polarization. Dielectric susceptibility of a substance. Electrical displacement. Dielectric constant of the medium. Calculation of the field strength in a homogeneous dielectric.
Conductors in an electric field.
The field inside the conductor and at its surface. Distribution of charges in a conductor. Electrical capacity of a solitary conductor. Mutual capacitance of two conductors. Capacitors. Energy of a charged conductor, capacitor and conductor system. The energy of the electrostatic field. Bulk energy density.
Constant electric current
Current strength. Current density. Conditions for the existence of a current. Outside forces. The electromotive force of the current source. Ohm's law for an inhomogeneous section of an electrical circuit. Kirchhoff rules. Work and power electric current... Joule-Lenz law. The classical theory of electrical conductivity of metals. Difficulties of the classical theory.
Electromagnetism
Magnetic field in a vacuum.
Magnetic interaction of direct currents. A magnetic field. Magnetic induction vector. Ampere's law. Magnetic field of the current. Bio-Savart-Laplace law and its application to calculation magnetic field straight conductor with current. Circular current magnetic field. The law of total current (circulation of the magnetic induction vector) for a magnetic field in a vacuum and its application to the calculation of the magnetic field of a toroid and a long solenoid. Magnetic flux. Ostrogradsky-Gauss theorem for a magnetic field. Vortex nature of the magnetic field The effect of a magnetic field on a moving charge. Lorentz force. The movement of charged particles in a magnetic field. Rotation of a circuit with a current in a magnetic field. The work of moving a conductor and a circuit with a current in a magnetic field.
Electromagnetic induction.
Phenomenon electromagnetic induction(Faraday's experiments). Lenz's rule. The law of electromagnetic induction and its derivation from the law of conservation of energy. The phenomenon of self-induction. Inductance. Currents during the closing and opening of an electrical circuit containing inductance. Energy of the coil with current. The volumetric energy density of the magnetic field.
Magnetic field in matter.
The magnetic moment of atoms. Types of magnets. Magnetization. Micro and macro currents. Elementary theory dia- and paramagnetism. The total current law for a magnetic field in a substance. Magnetic field strength. The magnetic permeability of the medium. Ferromagnets. Magnetic hysteresis. Curie point. Spin nature of ferromagnetism.
Maxwell's equations.
Faraday's and Maxwell's interpretations of the phenomenon of electromagnetic induction. Bias current. The system of Maxwell's equations in integral form.
Oscillatory motion
The concept of oscillatory processes. A unified approach to vibrations of different physical nature.
Amplitude, frequency, phase of harmonic oscillations. Addition of harmonic vibrations. Vector diagrams.
Pendulum, spring weight, oscillating circuit. Free damped oscillations. Differential equation damped oscillations Damping factor, logarithmic decrement, quality factor.
Forced oscillations with sinusoidal action. Amplitude and phase for forced oscillations. Resonance curves. Forced vibrations in electrical circuits.
Waves
The mechanism of wave formation in an elastic medium. Longitudinal and transverse waves. Plane sine wave. Running and standing waves. Phase velocity, wavelength, wavenumber. One-dimensional wave equation. Group velocity and dispersion of waves. Energy ratios. Umov's vector. Plane electromagnetic waves. Polarization of waves. Energy ratios. Poynting vector. Dipole radiation. Directional pattern
8 . Wave optics
Light interference.
Coherence and monochromaticity of light waves. Calculation of the interference pattern from two coherent sources. Jung's experience. Light interference in thin films. Interferometers.
Light diffraction.
Huygens-Fresnel principle. Fresnel zone method. Rectilinear light propagation. Fresnel diffraction at a round hole. Fraunhofer diffraction at one slit. Diffraction grating as a spectral device. The concept of a holographic method for obtaining and recovering an image.
Light polarization.
Natural and polarized light. Reflection polarization. Brewster's Law. Analysis of linearly polarized light. Malus' law. Double refraction. Artificial optical anisotropy. Electro-optical and magneto-optical effects.
Dispersion of light.
Regions of normal and anomalous dispersion. Electronic theory of dispersion of light.
Quantum nature radiation
Heat radiation.
Thermal radiation characteristics. Absorption capacity. Black body. Kirchhoff's law for thermal radiation. Stefan-Boltzmann law. Energy distribution in the blackbody spectrum. Wien's displacement law. Quantum hypothesis and Planck's formula.
The quantum nature of light.
External photoelectric effect and its laws. Einstein's equation for the external photoelectric effect. Photons. Mass and momentum of a photon. Light pressure. Lebedev's experiments. Quantum and wave explanation of light pressure. Corpuscular-wave dualism of light.
The completion of a space flight is considered to be landing on the planet. To date, only three countries have learned how to return spacecraft to Earth: Russia, the United States and China.
For planets with an atmosphere (Fig. 3.19), the landing problem is reduced mainly to solving three problems: overcoming high level overloads; protection against aerodynamic heating; control of the time of reaching the planet and the coordinates of the landing point.
Rice. 3.19. Scheme of spacecraft descent from orbit and landing on a planet with atmosphere:
N- turning on the brake motor; A- spacecraft descent from orbit; M- separation of the spacecraft from the orbiting spacecraft; V- the entry of the SA into the dense layers of the atmosphere; WITH - the beginning of the operation of the parachute landing system; D- landing on the surface of the planet;
1 - ballistic descent; 2 - gliding descent
When landing on a planet without atmosphere (Fig. 3.20, a, b) the problem of protection from aerodynamic heating is removed.
Spacecraft in orbit artificial satellite planets or those approaching a planet with an atmosphere to land on it has a large amount of kinetic energy associated with the spacecraft speed and mass, and potential energy due to the spacecraft position relative to the planet's surface.
Rice. 3.20. Descent and landing of spacecraft on a planet without atmosphere:
a- descent to the planet with a preliminary exit to the waiting orbit;
b- soft landing of a spacecraft with a braking motor and a landing gear;
I - hyperbolic trajectory of approach to the planet; II - orbital trajectory;
III - trajectory of descent from orbit; 1, 2, 3 - active areas of flight during braking and soft landing
When entering the dense layers of the atmosphere, a shock wave arises in front of the bow of the spacecraft, heating the gas to a high temperature. As it sinks into the atmosphere, the SA is decelerated, its speed decreases, and the hot gas heats up the SA more and more. The kinetic energy of the apparatus is converted into heat. In this case, most of the energy is removed into the surrounding space in two ways: most of the heat is removed into the surrounding atmosphere due to the action of strong shock waves and due to heat radiation from the heated surface of the SA.
The strongest shock waves occur with a blunt nose, which is why blunt forms are used for the SA, rather than the sharp ones characteristic of flight at low speeds.
With an increase in speeds and temperatures, most of the heat is transferred to the vehicle not due to friction against compressed layers of the atmosphere, but due to radiation and convection from the shock wave.
The following methods are used to remove heat from the CA surface:
- heat absorption by the heat-shielding layer;
- radiation cooling of the surface;
- the use of carry-over coatings.
Before entering the dense layers of the atmosphere, the spacecraft trajectory obeys the laws of celestial mechanics. In the atmosphere, in addition to gravitational forces, aerodynamic and centrifugal forces that change the shape of the trajectory of its movement. The force of attraction is directed to the center of the planet, the force of aerodynamic resistance is in the direction opposite to the velocity vector, the centrifugal and lift forces are perpendicular to the direction of motion of the SA. The force of aerodynamic drag reduces the speed of the vehicle, while the centrifugal and lift forces impart acceleration to it in the direction perpendicular to its motion.
The nature of the descent trajectory in the atmosphere is mainly determined by its aerodynamic characteristics. In the absence of lifting force from the SA, the trajectory of its motion in the atmosphere is called ballistic (the trajectory of the descent of the SA spaceships series "Vostok" and "Voskhod"), and in the presence of lifting force - either gliding (CA CA Soyuz and Apollo, as well as Space Shuttle), or ricocheting (CA CA Soyuz and Apollo). Motion in a planetary orbit does not impose high requirements on guidance accuracy when entering the atmosphere, since it is relatively easy to correct the trajectory by turning on the propulsion system for braking or accelerating. When entering the atmosphere at a speed exceeding the first cosmic velocity, errors in calculations are most dangerous, since a too steep descent can lead to the destruction of the spacecraft, and too shallow - to a distance from the planet.
At ballistic descent the vector of the resultant of the aerodynamic forces is directed directly opposite to the vector of the velocity of the vehicle. Descent along a ballistic trajectory does not require control. The disadvantage of this method is the large steepness of the trajectory, and, as a consequence, the entry of the apparatus into the dense layers of the atmosphere on high speed, which leads to strong aerodynamic heating of the apparatus and to overloads, sometimes exceeding 10g - close to the maximum permissible values for a person.
At aerodynamic descent The outer casing of the vehicle has, as a rule, a conical shape, and the axis of the cone makes a certain angle (angle of attack) with the velocity vector of the vehicle, due to which the resultant of aerodynamic forces has a component perpendicular to the velocity vector of the vehicle - the lifting force. Due to the lifting force, the vehicle descends more slowly, the trajectory of its descent becomes flatter, while the braking section is stretched both in length and in time, and the maximum overloads and the intensity of aerodynamic heating can be reduced several times, compared with ballistic braking, which makes the gliding the descent for people is safer and more comfortable.
The angle of attack during descent changes depending on the flight speed and the current air density. In the upper, rarefied layers of the atmosphere, it can reach 40 °, gradually decreasing with the descent of the apparatus. This requires the presence of a gliding flight control system on the SA, which complicates and makes the apparatus heavier, and in cases where it serves to launch only equipment that is capable of withstanding higher overloads than a person, ballistic braking is usually used.
The Space Shuttle orbital stage, which, when returning to Earth, performs the function of a descent vehicle, plans the entire descent section from the entry into the atmosphere to touching the landing strip landing gear, after which the braking parachute is deployed.
After the vehicle speed decreases to subsonic in the section of aerodynamic braking, then the descent of the SA can be carried out with the help of parachutes. Parachute in dense atmosphere extinguishes the speed of the device to almost zero and ensures its soft landing on the surface of the planet.
In the rarefied atmosphere of Mars, parachutes are less effective, therefore, at the final stage of descent, the parachute is unhooked and the landing rocket motors are turned on.
The landing manned spacecraft of the Soyuz TMA-01M series, designed for landing on land, also have solid-fuel brake motors that turn on a few seconds before touching the ground to ensure a safer and more comfortable landing.
The descent vehicle of the Venera-13 station, after descending by parachute to an altitude of 47 km, dropped it and resumed aerodynamic braking. Such a descent program was dictated by the peculiarities of the atmosphere of Venus, the lower layers of which are very dense and hot (up to 500 ° C), and parachutes made of cloth would not withstand such conditions.
It should be noted that in some projects of reusable spacecraft (in particular, single-stage vertical take-off and landing, for example, the Delta Clipper), it is assumed at the final stage of descent, after aerodynamic braking in the atmosphere, to also make a parachute-free motor landing on rocket engines. The design of the descent vehicles can differ significantly from each other depending on the nature of the payload and on the physical conditions on the surface of the planet on which the landing is made.
When landing on a planet without an atmosphere, the problem of aerodynamic heating is removed, but for landing, the speed is damped using a braking propulsion system, which must operate in a programmed thrust mode, and the mass of the fuel can significantly exceed the mass of the spacecraft itself.
ELEMENTS OF CONTINUOUS MEDIA MECHANICS
A medium is considered to be continuous, for which a uniform distribution of matter is characteristic - i.e. medium with the same density. These are liquids and gases.
Therefore, in this section, we will look at the basic laws that apply in these environments.
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