Elements of continuum mechanics. Elements of continuum mechanics Electric field in dielectrics
Plan
1. The concept of a continuous medium. General properties liquids and gases. Ideal and viscous liquid. Bernoulli equation. Laminar and turbulent flow liquids. Stokes formula. Poiseuille formula.
2. Elastic stresses. Energy of elastically deformed body.
Abstracts
1. The volume of a gas is determined by the volume of the vessel that the gas occupies. In liquids, unlike gases, the average distance between molecules remains almost constant, so the liquid has an almost constant volume. In mechanics, with a high degree of accuracy, liquids and gases are considered as continuous, continuously distributed in the part of space occupied by them. The density of a liquid depends little on pressure. The density of gases depends on pressure significantly. It is known from experience that in many problems the compressibility of a liquid and a gas 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, i.e., 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. It is opposed by a viscous liquid. Physical quantity, determined by the normal force acting from the side of the liquid per unit area, is called pressure R liquids. The unit of pressure is pascal (Pa): 1 Pa is equal to the pressure created by a force of 1 N, uniformly distributed over a surface normal to it with an area of 1 m 2 (1 Pa \u003d 1 N / m 2). Pressure at equilibrium of liquids (gases) obeys Pascal's law: the pressure in any place of a fluid at rest is the same in all directions, and the pressure is equally transmitted throughout the volume occupied by the fluid at rest.
Pressure changes linearly with altitude. Pressure P= rgh called hydrostatic. The pressure force on the lower layers of the liquid is greater than on the upper ones, therefore, a buoyant force acts on a body immersed in a liquid, determined by the law of Archimedes: a body immersed in a liquid (gas) is affected by an upward buoyant force equal to the weight liquid (gas) displaced by the body, where r is the density of the liquid, V is 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 fluid velocity vector at the corresponding points in space (Fig. 45). According to the pattern of streamlines, one can judge the direction and modulus of velocity in different points space, i.e., it is possible to determine the state of fluid motion. The part of the fluid bounded by streamlines is called the streamtube. The flow of a fluid 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 with time.
Consider any tube of current. We 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 cross 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 current tube is a constant value for this current tube. The relation is called the continuity equation for an incompressible fluid. - Bernoulli's equation - an expression of the law of conservation of energy in relation to the steady flow of an ideal fluid ( here r - static pressure (fluid pressure on the surface of the body flown around by it), the value is dynamic pressure, is hydrostatic pressure). For a horizontal current tube, the Bernoulli equation is written as , where left side called total pressure. - Torricelli's formula
Viscosity is the property of real liquids to resist the movement of one part of the liquid 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 internal friction force F is the greater, the larger the considered layer surface area S, and depends on how quickly the fluid flow velocity changes during the transition from layer to layer. The Dv/Dx value shows how quickly the speed changes when moving from layer to layer in the direction X, perpendicular to the direction of motion of the layers, and is called the velocity gradient. Thus, the modulus of the internal friction force is , where the coefficient of proportionality h , depending on the nature of the liquid is called dynamic viscosity(or just viscosity). The unit of viscosity is pascal second (Pa s) (1 Pa s \u003d 1 N s / m 2). The greater the viscosity, the more the liquid differs from the ideal one, the greater the forces of internal friction appear in it. 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 is especially dependent on temperature. Viscosity determination methods:
1) Stokes formula; 2) Poiseuille formula
2. The deformation is called elastic if, after the termination of the action of external forces, the body takes on its original dimensions and shape. Deformations that persist in the body after the termination of the action of external forces are called plastic. The force acting per unit cross-sectional area is called stress and is measured in pascals. A quantitative measure that characterizes the degree of deformation experienced by a body is its relative deformation. Relative change in the length of the rod (longitudinal deformation), relative transverse tension (compression), where d- rod diameter. Deformations e and e " always have different signs, where m is a positive coefficient depending on the properties of the material, called Poisson's ratio.
Robert Hooke experimentally found that for small deformations, elongation e and stress s are directly proportional to each other: , where the proportionality factor E is called Young's modulus.
Young's modulus is determined by the stress causing a relative elongation equal to one. Then Hooke's law can be written as k- coefficient of elasticity:elongation of the rod under elastic deformation is proportional to the acting on rod of strength. Potential energy of an elastically stretched (compressed) rod solids obey Hooke's law only for elastic deformations. The relationship between strain and stress is presented in the form of a stress diagram (Fig. 35). It can be seen from the figure that linear dependence s (e), established by Hooke, is carried out 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 residual deformations do not occur up to the elastic limit (s y). Beyond the elastic limit, residual deformations occur in the body and the graph describing the return of the body to its original state after the termination of the force will not be displayed as a curve VO, a parallel to it CF. The stress at which a noticeable residual deformation appears (~ \u003d 0.2%) is called the yield strength (s t) - point FROM on the curve. In the region of CD the deformation increases without increasing stress, i.e., the body, as it were, “flows”. This region is called the yield region (or plastic deformation region). Materials for which the yield region is significant are called viscous, for which it is practically absent - brittle. With further stretching (beyond the point D) body is destroyed. The maximum stress that occurs in the body before failure is called the ultimate strength (s p).
Plan
1. Elements of continuum mechanics. Stationary motion of an ideal fluid. Bernoulli equation.
2. Elastic stresses. Hooke's law.
Abstracts
1. The volume of a gas is determined by the volume of the vessel that the gas occupies. In liquids, unlike gases, the average distance between molecules remains almost constant, therefore the liquid has an almost constant volume. In mechanics, with a high degree of accuracy, liquids and gases are considered as continuous, continuously distributed in the part of space occupied by them. The density of a liquid depends little on pressure. The density of gases depends on pressure significantly. It is known from experience that in many problems the compressibility of a liquid and a gas 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, i.e., 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. A viscous fluid contradicts it. The physical quantity determined by the normal force acting from the liquid per unit area is called 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 \u003d 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 pressure force on the lower layers of the liquid is greater than on the upper ones, therefore, a buoyant force acts on a body immersed in a liquid, determined by Archimedes' law: on a body immersed in a liquid (gas), an upward buoyant force acts from this liquid, equal to the weight of the liquid (gas) displaced by the body, where r is the density of the liquid, V is 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 fluid velocity vector at the corresponding points in space (Fig. 45). From the pattern of streamlines, one can judge the direction and modulus of velocity at different points in space, i.e., one can determine the state of fluid motion. The part of the fluid bounded by streamlines is called the streamtube. The flow of a fluid 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 with time.
Consider any tube of current. We 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 cross 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 current tube is a constant value for this current tube. The relation is called the continuity equation for an incompressible fluid. 
- Bernoulli equation - expression of the law of conservation of energy as applied to the steady flow of an ideal fluid (here r - static pressure (fluid pressure on the surface of the body flown around by it), the value is dynamic pressure, is hydrostatic pressure). For a horizontal current tube, the Bernoulli equation is written as 
, where left side called total pressure. Toricelli's formula is written:
Viscosity is the property of real liquids to resist the movement of one part of the liquid 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 internal friction force F is the greater, the larger the considered layer surface area S, and depends on how quickly the fluid flow velocity changes during the transition from layer to layer. The Dv/Dx value shows how quickly the speed changes when moving from layer to layer in the direction X, perpendicular to the direction of motion of the layers, and is called the velocity gradient. In this way, modulus of internal friction force is , where the coefficient of proportionality h , which depends on the nature of the liquid, is called dynamic viscosity (or simply viscosity). Viscosity unit- pascal second (Pa s) (1 Pa s \u003d 1 N s / m 2). The greater the viscosity, the more the liquid differs from the ideal one, the greater the forces of internal friction appear in it. 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 is especially dependent on temperature. Viscosity determination methods:
1) Stokes formula 
; 2) Poiseuille formula
2. The deformation is called elastic if, after the termination of the action of external forces, the body takes on its original dimensions and shape. Deformations that persist in the body after the termination of the action of external forces are called plastic. The force acting per unit cross-sectional area is called stress and is measured in pascals. A quantitative measure that characterizes the degree of deformation experienced by a body is its relative deformation. Relative change in the length of the rod (longitudinal deformation), relative transverse tension (compression), where d- rod diameter. Deformations e and e " always have different signs, where m is a positive coefficient depending on the properties of the material, called Poisson's ratio.
Robert Hooke experimentally found that for small deformations, elongation e and stress s are directly proportional to each other: , where the proportionality factor E is Young's modulus.
Young's modulus is determined by the stress causing a relative elongation equal to one. Then Hooke's law can be written like this 
, where k- coefficient of elasticity: 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 of solid bodies 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 residual deformations do not occur up to the elastic limit (s y). Beyond the elastic limit, residual deformations occur in the body and the graph describing the return of the body to its original state after the termination of the force will not be displayed as a curve VO, a parallel to it CF. The stress at which a noticeable residual deformation appears (~ \u003d 0.2%) is called the yield strength (s t) - point FROM on the curve. In the region of CD the deformation increases without increasing stress, i.e., the body, as it were, “flows”. This region is called the yield region (or plastic deformation region). Materials for which the yield region is significant are called viscous, for which it is practically absent - brittle. With further stretching (beyond the point D) body is destroyed. The maximum stress that occurs in the body before failure is the tensile strength (s p).
Under the action of 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 termination of the action of forces, and the bodies restore their shape and volume.
Plastic deformations are called deformations that persist after the termination of the action of 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 experience only elastic deformations under the action of any forces are called perfectly elastic.
For such bodies, there is an unambiguous relationship between the acting forces and the elastic deformations caused by them.
We restrict ourselves to elastic deformations, which obey the law Hooke.
All solids can be divided into isotropic and anisotropic.
Isotropic bodies are bodies whose physical properties are the same in all directions.
Anisotropic bodies are bodies whose physical properties 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 the cubic system behave as isotropic if light propagates through them, but they are anisotropic if their elastic properties are considered.
In what follows, we confine ourselves to the study of isotropic bodies.
The most widespread in nature are metals with a polycrystalline structure.
Such metals consist of many tiny randomly oriented crystals.
As a result of plastic deformation, the randomness in the orientation of crystals can be broken.
After the termination of the action of 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 of 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 is stretched, then the resulting stresses are called tension
If the rod is compressed, then the resulting stresses are called pressure:
.
(7.2)
Consequently,
T = - R. (7.3)
If 
- the length of the undeformed rod, then after the application of force, it receives an elongation 
.
Then the length of the rod
. (7.4)
Attitude 
to 
, is called relative elongation, i.e.
. (7.5)
Based on experiments, Hooke established the 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 termination of the force, the length of the rod returns to its original (elastic deformation region).
Beyond the limits of elasticity, the deformation becomes partially or completely irreversible (plastic deformation). For most solids, linearity is maintained almost to the elastic limit. If the body continues to stretch, it will collapse.
The maximum force that can be applied to a body without breaking it is called tensile strength(P. 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 section.
,
(7.8)
where 
is the unit vector of the normal to the area dS.
.
(7.9)
To determine the mechanical stress in the medium, on an oppositely oriented site, at any point, it is enough to set the 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 is the area of the ABC face; n is the outer normal to it.
Consequently, the stress at each point of an elastically deformed body can be characterized by three vectors 
or nine of their projections on the X, Y, Z coordinate axes:
(7.12)
who are called elastic stress tensor.
| Parameter name | Meaning |
| Article subject: | ELEMENTS OF CONTINUOUS MEDIA MECHANICS |
| Rubric (thematic category) | Metals and Welding |
AND CLASSIFICATION OF DRILLING METHODS
ROCK DESTRUCTION METHODS
the main and most widely used method of rock destruction during well drilling is currently mechanical. In this method, rock cutting tools are drill bits and crowns. The rock cutting tool is rotated in several ways: rotary, turbine and with the help electric drill- all these methods are a kind rotational method, in which the formation of a well occurs due to the continuous rotation of the bit and its penetration into the rock under the action of an axial load.
In addition to the rotational method, there is impact method- here the well is formed due to the destruction of the rock under the impact of a wedge-shaped bit. The combination of rotary and percussion drilling methods creates combined method(shock-rotational).
The destruction of the rock is carried out as follows:
1. By cutting - during rotary drilling with chisels and crowns of the cutting type.
2. Crushing - during percussive drilling with wedge-shaped bits and during rotary drilling - with cone bits of "pure" rolling.
3. By shearing - during rotary drilling of a well with cone bits of a shearing type.
4. Abrasion - during rotary drilling with bits of cutting and cone type at low specific loads on the bit and a large number revolutions.
Mechanical properties of a solid body- these are its specific features, manifested during mechanical processes, due to the nature and internal structure body.
Deformation It is customary to call the process of changing the size or shape of a solid body under the action of external forces.
Deformation - it is the relative amount of change in the size or shape of the body.
The resistance of the body to deformation at the considered point is usually characterized by the ratio:
where is the resultant of internal forces on the elementary area of the section,
The area over which the forces act
Voltage at a point (vector value).
elastic (reversible) deformation will be in the event that when the external forces are removed, the dimensions and shape of the body are completely restored. In this case internal forces perform work equal to the work of external forces, opposite in sign.
Plastic (irreversible) deformation will be in the event that when the external forces are removed, the dimensions and shape of the body are not restored. In this case, of course, the work spent on deforming the body is greater than the work of restoration.
Body destruction occurs when, in the process of its deformation, there is a break in the bonds that cause the solid body itself.
In the absence of irreversible deformation in the process of destruction of a solid body, the destruction is usually called fragile.
Plastic destruction of the body is characterized by significant irreversible deformation.
strength It is customary to call the ability of a solid body to resist destruction from the action of external forces. The strength of solids is characterized by the magnitude of ultimate stresses in the dangerous section of the body.
The behavior of a deformed solid must be described by the field test method, the model testing method, and the calculation method.
It should be noted that there is no exact mathematical description of the state of a solid body, which makes it difficult to analytically characterize the mechanical properties of rocks.
The natural test method is a reliable, but time-consuming, method for testing models is carried out using the theory of similarity and simulation in mechanics. The third method (calculation) is the least time-consuming and the least accurate.
For various groups of bodies, idealized mathematical models, which includes only the most essential features groups.
The main models include:
1. elastic body, or Hooke's body (is deformed elastically until failure).
2. A plastic body, or a San Venant body (it deforms elastically up to the limit stress, and then it deforms plastically under a constant load).
3. A viscous body, or Newton's body (it deforms like a viscous fluid).
In accordance with the models, groups of elastic, plastic, rheological (viscous) and strength properties are distinguished.
The considered methods cannot replace the extreme importance of studying the essence of the processes of deformation and destruction of solids (experiments and forecasting methods are necessary).
ELEMENTS OF CONTINUOUS MEDIA MECHANICS - concept and types. Classification and features of the category "ELEMENTS OF CONTINUOUS MEDIA MECHANICS" 2017, 2018.
