Everything that happens in nature is based on the action of various forces - the law of the thief is a confirmation. This is one of the fundamental phenomena of science.

This process is the determining link of compression processes, bends, stretching and other modifications of materials of various structures.

Let's discern what this law is how to apply the thrust rule in practice, and whether it is always performed.

Definition and Formula of the Dungal Law

Long people tried to explain the origin of compression and stretching phenomena. The lack of knowledge caused the accumulation of experimental data. Actually, his theorem English Testor GUK opened from his observations and experiments. Only later, after the death of a scientist, contemporaries will call them an axiom - the law of the throat.

The researcher noticed that with each elastic impact on the object, the force appears, which returns it into the original form. This was the beginning of experiments.

Axom of the throat says:

With very small elastic impacts, force is created proportional to the change in the object, but the opposite sign at the absolute value of the movement of its particles.

Mathematically, this definition can be written as follows:

F X.= F. aplative= — k * X.,

where in the left part indicates:

force acting on the body;

x. - displacement of the body (m);

k. - The deformation coefficient, depending on the properties of the object.

Unit of measurement, like any other force, is Newton.

By the way, k. It is also called the rigidity of the body, it is measured in H / m. The rigidity is due not to the external parameters of the object, but depends on its material.

True, it is worth considering that his law is fair only for elastic deformations.

The power of elasticity

The wording is based on the definition of the force of elasticity. What is its difference from other impacts on the body?

In fact, the force of elasticity can occur at any point of the body with its elastic deformation. What is understood by such an impact? This change in the body shape, in which the object is returned to the source view after a certain period of time.

And this in turn occurs due to the molecular effect of particles: with any deformation, the distance between the molecules of the object occurs, and the Coulomb forces of attraction or repulsion seek to return the body to its original position.

The simplest model demonstrating the action of the forces of elasticity is a spring pendulum.

What formula expresses axioms established by scientists in this case?

Here, the thrust of the bitter will be recorded in the form:

ε = α * S.,

where ε is the relative elongation of the body (its value is equal to the ratio of elongation to movement);

α is a proportionality coefficient (inversely proportional to the Young E module);

S is the mechanical voltage of the object (its value is equal to the ratio of the force of elasticity to the area of \u200b\u200bthe body cross section).

Considering the foregoing, the equation can be written like this:

Δx./ x.= F. aplative/ E * S.,

where Δx is the maximum shift during deformation.

It is worth converting this expression, then we obtain the following:

F. aplative = (E * S./ x.) Δx.= k * ΔX.

Since the strength of elasticity is opposite to external influence, then briefly the law is read in this way:

F. aplative= — k * ΔX.

It is not in vain in vain are small in the magnitude of the deformation: with them Δx ̴ x, therefore, F is \u003d - k * x.

Under what conditions the law is performed

Now let's see what the boundaries of the applicability of this expression, and in what conditions it is generally performed.

It should be known that the main condition is:

s.= E * E.,

where to the left in the equation is a voltage arising during deformation, and in the right part of the Jung and extension module.

Moreover, E depends on the characteristics of the object particles, but not from its form parameters, and the second factor is taken by the module.

In general, the axiom of the throat is valid for many situations.

So, with an elastic bending of the spring lying on two supports, the mathematical record of the theorem looks like this:

F. aplative= — m * G.

F. aplative= — k * X.

In other situations (with a crash, various pendulums and other deforming processes), the impact of the forces on the object is similar.

How to apply the law of elastic deformation in practice

This law (generalized for many situations) is basic in dynamics and statics of bodies, therefore its applicability is carried out in areas where it is necessary to calculate the rigidity and voltages of deformation of objects.

First of all, the hand rule must be applied in construction and technician. So, the workers should know exactly what maximum cargo can raise the tower crane or what load will withstand the foundation of the future building.

None of the trains do without deformation of stretching and compression, so the bouquet law is valid for these situations. In addition, the mechanism and principle of operation of any dynamometers that are equipped with some parts of the technical equipment are also based on this wonderful law.

The bike law is performed in all objects that are analogues of the Spring Pendulum model.

In ordinary life, at home, you can see the applicability of this law in the springs of some mechanisms.

Thus, the law of the throat is applicable in many spheres of human life. It is one of the basic phenomena, which keeps the existence of his life on the planet.

Conclusion

Summing up, it should be noted that the law of a bitch is a universal assistant in tasks with solutions for deformation of objects not only in student books on the rustic, but also in various engineering areas.

It is these simple tasks that help scientists and masters create new technical models necessary in the context of modern technical progress.

The law of the knuckles was opened in the XVII century by the Englishman Robert Ducky. This discovery of springs tension is one of the laws of the theory of elasticity and performs an important role in science and technology.

Definition and Formula of the Dungal Law

The wording of this law is as follows: the force of elasticity, which appears at the time of deformation of the body is proportional to the elongation of the body and is directed opposite to the movement of particles of this body relative to other particles during deformation.

The mathematical record of the law looks like this:

Fig. 1. Formula of the Dungal Law

where Fupr.- accordingly the force of elasticity, x. - body lengthening (the distance to which the original body length changes), and k. - The coefficient of proportionality, called the rigidity of the body. The force is measured in Newton, and the elongation of the body is in meters.

To disclose the physical meaning of rigidity, it is necessary to substitute a unit in the formula for the law of the thief, in which the elongation is measured - 1 m, having received an expression for k in advance.

Fig. 2. Body rigidity formula

This formula shows that the rigidity of the body is numerically equal to the strength of elasticity, which occurs in the body (spring), when it is deformed by 1 m. It is known that the rigidity of the spring depends on its shape, size and material from which this body produced.

The power of elasticity

Now that it is known which formula expresses the law of the throat, it is necessary to understand its primary value. The primary value is the force of elasticity. It appears at a certain point when the body begins to deform, for example, when the spring is compressed or stretched. It is aimed at the opposite direction from gravity. When the strength of elasticity and the strength of gravity acting on the body becomes equal, support and body stop.

The deformation is irreversible changes that occur with the sizes of the body and its shape. They are associated with the movement of particles relative to each other. If a person is sitting in a soft chair, a deformation will occur with the chair, that is, its characteristics will change. It happens different types: bending, stretching, compression, shift, twist.

Since the strength of elasticity refers to its origin to electromagnetic forces, it should be known that it arises due to the fact that molecules and atoms are the smallest particles from which all the bodies are attracted to each other and repel each other. If the distance between the particles is very little, it means that the repulsion force affects them. If this distance is to increase, then the force of attraction will be valid. Thus, the difference of the attraction and repulsion forces is manifested in the forces of elasticity.

The strength of elasticity includes the power of the reaction of the support and weight of the body. The reaction force is of particular interest. This is such a force that acts on the body when it is put on some surface. If the body is suspended, then the force acting on it is called the force of tensioning the thread.

Features of the strength of elasticity

As we have already found out, the strength of elasticity occurs during deformation, and it is aimed at restoring the initial forms and sizes is strictly perpendicular to the deformable surface. The forces of elasticity also have a number of features.

  • they arise during deformation;
  • they appear in two deformable bodies at the same time;
  • they are perpendicular to the surface, with respect to which the body is deformed.
  • they are opposite to the direction of bodies of body particles.

Application of the Law in Practice

The bike law is used both in technical and high-tech devices and in nature itself. For example, the strengths of elasticity are found in clock mechanisms, in shock absorbers in transport, in ropes, rubber bands, and even in human bones. The principle of the Law of the Thick is the basis of a dynamometer - the device with which the force is measured.

If on the body to affect some force, then its size and (or) the form changes. This process is called deformation of the body. In bodies subjected to deformations, the forces of elasticity arise that balancing external forces.

Types of deformation

All deformations can be divided into two types: elastic deformations and plastic.

Definition

Elastic Called deformation if, after removing the load, the former sizes of the body and its form are completely restored.

Definition

Plastic They consider the deformation at which that appeared, due to deformation, changes in size and body shape, after removal of the load are partially restored.

The character of deformation depends on

  • the values \u200b\u200band time of exposure to external load;
  • body material;
  • body states (temperature, processing methods, etc.).

There is no sharp boundary between elastic and plastic deformations. In a large number of cases, small and short-term deformations can be considered elastic.

The wording of the law of a gouge

It is empirically obtained that the greater deformation it is necessary to obtain, the greater deforming force should be attached to the body. The magnitude of the deformation ($ \\ delta l $) can be judged by the amount of force:

\\ [\\ Delta L \u003d \\ FRAC (F) (K) \\ Left (1 \\ Right), \\]

the expression (1) means that the absolute value of the elastic deformation is directly proportional to the applied force. This statement is the content of the bike law.

In deformation of the elongation (compression) of the body, equality is performed:

where $ F $ is a deforming force; $ L_0 $ - the initial length of the body; $ l $ - body length after deformation; $ k $ - coefficient of elasticity (stiffness coefficient, rigidity), $ \\ left \u003d \\ frac (H) (m) $. The coefficient of elasticity depends on the material of the body, its size and form.

Since the strengths of elasticity ($ F_U $) arise in a deformed body, which seek to restore the former sizes and shape of the body, then often the law of the throat formulates relative to the forces of elasticity:

The bike law works well for deformations that occur in rods made of steel, cast iron, and other solids, in springs. Fair the leg of the thread for stretching and compression deformations.

Law Thick for Small Deformations

The force of elasticity depends on the change in the distance between the parts of the same body. It should be remembered that the bike law is performed only for small deformations. For large deformations, the force of elasticity is not proportional to the measurement of length, with a further increase in the deformal effect of the body can be collected.

If the deformation of the body is small, then the forces of elasticity can be determined by acceleration, which these forces reports to bodies. If the body is still, the modulus of the strength of the elasticity is found from the equality zero vector sum of the forces that act on the body.

The leg of the bitter can be recorded not only about the forces, but it is often formulated for this value as the voltage ($ \\ sigma \u003d \\ FRAC (F) (S) $ is the force that acts on the unit cross-sectional area), then for small deformations:

\\ [\\ Sigma \u003d E \\ FRAC (\\ Delta L) (L) \\ \\ Left (4 \\ Right), \\]

where $ Е $ is a Jung module; $ \\ \\ FRAC (\\ Delta L) (L) $ is a relative elongation of the body.

Examples of tasks with the solution

Example 1.

The task. To the steel cable of a long $ L $, a diameter of $ d $ joined the load weighing $ M $. What is the voltage in the cable ($ \\ sigma $), as well as its absolute elongation ($ \\ Delta L $)?

Decision. Make a drawing.

In order to find the force of elasticity, consider the forces that act on the body suspended to the cable, since the force of elasticity will be equal to the magnitude of the tension force ($ \\ overline (n) $). On the second law of Newton we have:

In the projection on the axis y of equation (1.1) we get:

According to the third law of Newton, the body acts on a cable with a force equal to the value of $ \\ overline (n) $, the cable, acts on the body with the force of $ \\ overline (f) $, equal to $ \\ overline (\\ n,) $ but the opposite Directions, so deforming cable force ($ \\ overline (F) $) is equal to:

\\ [\\ Overline (f) \u003d - \\ OVERLINE (N \\) \\ left (1.3 \\ Right). \\]

Under the influence of deforming force in the cable, the force of elasticity arises, which is equal in size:

Voltage in the cable ($ \\ sigma $) will find as:

\\ [\\ Sigma \u003d \\ FRAC (FRU) (S) \u003d \\ FRAC (MG) (S) \\ Left (1.5 \\ Right). \\]

Size S is a cross-sectional area of \u200b\u200ba cable:

\\ [\\ Sigma \u003d \\ FRAC (4mg \\) ((\\ pi d) ^ 2) \\ left (1.7 \\ Right). \\]

By the law of the throat:

\\ [\\ Sigma \u003d E \\ FRAC (\\ Delta L) (L) \\ Left (1.8 \\ RIGHT), \\]

\\ [\\ FRAC (\\ Delta L) (L) \u003d \\ FRAC (\\ Sigma) (E) \\ To \\ Delta L \u003d \\ FRAC (\\ Sigma L) (E) \\ To \\ Delta L \u003d \\ FRAC (4mgl \\) ((\\ pi d) ^ 2e). \\]

Answer. $ \\ sigma \u003d \\ FRAC (4mg \\) ((\\ pi d) ^ 2); \\ \\ Delta L \u003d \\ FRAC (4mgl \\) ((\\ pi d) ^ 2e) $

Example 2.

The task. What is the absolute deformation of the first spring of the two successively connected springs (Fig. 2), if the coefficients of the springs are equal to: $ k_1 \\ and \\ k_2 $, and the elongation of the second spring is $ \\ delta x_2?

Decision. If a system of sequentially connected springs is in equilibrium state, then the power of the data of the springs is the same:

By the law of the throat:

According to (2.1) and (2.2), we have:

Express the first spring lengthening from (2.3):

\\ [\\ Delta X_1 \u003d \\ FRAC (k_2 \\ Delta x_2) (k_1). \\]

Answer. $ \\ Delta X_1 \u003d \\ FRAC (K_2 \\ Delta x_2) (k_1) $.

Definition

Deformations You are called any changes in the shape, size and volume of the body. The deformation determines the final result of the movement of parts of the body relative to each other.

Definition

Elastic deformations Called deformations that are completely disappearing after eliminating external forces.

Plastic deformations Called deformations, fully or partially persist after the cessation of external forces.

The ability to elastic and plastic deformations depends on the nature of the substance from which the body consists of the conditions in which it is located; ways to manufacture it. For example, if you take different grades of iron or steel, then they can detect completely different elastic and plastic properties. At ordinary room temperatures, iron is very soft, plastic material; Hardened steel, on the contrary, is solid, elastic material. The plasticity of many materials is a condition for processing them, for the manufacture of the necessary parts from them. Therefore, it is considered one of the most important technical properties of a solid.

During the deformation of the solid, the particles (atoms, molecules or ions) are displaced from the initial equilibrium positions into new provisions. This changes the power interactions between individual particles of the body. As a result, in the deformed body arises internal forces that prevent its deformation.

There are deformations of stretching (compression), shear, bending, twisting.

Forces of elasticity

Definition

Forces of elasticity - These are the forces arising in the body with its elastic deformation and directed towards the opposite displacement of particles during deformation.

Elastic strengths have electromagnetic nature. They interfere with deformations and are directed perpendicular to the surface of contacting interacting bodies, and if such bodies interact as springs, threads, the strengths of elasticity are directed along their axis.

The force of elasticity acting on the body from the support is often called the power of the support reaction.

Definition

Stretching deformation (linear deformation) - This is a deformation at which only one linear body size changes. Its quantitative characteristics are absolute and relative elongation.

Absolute elongation:

where and the length of the body in a deformed and undeformed state, respectively.

Relative extension:

Law Guka.

Small and short-term deformations with a sufficient degree of accuracy can be considered as elastic. For such deformations, the bouquet law is valid:

where the projection of the force on the axis of the rigidity of the body, depending on the size of the body and the material from which it is manufactured, the stiffness unit in the system SI N / M.

Examples of solving problems

Example 1.

The task Spring rigidity N / M in an unloaded state has a length of 25 cm. What will be the length of the spring, if it is cut to the load weighing 2 kg?
Decision Make a drawing.

The load suspended on the spring, the strength of elasticity acts.

Designing this vector equality to the coordinate axis, we get:

By the law of the thief force of elasticity:

therefore, you can write:

where did the deformal spring length:

We translate into the system the length of the non-deformed spring cm m.

Substituting in the formula numerical values \u200b\u200bof physical quantities, calculate:
Answer The length of the deformed spring will be 29 cm.

Example 2.

The task The horizontal surface move the body weighing 3 kg using the rigidity of N / m. How long the spring is lengthened, if under its action with an equilibrium movement for 10 with the body rate changed from 0 to 20 m / s? Friction neglect.
Decision Make a drawing.

The body is valid, the reaction force of the support and the strength of the elasticity of the spring.

Ministry of Education Ar Crimea

Tavrichesky National University. Vernadsky

Research of the physical law

Law Guka.

Completed: Student 1 course

physical faculty gr. F-111.

Potapov Eugene

Simferopol-2010.

Plan:

    The relationship between what phenomena or values \u200b\u200bexpresses the law.

    Formulation of the Law

    Mathematical expression of the law.

    How the law was opened: on the basis of experimental data or theoretically.

    Experienced facts on the basis of which the law was formulated.

    Experiments confirming the justice of the law formulated on the basis of the theory.

    Examples of the use of the law and accounting of the action of the law in practice.

    Literature.

The relationship between what phenomena or values \u200b\u200bexpresses the law:

The leg of the bitter binds such phenomena as the voltage and deformation of the solid body, the module of the force of elasticity and elongation. The modulus of the force of elasticity arising during body deformation is proportional to its elongation. The elongation is the characteristic of the deformity of the material, assessed to increase the length of the sample from this material with tension. The force of elasticity is the force arising during deformation of the body and opposing this deformation. Voltage is a measure of the internal forces arising in a deformable body under the influence of external influences. Deformation - change in the mutual position of body particles associated with their movement relative to each other. These concepts are associated with the so-called stiffness coefficient. It depends on the elastic properties of the material and sizes of the body.

Formulation of the Law:

The bouquet law is the equation of the theory of elasticity, binding the stress and deformation of the elastic medium.

The wording of the law is the strength of elasticity is directly proportional to the deformation.

Mathematical expression of the law:

For a thin tensile rod, the law of the throat is:

Here F. Rod tension force, δ l. - his elongation (compression), and k. called coefficient of elasticity (or rigidity). The minus in the equation indicates that the strength of the tension is always directed towards opposite deformation.

If you enter a relative elongation

and normal voltage in cross section

t about the law of the throat will be recorded so

In this form, it is fair for any small volumes of substance.

In the general case, voltage and deformation are second grade tensors in three-dimensional space (9 components have 9). Binding their tensor of elastic constant is the fourth rank tensor C. ijkl and contains 81 coefficients. Due to the symmetry of the tensor C. ijkl As well as stress and strain tensors, independent are only 21 permanent. The bike law is as follows:

where σ. iJ. - Tensor stresses, -tores of deformations. For isotropic material tensor C. ijkl Contains only two independent coefficients.

How the law was opened: on the basis of experienced data or theoretically:

The law was opened in 1660 by the English scientist Robert Golk (huof) on the basis of observations and experiments. Opening, as the GUK claimed in his essay, "De Potentia Restitutiva", published in 1678, was made for them 18 years before that time, and in 1676 it was placed in his other book under the guise of the anatram "CeiiiInoSsttUV", meaning "UT Tensio Sic Vis" . According to the author's explanation, the above law of proportionality is applied not only to metals, but also to wood, stones, rogues, bones, glass, silly, hair, and so on.

Experienced facts based on which the law was formulated:

The story about it is silent ..

Experiments confirming the justice of the law formulated on the basis of theory:

The law is formulated on the basis of experienced data. Indeed, when stretching the body (wire) with a certain stiffness coefficient k. at a distance Δ. l, That their work will be equal to the power module, stretching the body (wire). This ratio will be performed, however, not for all deformations, but for small. With large deformations, the law of the throat ceases to act, the body is destroyed.

Examples of the use of the law and accounting of the action of the law in practice:

As follows from the law of the throat, on the extension of the spring, one can judge the strength acting on it. This fact is used to measure forces using a dynamometer - springs with a linear scale, patient to different values \u200b\u200bof forces.

Literature.

1. Internet resources: - Wikipedia website (http://ru.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%93%D1%83 % D0% Ba% D0% B0).

2. Textbook on physics Pryrickin A.V. Grade 9.

3. Textbook on physics V.A. Kasyanov Grade 10.

4. Lectures on the mechanics of Ryabushkin D.S.

The coefficient of elasticity

The coefficient of elasticity (Sometimes referred to as a bitch coefficient, a coefficient of rigidity or a rigidity of the spring) - a coefficient bonding in the arbitration of the throat of the elastic body and arising from this extension of elasticity. It is used in solid mechanics in the elasticization section. Denotes letter k.sometimes D. or c.. It has the dimension of N / M or kg / C2 (in SI), din / cm or g / c2 (in the SGS).

The coefficient of elasticity is numerically equal to the force that must be applied to the spring so that its length changed by a unit of distances.

Definition and properties

The coefficient of elasticity by definition is equal to the strength of the elasticity divided by the change in the spring length: k \u003d f e / δ l. (\\ displaystyle k \u003d f _ (\\ MathRM (E)) / \\ Delta L.) The coefficient of elasticity depends on both the properties of the material and the size of the elastic body. So, for the elastic rod, the dependence on the size of the rod (cross-sectional area S (\\ displayStyle S) and the length L (\\ displayStyle L)) can be distinguished by writing the elastic coefficient as k \u003d e ⋅ S / L. (\\ displaystyle k \u003d e \\ cdot S / L.) The value of E (\\ DisplayStyle E) is called the Young Module and, unlike the coefficient of elasticity, depends only on the properties of the rod material.

Stiffness of deformable bodies when they are connected

Parallel springs connection. Serial connection springs.

When connecting several elastically deformable bodies (hereinafter referred to as the Spring), the overall rigidity of the system will change. With a parallel compound, rigidity increases, with a sequential one, decreases.

Parallel connection

With parallel compound N (\\ displayStyle N) springs with toughs, equal to k 1, k 2, k 3 ,. . . , kn, (\\ displaystyle k_ (1), k_ (2), k_ (3), ..., k_ (n),) the rigidity of the system is equal to the amount of gods, that is, k \u003d k 1 + k 2 + k 3 + . . . + k n. (\\ displaystyle k \u003d k_ (1) + k_ (2) + k_ (3) + ... + k_ (n).)

Evidence

In a parallel connection, there is N (\\ DisplayStyle N) springs with toughness k 1, k 2 ,. . . , k n. (\\ displaystyle k_ (1), k_ (2), ..., k_ (n).) From the III of the Newton law, F \u003d F 1 + F 2 +. . . + F n. (\\ DisplayStyle F \u003d F_ (1) + F_ (2) + ... + F_ (N).) (The force F (\\ DisplayStyle F) is applied to them. At the same time, the power of F 1 is applied to the spring 1, (\\ displaystyle F_ (1),) to spring 2 Power F 2, (\\ displaystyle F_ (2),) ..., to spring N (\\ DisplayStyle N) force F n. (\\ DisplayStyle F_ (N).))

Now from the law of the throat (f \u003d - k x (\\ displaystyle f \u003d -kx), where X is elongation) withdraw: f \u003d k x; F 1 \u003d k 1 x; F 2 \u003d k 2 x; . . . ; F n \u003d k n x. (\\ displaystyle f \u003d kx; f_ (1) \u003d k_ (1) x; f_ (2) \u003d k_ (2) x; ...; f_ (n) \u003d k_ (n) x.) We will substitute these expressions into equality (1): kx \u003d k 1 x + k 2 x +. . . + k n x; (\\ displaystyle kx \u003d k_ (1) x + k_ (2) x + ... + k_ (n) x;) Reducing on x, (\\ displayStyle X,) We obtain: k \u003d k 1 + k 2 +. . . + k n, (\\ displaystyle k \u003d k_ (1) + k_ (2) + ... + k_ (n),) as required to prove.

Serial connection

With a sequential connection of N (\\ DisplayStyle N) springs with toughs, equal to k 1, k 2, k 3 ,. . . , kn, (\\ displaystyle k_ (1), k_ (2), k_ (3), ..., k_ (n),) general stiffness is determined from the equation: 1 / k \u003d (1 / k 1 + 1 / k 2 + 1 / k 3 +... + 1 / Kn). (\\ displaystyle 1 / k \u003d (1 / k_ (1) + 1 / k_ (2) + 1 / k_ (3) + ... + 1 / k_ (n)).)

Evidence

In a sequential connection, there is N (\\ DisplayStyle N) springs with the toughness k 1, k 2 ,. . . , k n. (\\ DisplayStyle k_ (1), k_ (2), ..., k_ (n).) From the bike law (F \u003d - KL (\\ DisplayStyle F \u003d -KL), where L is elongation) it follows that F \u003d k ⋅ l. (\\ displaystyle f \u003d k \\ cdot l.) The amount of elongation of each spring is equal to the total elongation of the entire compound L 1 + L 2 +. . . + L n \u003d l. (\\ displaystyle L_ (1) + L_ (2) + ... + L_ (n) \u003d l.)

On each spring there is one and the same force f. (\\ DisplayStyle F.) According to the law of the throat, F \u003d L 1 ⋅ K 1 \u003d L 2 ⋅ K 2 \u003d. . . \u003d L n ⋅ k n. (\\ displaystyle f \u003d l_ (1) \\ Cdot k_ (1) \u003d L_ (2) \\ Cdot k_ (2) \u003d ... \u003d L_ (n) \\ Cdot k_ (n).) From previous expressions, we derive: L \u003d F / k, l 1 \u003d f / k 1, l 2 \u003d f / k 2 ,. . . , L n \u003d f / k n. (\\ displaystyle l \u003d f / k, \\ quad l_ (1) \u003d f / k_ (1), \\ quad l_ (2) \u003d f / k_ (2), \\ quad ..., \\ quad l_ (n) \u003d F / k_ (n).) Substitting these expressions in (2) and dividing on F, (\\ displayStyle F,) we obtain 1 / k \u003d 1 / k 1 + 1 / k 2 +. . . + 1 / k n, (\\ displaystyle 1 / k \u003d 1 / k_ (1) + 1 / k_ (2) + ... + 1 / k_ (n),) as required to prove.

Stiffness of some deformable bodies

Rod of permanent cross section

A homogeneous constant cross section rod, elastically deforming along the axis, has a rigidity ratio

K \u003d e s l 0, (\\ displaystyle k \u003d (\\ FRAC (E \\, S) (L_ (0))),) E. - the Jung module, depending only on the material from which the rod is made; S. - cross-sectional area; L. 0 - Length.

Cylindrical twisted spring

Twisted cylindrical compression spring.

Twisted cylindrical compression spring or stretching, wound from cylindrical wire and elastically deforming along the axis, has a rigidity coefficient

K \u003d g ⋅ d d 4 8 ⋅ d f 3 ⋅ n, (\\ displaystyle k \u003d (\\ FRAC (G \\ Cdot D _ (\\ MathRM (D)) ^ (4)) (8 \\ Cdot D _ (\\ MathRM (F )) ^ (3) \\ CDOT N)),) d. - wire diameter; d. F - winding diameter (measured from the axis of the wire); n. - number of turns; G. - shift module (for conventional steel G. ≈ 80 GPa, for spring steel G. ≈ 78.5 GPa, for copper ~ 45 GPa).

Sources and notes

  1. Elastic deformation (rus.). Archived June 30, 2012.
  2. Dieter Meschede, Christian Gerthsen. Physik. - Springer, 2004. - P. 181 ..
  3. Bruno Assmann. Technische Mechanik: Kinematik und Kinetik. - Oldenbourg, 2004. - P. 11 ..
  4. Dynamics, strength of elasticity (Rus.). Archived June 30, 2012.
  5. Mechanical properties of tel (rus.). Archived June 30, 2012.

10.Art of a thread of a tension-compression. Module of elasticity (Jung module).

With axial stretching or compression to the limit of proportionality σ pr. fair a bike law, i.e. The law on directly proportional to the relationship between normal voltages and longitudinal relative deformations :


(3.10)

or

(3.11)

Here e - the proportionality coefficient in the law of the throat has the dimension of the voltage and is called module elasticity of the first kindcharacterizing the elastic properties of the material or jung module.

The relative longitudinal deformation is called the ratio of the absolute longitudinal deformation of the site

Rod to the length of this area before deformation:


(3.12)

The relative transverse deformation will be equal to:  "\u003d \u003d b / b, where B \u003d B 1 - b.

The ratio of relative transverse deformation  "to the relative longitudinal deformation , taken by the module, there is a permanent value for each material and is called the Poisson coefficient:


Determination of the absolute deformation of the site of the bar

In formula (3.11) instead and Substitute expressions (3.1) and (3.12):



From here we obtain the formula to determine the absolute elongation (or shortening) of the rod section length:


(3.13)

In formula (3.13), the work of Ea is called rigidity of a bar when stretching or compression, which is measured in the KN, or in the MN.

According to this formula, absolute deformation is determined if the longitudinal force is constant on the site. In the case when on the site the longitudinal force of the variable, it is determined by the formula:


(3.14)

where n (x) is the function of the longitudinal force along the length of the site.

11. Coefficient of transverse deformation (Poisson coefficient

12. Determination of movements during stretching compression. The law of the knuckle for the bar. Determining the movements of the cross sections of the bar

Determine the horizontal movement of the point but Bruss axis (Fig. 3.5) - U A: It is equal to the absolute deformation of the part of the bar butd.concluded between the sealing and cross section conducted through the point, i.e.

In turn, lengthening the site butd. Consists of extensions of individual cargo sites 1, 2 and 3:

Longitudinal forces on the sections under consideration:




Hence,






Then

Similarly, it is possible to determine the movement of any cross section of the bar and formulate the following rule:

moving any section j.the rod during stretching compression is defined as the sum of absolute deformations. n.freight plots concluded between considered and fixed (fixed) cross sections, i.e.


(3.16)

The hardness condition of the bar is recorded as follows:


, (3.17)

where

- The largest movement of the cross section, taken by the module from the movement of the movements; u - the allowable value of the cross section for this design or its element installed in the norms.

13. Determination of the mechanical characteristics of materials. Tensile test. Compression test.

For quantifying the basic properties of materials like


The rule is experimentally determined by the tensile diagram in the coordinates  and  (Fig. 2.9), characteristic points are noted in the diagram. We give their definition.

The greatest voltage to which the material follows the law of the throat is called limit proportionalityP . Within the law of the TANGEANS of the angle of inclination direct  \u003d f. () to the axis  is determined by the magnitude E..

Elastic properties of the material are saved to voltage  W. called the limit of elasticity. Under the limit of elasticity  W. It is understood as the greatest tension, to which the material does not receive residual deformations, i.e. After full unloading, the last point of the diagram coincides with the starting point 0.

Value  T. called the yield strength material. Under the yield strength is understood that the voltage at which the growth of deformations occurs without a noticeable increase in the load. If it is necessary to distinguish the yield strength during stretching and compression  T. accordingly is replaced by  Tr. and  TC . At high voltages  T. Plastic deformations are developing in the body of construction  P which do not disappear when removing the load.

The ratio of the maximum force that the sample is capable of withstanding, its initial cross-sectional area is the name of the strength, or time resistance, and is indicated by,  BP (with compression  Sun).

When performing practical calculations, the real diagram (Fig. 2.9) is simplified, and for this purpose various approximating charts are used. To solve problems with regard to elasticplastic properties of structural materials is most often used diagram Prandtla. According to this diagram, the voltage varies from zero to the yield strength by the leg of the throat  \u003d E. , and then with growth ,  \u003d  T. (Fig. 2.10).

The ability of materials to receive residual deformations is called plasticity. In fig. 2.9 The characteristic diagram for plastic materials was presented.


Fig. 2.10 Fig. 2.11

The opposite property of plasticity is the property fragility. The ability of the material to collapse without the formation of noticeable residual deformations. The material with this property is called fragile. Fragile materials include cast iron, high carbon steel, glass, brick, concrete, natural stones. A characteristic diagram of deformation of fragile materials is shown in Fig. 2.11.

1. What is called body deformation? How is the law of the thief?

Vakhit Shaviliev

The deformations are called any changes in the shape, size and volume of the body. The deformation determines the final result of the movement of parts of the body relative to each other.
Elastic deformations are called deformations that are completely disappearing after eliminating external forces.
Plastic deformations are called deformations, fully or partially persist after the cessation of external forces.
The forces of elasticity are the forces arising in the body with its elastic deformation and directed towards opposite the displacement of particles during deformation.
Law Guka.
Small and short-term deformations with a sufficient degree of accuracy can be considered as elastic. For such deformations, the bouquet law is valid:
The force of elasticity arising during deformation of the body is directly proportional to the absolute elongation of the body and is directed towards the opposite dispersion of body particles:
\
where F_X is the projection of the force on the X axis, K-rigidity of the body, depending on the size of the body and the material from which it is manufactured, the stiffness unit in the system SI n / m.
http://ru.solverbook.com/spravochnik/mexanika/dinamika/deformacii-sily-uprugosti/

Vary Guseva

The deformation is a change in the shape or volume of the body. Types of deformation - stretching or compression (examples: stretch elastic or squeeze, accordion), bend (the board under a person rushed, the sheet of paper was curved), twist (work with scolding, squeezing with hands), shift (when braking the car tires are deformed due to the friction force ).
Dungal Law: The strength of elasticity arising in the body in its deformation is directly proportional to the magnitude of this deformation
or
The force of elasticity arising in the body during its deformation is directly proportional to the magnitude of this deformation.
Formula of the Dungal Law: Fur \u003d KX

The law of a bitch. Can I express the formula f \u003d -kh or f \u003d kh?

⚓ otd ☸

The bouquet law is the equation of the theory of elasticity, binding the stress and deformation of the elastic medium. Opened in 1660 by the English scientist Robert Golo (huko) (English Robert Hoke). Since the bike law is recorded for small stresses and deformations, it has a form of simple proportionality.

For a thin tensile rod, the law of the throat is:
Here f the power of the tension of the rod, ΔL is its elongation (compression), and K is called the elastic coefficient (or rigidity). The minus in the equation indicates that the strength of the tension is always directed towards opposite deformation.

The coefficient of elasticity depends on both the properties of the material and the size of the rod. You can select the dependence on the sizes of the rod (cross-sectional area S and length L) explicitly, writing the elastic coefficient as
The value of E is called the Jung module and depends on the properties of the body.

If you enter a relative elongation
and normal voltage in cross section
then the bike law is recorded as
In this form, it is fair for any small volumes of substance.
[edit]
Generalized Law Guka.

In the general case, voltage and deformation are second grade tensors in three-dimensional space (they have 9 components). The binding their tensor of elastic constants is the fourth grade tensor Cijkl and contains 81 coefficients. Due to the symmetry of Cijkl tensor, as well as stress and strain strain, independent are only 21 permanent. The bike law is as follows:
For isotropic material, the Cijkl tensor contains only two independent coefficients.

It should be borne in mind that the bike law is performed only at small deformations. Upon exceeding the limit of proportionality, the connection between stresses and deformations becomes nonlinear. For many environments, the bike law is not applicable even with small deformations.
[edit]

in short, you can also, and so, depending on what you want to indicate in the end: just the module of the thickness of the throat or also the direction of this force. Strictly speaking, of course, -kx, since the thickness of the throat is directed against the positive increase in the coordinate of the spring end.