Elastic force Hooke's law message. Derivation of Hooke's law for various types of deformation

The coefficient E in this formula is called Young's modulus. Young's modulus depends only on the properties of the material and does not depend on the size and shape of the body. For different materials, Young's modulus varies widely. For steel, for example, E ≈ 2·10 11 N/m 2 , and for rubber E ≈ 2·10 6 N/m 2 , that is, five orders of magnitude less.

Hooke's law can be generalized to the case of more complex deformations. For example, when bending deformation the elastic force is proportional to the deflection of the rod, the ends of which lie on two supports (Fig. 1.12.2).

Figure 1.12.2. Bend deformation.

The elastic force acting on the body from the side of the support (or suspension) is called ground reaction force. When the bodies come into contact, the support reaction force is directed perpendicular contact surfaces. That's why it's often called strength normal pressure. If a body lies on a horizontal stationary table, the support reaction force is directed vertically upward and balances the force of gravity: The force with which the body acts on the table is called body weight.

In technology, spiral-shaped springs(Fig. 1.12.3). When springs are stretched or compressed, elastic forces arise, which also obey Hooke's law. The coefficient k is called spring stiffness. Within the limits of applicability of Hooke's law, springs are capable of greatly changing their length. Therefore, they are often used to measure forces. A spring whose tension is measured in units of force is called dynamometer. It should be borne in mind that when a spring is stretched or compressed, complex torsional and bending deformations occur in its coils.

Figure 1.12.3. Spring extension deformation.

Unlike springs and some elastic materials (for example, rubber), the tensile or compressive deformation of elastic rods (or wires) obeys Hooke's linear law within very narrow limits. For metals, the relative deformation ε = x / l should not exceed 1%. With large deformations, irreversible phenomena (fluidity) and destruction of the material occur.

§ 10. Elastic force. Hooke's law

Types of deformations

Deformation called a change in the shape, size or volume of the body. Deformation can be caused by external forces applied to the body.
Deformations that completely disappear after the action of external forces on the body ceases are called elastic, and deformations that persist even after external forces have ceased to act on the body - plastic.
Distinguish tensile strain or compression(unilateral or comprehensive), bending, torsion And shift.

Elastic forces

When a solid body is deformed, its particles (atoms, molecules, ions) located at the nodes of the crystal lattice are displaced from their equilibrium positions. This displacement is counteracted by the interaction forces between particles of a solid body, which keep these particles at a certain distance from each other. Therefore, with any type of elastic deformation, internal forces arise in the body that prevent its deformation.

The forces that arise in a body during its elastic deformation and are directed against the direction of displacement of the particles of the body caused by the deformation are called elastic forces. Elastic forces act in any section of a deformed body, as well as at the point of its contact with the body causing deformation. In the case of unilateral tension or compression, the elastic force is directed along the straight line along which the external force acts, causing deformation of the body, opposite to the direction of this force and perpendicular to the surface of the body. The nature of elastic forces is electrical.

We will consider the case of the occurrence of elastic forces during unilateral tension and compression of a solid body.

Hooke's law

The connection between the elastic force and the elastic deformation of a body (at small deformations) was experimentally established by Newton's contemporary, the English physicist Hooke. The mathematical expression of Hooke's law for unilateral tension (compression) deformation has the form

where f is the elastic force; x - elongation (deformation) of the body; k is a proportionality coefficient depending on the size and material of the body, called rigidity. The SI unit of stiffness is newton per meter (N/m).

Hooke's law for one-sided tension (compression) is formulated as follows: The elastic force arising during deformation of a body is proportional to the elongation of this body.

Let's consider an experiment illustrating Hooke's law. Let the axis of symmetry of the cylindrical spring coincide with the straight line Ax (Fig. 20, a). One end of the spring is fixed in the support at point A, and the second is free and the body M is attached to it. When the spring is not deformed, its free end is located at point C. This point will be taken as the origin of the coordinate x, which determines the position of the free end of the spring.

Let's stretch the spring so that its free end is at point D, the coordinate of which is x>0: At this point the spring acts on the body M with an elastic force

Let us now compress the spring so that its free end is at point B, whose coordinate is x<0. В этой точке пружина действует на тело М упругой силой

It can be seen from the figure that the projection of the elastic force of the spring onto the Ax axis always has a sign opposite to the sign of the x coordinate, since the elastic force is always directed towards the equilibrium position C. In Fig. 20, b shows a graph of Hooke's law. The values of elongation x of the spring are plotted on the abscissa axis, and the elastic force values are plotted on the ordinate axis. The dependence of fx on x is linear, so the graph is a straight line passing through the origin of coordinates.

Let's consider another experiment.
Let one end of a thin steel wire be fixed to a bracket, and a load suspended from the other end, the weight of which is an external tensile force F acting on the wire perpendicular to its cross section (Fig. 21).

The action of this force on the wire depends not only on the force modulus F, but also on the cross-sectional area of the wire S.

Under the influence of an external force applied to it, the wire is deformed and stretched. If the stretch is not too great, this deformation is elastic. In an elastically deformed wire, an elastic force f unit arises.
According to Newton's third law, the elastic force is equal in magnitude and opposite in direction to the external force acting on the body, i.e.

f up = -F (2.10)

The state of an elastically deformed body is characterized by the value s, called normal mechanical stress(or, for short, just normal voltage). Normal stress s is equal to the ratio of the modulus of the elastic force to the cross-sectional area of the body:

s=f up /S (2.11)

Let the initial length of the unstretched wire be L 0 . After applying force F, the wire stretched and its length became equal to L. The value DL=L-L 0 is called absolute wire elongation. Size

called relative body elongation. For tensile strain e>0, for compressive strain e<0.

Observations show that for small deformations the normal stress s is proportional to the relative elongation e:

Formula (2.13) is one of the types of writing Hooke’s law for unilateral tension (compression). In this formula, the relative elongation is taken modulo, since it can be both positive and negative. The proportionality coefficient E in Hooke's law is called the longitudinal modulus of elasticity (Young's modulus).

Let us establish the physical meaning of Young's modulus. As can be seen from formula (2.12), e=1 and L=2L 0 with DL=L 0 . From formula (2.13) it follows that in this case s=E. Consequently, Young's modulus is numerically equal to the normal stress that should arise in the body if its length is doubled. (if Hooke's law were true for such a large deformation). From formula (2.13) it is also clear that in the SI Young’s modulus is expressed in pascals (1 Pa = 1 N/m2).

Tension diagram

Using formula (2.13), from the experimental values of the relative elongation e, one can calculate the corresponding values of the normal stress s arising in the deformed body and construct a graph of the dependence of s on e. This graph is called stretch diagram. A similar graph for a metal sample is shown in Fig. 22. In section 0-1, the graph looks like a straight line passing through the origin. This means that up to a certain stress value, the deformation is elastic and Hooke’s law is satisfied, i.e., the normal stress is proportional to the relative elongation. The maximum value of normal stress s p, at which Hooke’s law is still satisfied, is called limit of proportionality.

With a further increase in load, the dependence of stress on relative elongation becomes nonlinear (section 1-2), although the elastic properties of the body are still preserved. The maximum value s of normal stress, at which residual deformation does not yet occur, is called elastic limit. (The elastic limit exceeds the proportionality limit by only hundredths of a percent.) Increasing the load above the elastic limit (section 2-3) leads to the fact that the deformation becomes residual.

Then the sample begins to elongate at almost constant stress (section 3-4 of the graph). This phenomenon is called material fluidity. The normal stress s t at which the residual deformation reaches a given value is called yield strength.

At stresses exceeding the yield strength, the elastic properties of the body are restored to a certain extent, and it again begins to resist deformation (section 4-5 of the graph). The maximum value of normal stress spr, above which the sample ruptures, is called tensile strength.

Energy of an elastically deformed body

Substituting the values of s and e from formulas (2.11) and (2.12) into formula (2.13), we obtain

f up /S=E|DL|/L 0 .

whence it follows that the elastic force fуn, arising during deformation of the body, is determined by the formula

f up =ES|DL|/L 0 . (2.14)

Let us determine the work A def performed during deformation of the body, and the potential energy W of the elastically deformed body. According to the law of conservation of energy,

W=A def. (2.15)

As can be seen from formula (2.14), the modulus of the elastic force can change. It increases in proportion to the deformation of the body. Therefore, to calculate the work of deformation, it is necessary to take the average value of the elastic force , equal to half of its maximum value:

= ES|DL|/2L 0 . (2.16)

Then determined by the formula A def = |DL| deformation work

A def = ES|DL| 2 /2L 0 .

Substituting this expression into formula (2.15), we find the value of the potential energy of an elastically deformed body:

W=ES|DL| 2 /2L 0 . (2.17)

For an elastically deformed spring ES/L 0 =k is the spring stiffness; x is the extension of the spring. Therefore, formula (2.17) can be written in the form

W=kx 2 /2. (2.18)

Formula (2.18) determines the potential energy of an elastically deformed spring.

Questions for self-control:

 What is deformation?

 What deformation is called elastic? plastic?

 Name the types of deformations.

 What is elastic force? How is it directed? What is the nature of this force?

 How is Hooke's law formulated and written for unilateral tension (compression)?

 What is rigidity? What is the SI unit of hardness?

 Draw a diagram and explain an experiment that illustrates Hooke's law. Draw a graph of this law.

 After making an explanatory drawing, describe the process of stretching a metal wire under load.

 What is normal mechanical stress? What formula expresses the meaning of this concept?

 What is called absolute elongation? relative elongation? What formulas express the meaning of these concepts?

 What is the form of Hooke's law in a record containing normal mechanical stress?

 What is called Young's modulus? What is its physical meaning? What is the SI unit of Young's modulus?

 Draw and explain the stress-strain diagram of a metal specimen.

 What is called the limit of proportionality? elasticity? turnover? strength?

 Obtain formulas that determine the work of deformation and potential energy of an elastically deformed body.

As you know, physics studies all the laws of nature: from the simplest to the most general principles of natural science. Even in those areas where it would seem that physics is not able to understand, it still plays a primary role, and every smallest law, every principle - nothing escapes it.

In contact with

It is physics that is the basis of the foundations; it is this that lies at the origins of all sciences.

Physics studies the interaction of all bodies, both paradoxically small and incredibly large. Modern physics is actively studying not just small, but hypothetical bodies, and even this sheds light on the essence of the universe.

Physics is divided into sections, this simplifies not only the science itself and its understanding, but also the study methodology. Mechanics deals with the movement of bodies and the interaction of moving bodies, thermodynamics deals with thermal processes, electrodynamics deals with electrical processes.

Why should mechanics study deformation?

When talking about compression or tension, you should ask yourself the question: which branch of physics should study this process? With strong distortions, heat can be released, perhaps thermodynamics should deal with these processes? Sometimes when liquids are compressed, it begins to boil, and when gases are compressed, liquids are formed? So, should hydrodynamics understand deformation? Or molecular kinetic theory?

It all depends on the force of deformation, on its degree. If the deformable medium (material that is compressed or stretched) allows, and the compression is small, it makes sense to consider this process as the movement of some points of the body relative to others.

And since the question is purely related, it means that the mechanics will deal with it.

Hooke's law and the condition for its fulfillment

In 1660, the famous English scientist Robert Hooke discovered a phenomenon that can be used to mechanically describe the process of deformation.

In order to understand under what conditions Hooke's law is satisfied, Let's limit ourselves to two parameters:

Wednesday;
force.

There are media (for example, gases, liquids, especially viscous liquids close to solid states or, conversely, very fluid liquids) for which it is impossible to describe the process mechanically. Conversely, there are environments in which, with sufficiently large forces, the mechanics stop “working.”

Important! To the question: “Under what conditions is Hooke’s law true?”, a definite answer can be given: “At small deformations.”

Hooke's Law, definition: The deformation that occurs in a body is directly proportional to the force that causes that deformation.

Naturally, this definition implies that:

compression or stretching is small;
elastic object;
it consists of a material in which there are no nonlinear processes as a result of compression or tension.

Hooke's Law in Mathematical Form

Hooke's formulation, which we cited above, makes it possible to write it in the following form:

where is the change in the length of the body due to compression or stretching, F is the force applied to the body and causes deformation (elastic force), k is the elasticity coefficient, measured in N/m.

It should be remembered that Hooke's law valid only for small stretches.

We also note that it has the same appearance when stretched and compressed. Considering that force is a vector quantity and has a direction, then in the case of compression, the following formula will be more accurate:

But again, it all depends on where the axis relative to which you are measuring will be directed.

What is the fundamental difference between compression and extension? Nothing if it is insignificant.

The degree of applicability can be considered as follows:

Let's pay attention to the graph. As we can see, with small stretches (the first quarter of the coordinates), for a long time the force with the coordinate has a linear relationship (red line), but then the real relationship (dotted line) becomes nonlinear, and the law ceases to be true. In practice, this is reflected by such strong stretching that the spring stops returning to its original position and loses its properties. With even more stretching a fracture occurs and the structure collapses material.

With small compressions (third quarter of the coordinates), for a long time the force with the coordinate also has a linear relationship (red line), but then the real relationship (dotted line) becomes nonlinear, and everything stops working again. In practice, this results in such strong compression that heat begins to be released and the spring loses its properties. With even greater compression, the coils of the spring “stick together” and it begins to deform vertically and then completely melt.

As you can see, the formula expressing the law allows you to find the force, knowing the change in the length of the body, or, knowing the elastic force, measure the change in length:

Also, in some cases, you can find the elasticity coefficient. To understand how this is done, consider an example task:

A dynamometer is connected to the spring. It was stretched by applying a force of 20, due to which it became 1 meter long. Then they released her, waited until the vibrations stopped, and she returned to her normal state. In normal condition, its length was 87.5 centimeters. Let's try to find out what material the spring is made of.

Let's find the numerical value of the spring deformation:

From here we can express the value of the coefficient:

Looking at the table, we can find that this indicator corresponds to spring steel.

Trouble with elasticity coefficient

Physics, as we know, is a very precise science; moreover, it is so precise that it has created entire applied sciences that measure errors. A model of unwavering precision, she cannot afford to be clumsy.

Practice shows that the linear dependence we considered is nothing more than Hooke's law for a thin and tensile rod. Only as an exception can it be used for springs, but even this is undesirable.

It turns out that the coefficient k is a variable value that depends not only on what material the body is made of, but also on the diameter and its linear dimensions.

For this reason, our conclusions require clarification and development, because otherwise, the formula:

can be called nothing more than a dependence between three variables.

Young's modulus

Let's try to figure out the elasticity coefficient. This parameter, as we found out, depends on three quantities:

material (which suits us quite well);
length L (which indicates its dependence on);
area S.

Important! Thus, if we manage to somehow “separate” the length L and area S from the coefficient, then we will obtain a coefficient that completely depends on the material.

What we know:

the larger the cross-sectional area of the body, the greater the coefficient k, and the dependence is linear;
the greater the body length, the lower the coefficient k, and the dependence is inversely proportional.

This means that we can write the elasticity coefficient in this way:

where E is a new coefficient, which now precisely depends solely on the type of material.

Let us introduce the concept of “relative elongation”:

It should be recognized that this value is more meaningful than , since it reflects not just how much the spring was compressed or stretched, but how many times this happened.

Since we have already “introduced” S into the game, we will introduce the concept of normal stress, which is written as follows:

Important! The normal stress is the fraction of the deforming force on each element of the sectional area.

Hooke's law and elastic deformations

Conclusion

Let us formulate Hooke's law for tension and compression: For small compressions, normal stress is directly proportional to elongation.

The coefficient E is called Young's modulus and depends solely on the material.

The force of resistance of an elastic substance to linear stretching or compression is directly proportional to the relative increase or decrease in length.

Imagine that you grabbed one end of an elastic spring, the other end of which is fixed motionless, and began to stretch or compress it. The more you compress or stretch a spring, the more it resists this. It is on this principle that any spring scale is designed - be it a steelyard (in which the spring is stretched) or a platform spring scale (the spring is compressed). In any case, the spring resists deformation under the influence of the weight of the load, and the force of gravitational attraction of the weighed mass to the Earth is balanced by the elastic force of the spring. Thanks to this, we can measure the mass of the object being weighed by the deviation of the end of the spring from its normal position.

The first truly scientific study of the process of elastic stretching and compression of matter was undertaken by Robert Hooke. Initially, in his experiment, he did not even use a spring, but a string, measuring how much it extended under the influence of various forces applied to one end, while the other end was rigidly fixed. He managed to find out that up to a certain limit, the string stretches strictly proportionally to the magnitude of the applied force, until it reaches the limit of elastic stretching (elasticity) and begins to undergo irreversible nonlinear deformation ( cm. below). In equation form, Hooke's law is written in the following form:

Where F— elastic resistance force of the string, x- linear tension or compression, and k- so-called elasticity coefficient. The higher k, the stiffer the string and the more difficult it is to stretch or compress. The minus sign in the formula indicates that the string is resisting deformation: when stretched, it tends to shorten, and when compressed, it tends to straighten.

Hooke's law formed the basis of a branch of mechanics called theory elasticity. It turned out that it has much wider applications, since atoms in a solid behave as if they were connected to each other by strings, that is, elastically fixed in a three-dimensional crystal lattice. Thus, with slight elastic deformation of an elastic material, the acting forces are also described by Hooke’s law, but in a slightly more complex form. In the theory of elasticity, Hooke's law takes the following form:

σ /η = E

Where σ — mechanical stress(specific force applied to the cross-sectional area of the body), η - relative elongation or compression of the string, and E - so-called Young's modulus, or elastic modulus, playing the same role as the elasticity coefficient k. It depends on the properties of the material and determines how much the body will stretch or contract during elastic deformation under the influence of a single mechanical stress.

In fact, Thomas Young is much better known in science as one of the proponents of the theory of the wave nature of light, who developed a convincing experiment with splitting a light beam into two beams to confirm it ( cm. The principle of complementarity and interference), after which no one had any doubts about the correctness of the wave theory of light (although Jung was never able to fully put his ideas into a strict mathematical form). Generally speaking, Young's modulus is one of three quantities that describe the response of a solid material to an external force applied to it. The second is displacement modulus(describes how much a substance is displaced under the influence of a force applied tangentially to a surface), and the third - Poisson's ratio(describes how much a solid thins when stretched). The latter is named after the French mathematician Simeon-Denis Poisson (1781-1840).

Of course, Hooke's law, even in the form improved by Jung, does not describe everything that happens to a solid under the influence of external forces. Imagine a rubber band. If you do not stretch it too much, a return force of elastic tension will arise from the rubber band, and as soon as you release it, it will immediately come together and take its previous shape. If you stretch the rubber band further, sooner or later it will lose its elasticity, and you will feel that the tensile strength has weakened. So you have crossed the so-called elastic limit material. If you pull the rubber further, after some time it will completely break and the resistance will disappear completely - you have crossed the so-called breaking point.

In other words, Hooke's law only applies to relatively small compressions or stretches. As long as a substance retains its elastic properties, the forces of deformation are directly proportional to its magnitude, and you are dealing with a linear system - for every equal increment of applied force there corresponds an equal increment of deformation. It's worth re-tightening the tires elastic limit, and the interatomic bonds-springs inside the substance first weaken and then break - and Hooke’s simple linear equation ceases to describe what is happening. In this case, it is customary to say that the system has become nonlinear. Today, the study of nonlinear systems and processes is one of the main directions in the development of physics.

Robert Hooke, 1635—1703

English physicist. Born in Freshwater on the Isle of Wight, the son of a priest, he graduated from Oxford University. While still at the university, he worked as an assistant in the laboratory of Robert Boyle, helping the latter build a vacuum pump for the installation in which the Boyle-Mariotte law was discovered. Being a contemporary of Isaac Newton, he actively participated with him in the work of the Royal Society, and in 1677 he took up the post of scientific secretary there. Like many other scientists of his time, Robert Hooke was interested in a wide variety of areas of the natural sciences and contributed to the development of many of them. In his monograph “Micrography” ( Micrographia) he published many sketches of the microscopic structure of living tissues and other biological specimens and was the first to introduce the modern concept of a "living cell". In geology, he was the first to recognize the importance of geological strata and the first in history to engage in the scientific study of natural disasters ( cm. Uniformitarianism). He was one of the first to hypothesize that the force of gravitational attraction between bodies decreases in proportion to the square of the distance between them, and this is a key component of Newton’s Law of Universal Gravitation, and the two compatriots and contemporaries disputed each other’s right to be called its discoverer until the end of their lives. Finally, Hooke developed and personally built a number of important scientific measuring instruments - and many are inclined to see this as his main contribution to the development of science. In particular, he was the first to think of placing a crosshair made of two thin threads in the eyepiece of a microscope, the first to propose taking the freezing temperature of water as zero on the temperature scale, and also invented a universal joint (gimbal joint).

Types of deformations

Deformation called a change in the shape, size or volume of the body. Deformation can be caused by external forces applied to the body. Deformations that completely disappear after the action of external forces on the body ceases are called elastic, and deformations that persist even after external forces have ceased to act on the body - plastic. Distinguish tensile strain or compression(unilateral or comprehensive), bending, torsion And shift.

Elastic forces

We will consider the case of the occurrence of elastic forces during unilateral tension and compression of a solid body.

Hooke's law

Hooke's law for one-sided tension (compression) is formulated as follows: The elastic force arising during deformation of a body is proportional to the elongation of this body.

Let's stretch the spring so that its free end is at point D, the coordinate of which is x > 0: At this point the spring acts on the body M with an elastic force

Let us now compress the spring so that its free end is at point B, whose coordinate is x

Let's consider another experiment.

Let one end of a thin steel wire be fixed to a bracket, and a load suspended from the other end, the weight of which is an external tensile force F acting on the wire perpendicular to its cross section (Fig. 21).

The action of this force on the wire depends not only on the force modulus F, but also on the cross-sectional area of the wire S.

Under the influence of an external force applied to it, the wire is deformed and stretched. If the stretch is not too great, this deformation is elastic. In an elastically deformed wire, an elastic force f unit arises. According to Newton's third law, the elastic force is equal in magnitude and opposite in direction to the external force acting on the body, i.e.

f up = -F (2.10)

s = f up /S (2.11)

Let the initial length of the unstretched wire be L 0 . After applying force F, the wire stretched and its length became equal to L. The quantity DL = L - L 0 is called absolute wire elongation. The quantity e = DL/L 0 (2.12) is called relative body elongation. For tensile strain e>0, for compressive strain e< 0.

Observations show that for small deformations the normal stress s is proportional to the relative elongation e:

s = E|e|. (2.13)

Let us establish the physical meaning of Young's modulus. As can be seen from formula (2.12), e = 1 and L = 2L 0 for DL = L 0 . From formula (2.13) it follows that in this case s = E. Consequently, Young’s modulus is numerically equal to the normal stress that should arise in the body if its length is doubled. (if Hooke's law were true for such a large deformation). From formula (2.13) it is also clear that in the SI Young’s modulus is expressed in pascals (1 Pa = 1 N/m2).

Ministry of Education of the Autonomous Republic of Crimea

Tauride National University named after. Vernadsky

Study of physical law

HOOKE'S LAW

Completed by: 1st year student

Faculty of Physics gr. F-111

Potapov Evgeniy

Simferopol-2010

Plan:

The connection between what phenomena or quantities is expressed by the law.

Statement of the law

Mathematical expression of the law.

How was the law discovered: based on experimental data or theoretically?

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

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

Examples of using the law and taking into account the effect of the law in practice.

Literature.

The relationship between what phenomena or quantities is expressed by the law:

Hooke's law relates phenomena such as stress and deformation of a solid, elastic modulus and elongation. The modulus of the elastic force arising during deformation of a body is proportional to its elongation. Elongation is a characteristic of the deformability of a material, assessed by the increase in the length of a sample of this material when stretched. Elastic force is a force that arises during deformation of a body and counteracts this deformation. Stress is a measure of internal forces that arise in a deformable body under the influence of external influences. Deformation is a change in the relative position of particles of a body associated with their movement relative to each other. These concepts are related by the so-called stiffness coefficient. It depends on the elastic properties of the material and the size of the body.

Statement of the law:

Hooke's law is an equation of the theory of elasticity that relates stress and deformation of an elastic medium.

The formulation of the law is that the elastic force is directly proportional to the deformation.

Mathematical expression of the law:

For a thin tensile rod, Hooke's law has the form:

Here F rod tension force, Δ l- its elongation (compression), and k called elasticity coefficient(or rigidity). The minus in the equation indicates that the tension force is always directed in the direction opposite to the deformation.

If you enter the relative elongation

abnormal stress in cross section

then Hooke's law will be written like this

In this form it is valid for any small volumes of matter.

In the general case, stress and strain are tensors of the second rank in three-dimensional space (they have 9 components each). The tensor of elastic constants connecting them is a tensor of the fourth rank C ijkl and contains 81 coefficients. Due to the symmetry of the tensor C ijkl, as well as stress and strain tensors, only 21 constants are independent. Hooke's law looks like this:

where σ ij- stress tensor, - strain tensor. For an isotropic material, the tensor C ijkl contains only two independent coefficients.

How was the law discovered: based on experimental data or theoretically:

The law was discovered in 1660 by the English scientist Robert Hooke (Hook) based on observations and experiments. The discovery, as stated by Hooke in his essay “De potentia restitutiva”, published in 1678, was made by him 18 years earlier, and in 1676 it was placed in another of his books under the guise of the anagram “ceiiinosssttuv”, meaning “Ut tensio sic vis” . According to the author's explanation, the above law of proportionality applies not only to metals, but also to wood, stones, horn, bones, glass, silk, hair, etc.

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

History is silent about this..

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

The law is formulated on the basis of experimental data. Indeed, when stretching a body (wire) with a certain stiffness coefficient k to a distance Δ l, then their product will be equal in magnitude to the force stretching the body (wire). This relationship will hold true, however, not for all deformations, but for small ones. With large deformations, Hooke's law ceases to apply and the body collapses.

Examples of using the law and taking into account the effect of the law in practice:

As follows from Hooke's law, the elongation of a spring can be used to judge the force acting on it. This fact is used to measure forces using a dynamometer - a spring with a linear scale calibrated for different force values.

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 Peryshkin A.V. 9th grade

3. textbook on physics V.A. Kasyanov 10th grade

4. lectures on mechanics Ryabushkin D.S.