Under what deformations does Hooke's law hold? Deformations

DEFINITION

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

DEFINITION

Elastic deformations are called deformations that completely disappear after the removal of external forces.

Plastic deformations are called deformations that remain fully or partially after the cessation of external forces.

The ability to elastic and plastic deformations depends on the nature of the substance of which the body is composed, the conditions in which it is located; methods of its manufacture. For example, if you take different types of iron or steel, you can find completely different elastic and plastic properties in them. At normal room temperatures, iron is a very soft, ductile material; hardened steel, on the contrary, is a hard, elastic material. The plasticity of many materials is a condition for their processing and for the manufacture of the necessary parts from them. Therefore, it is considered one of the most important technical properties of a solid.

When a solid body is deformed, particles (atoms, molecules or ions) are displaced from their original equilibrium positions to new positions. In this case, the force interactions between individual particles of the body change. As a result, internal forces arise in the deformed body, preventing its deformation.

There are tensile (compressive), shear, bending, and torsional deformations.

Elastic forces

DEFINITION

Elastic forces– these are the forces that arise in a body during its elastic deformation and are directed in the direction opposite to the displacement of particles during deformation.

Elastic forces are of an electromagnetic nature. They prevent deformations and are directed perpendicular to the contact surface of interacting bodies, and if bodies such as springs or threads interact, then the elastic forces are directed along their axis.

The elastic force acting on the body from the support is often called the support reaction force.

DEFINITION

Tensile strain (linear strain) is a deformation in which only one linear dimension of the body changes. Its quantitative characteristics are absolute and relative elongation.

Absolute elongation:

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

Elongation:

Hooke's law

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

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

Examples of problem solving

EXAMPLE 1

Exercise	A spring with stiffness N/m in an unloaded state has a length of 25 cm. What will be the length of the spring if a load weighing 2 kg is suspended from it?
Solution	Let's make a drawing. An elastic force also acts on a load suspended on a spring. By designing it vector equality to the coordinate axis, we get: According to Hooke's law, elastic force: so we can write: where does the length of the deformed spring come from: Let us convert the length of the undeformed spring, cm, to the SI system. Substituting numerical values ​​into the formula physical quantities, let's calculate:
Answer	The length of the deformed spring will be 29 cm.

EXAMPLE 2

Exercise	A body weighing 3 kg is moved along a horizontal surface using a spring with stiffness N/m. How much will the spring lengthen if, under its action, with uniformly accelerated motion, the speed of the body changes from 0 to 20 m/s in 10 s? Ignore friction.
Solution	Let's make a drawing. The body is acted upon by the reaction force of the support and the elastic force of the spring.

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, 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 stretching and compression of a solid body.

Hooke's law

The relationship between elastic force and elastic deformation 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

It can be seen from the figure that the projection of the elastic force of the spring onto the Ax axis always has the sign opposite sign coordinates x, 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 area cross section 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 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)

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 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 them. 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

and normal stress in the 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 graduated to different meanings strength

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.

Elasticity coefficient

Elasticity coefficient(sometimes called Hooke's coefficient, stiffness coefficient or spring stiffness) - a coefficient that relates in Hooke's law the elongation of an elastic body and the elastic force resulting from this elongation. It is used in solid mechanics in the section of elasticity. Denoted by the letter k, Sometimes D or c. It has the dimension N/m or kg/s2 (in SI), dyne/cm or g/s2 (in GHS).

The elasticity coefficient is numerically equal to the force that must be applied to the spring in order for its length to change per unit distance.

Definition and properties

The elasticity coefficient, by definition, is equal to the elastic force divided by the change in spring length: k = F e / Δ l. (\displaystyle k=F_(\mathrm (e) )/\Delta l.) The elasticity coefficient depends both on the properties of the material and on the dimensions of the elastic body. Thus, for an elastic rod, we can distinguish the dependence on the dimensions of the rod (cross-sectional area S (\displaystyle S) and length L (\displaystyle L)), writing the elasticity coefficient as k = E ⋅ S / L. (\displaystyle k=E\cdot S/L.) The quantity E (\displaystyle E) is called Young's modulus and, unlike the elasticity coefficient, depends only on the properties of the material of the rod.

Stiffness of deformable bodies when they are connected

Parallel connection of springs. Series connection of springs.

When connecting several elastically deformable bodies (hereinafter referred to as springs for brevity), the overall rigidity of the system will change. With a parallel connection, the stiffness increases, with a series connection it decreases.

Parallel connection

With a parallel connection of n (\displaystyle n) springs with stiffnesses equal to k 1 , k 2 , k 3 , . . . , k n , (\displaystyle k_(1),k_(2),k_(3),...,k_(n),) the rigidity of the system is equal to the sum of the rigidities, that is, k = k 1 + k 2 + k 3 + . . . +kn. (\displaystyle k=k_(1)+k_(2)+k_(3)+...+k_(n).)

Proof

In a parallel connection there are n (\displaystyle n) springs with stiffnesses k 1 , k 2 , . . . ,kn. (\displaystyle k_(1),k_(2),...,k_(n).) From Newton's III law, F = F 1 + F 2 + . . . +Fn. (\displaystyle F=F_(1)+F_(2)+...+F_(n).) (A force F is applied to them (\displaystyle F). At the same time, a force F 1 is applied to spring 1, (\displaystyle F_(1),) to spring 2 force F 2 , (\displaystyle F_(2),) ... , to spring n (\displaystyle n) force F n (\displaystyle F_(n).))

Now from Hooke’s law (F = − k x (\displaystyle F=-kx), where x is the elongation) we derive: F = k x ; F 1 = k 1 x ; F 2 = k 2 x ; . . . ; F n = k n x . (\displaystyle F=kx;F_(1)=k_(1)x;F_(2)=k_(2)x;...;F_(n)=k_(n)x.) Substitute these expressions into the equality (1): k x = k 1 x + k 2 x + . . . + k n x ; (\displaystyle kx=k_(1)x+k_(2)x+...+k_(n)x;) reducing by x, (\displaystyle x,) we get: k = k 1 + k 2 + . . . + k n , (\displaystyle k=k_(1)+k_(2)+...+k_(n),) which is what needed to be proven.

Serial connection

With a series connection of n (\displaystyle n) springs with stiffnesses equal to k 1 , k 2 , k 3 , . . . , k n , (\displaystyle k_(1),k_(2),k_(3),...,k_(n),) the total stiffness is determined from the equation: 1 / k = (1 / k 1 + 1 / k 2 + 1 / k 3 + . (\displaystyle 1/k=(1/k_(1)+1/k_(2)+1/k_(3)+...+1/k_(n)).)

Proof

In a series connection there are n (\displaystyle n) springs with stiffnesses k 1 , k 2 , . . . ,kn. (\displaystyle k_(1),k_(2),...,k_(n).) From Hooke’s law (F = − k l (\displaystyle F=-kl) , where l is the elongation) it follows that F = k ⋅ l . (\displaystyle F=k\cdot l.) The sum of the elongations of each spring is equal to the total elongation of the entire connection l 1 + l 2 + . . . + l n = l . (\displaystyle l_(1)+l_(2)+...+l_(n)=l.)

Each spring is subject to the same force F. (\displaystyle F.) According to Hooke's law, F = l 1 ⋅ k 1 = l 2 ⋅ k 2 = . . . = l n ⋅ k n . (\displaystyle F=l_(1)\cdot k_(1)=l_(2)\cdot k_(2)=...=l_(n)\cdot k_(n).) From the previous expressions we deduce: l = F / k, l 1 = F / k 1, l 2 = F / k 2, . . . , l n = F / k n . (\displaystyle l=F/k,\quad l_(1)=F/k_(1),\quad l_(2)=F/k_(2),\quad ...,\quad l_(n)= F/k_(n).) Substituting these expressions into (2) and dividing by F, (\displaystyle F,) we get 1 / k = 1 / k 1 + 1 / k 2 + . . . + 1 / k n , (\displaystyle 1/k=1/k_(1)+1/k_(2)+...+1/k_(n),) which is what needed to be proven.

Stiffness of some deformable bodies

Constant cross-section rod

A homogeneous rod of constant cross-section, elastically deformed along the axis, has a stiffness coefficient

K = E S L 0 , (\displaystyle k=(\frac (E\,S)(L_(0))),) E- Young's modulus, which depends only on the material from which the rod is made; S- cross-sectional area; L 0 - length of the rod.

Cylindrical coil spring

Twisted cylindrical compression spring.

A twisted cylindrical compression or tension spring, wound from a cylindrical wire and elastically deformed along the axis, has a stiffness coefficient

K = G ⋅ d D 4 8 ⋅ d F 3 ⋅ n , (\displaystyle k=(\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 wire axis); n- number of turns; G- shear modulus (for ordinary steel G≈ 80 GPa, for spring steel G≈ 78.5 GPa, for copper ~ 45 GPa).

Sources and notes

1. Elastic deformation (Russian). 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, Elastic force (Russian). Archived June 30, 2012.
5. Mechanical properties of bodies (Russian). Archived June 30, 2012.

10. Hooke's law in tension-compression. Modulus of elasticity (Young's modulus).

Under axial tension or compression to the limit of proportionality σ pr Hooke's law is valid, i.e. law on the directly proportional relationship between normal stresses  and longitudinal relative deformations :

(3.10)

or

(3.11)

Here E - the proportionality coefficient in Hooke's law has the dimension of voltage and is called modulus of elasticity of the first kind, characterizing the elastic properties of the material, or Young's modulus.

Relative longitudinal strain is the ratio of the absolute longitudinal strain of the section

rod to the length of this section before deformation:

(3.12)

The relative transverse deformation will be equal to: " = = b/b, where b = b 1 – b.

The ratio of the relative transverse deformation " to the relative longitudinal deformation , taken modulo, is a constant value for each material and is called Poisson's ratio:

Determination of the absolute deformation of a section of timber

In formula (3.11) instead And Let's substitute expressions (3.1) and (3.12):

From here we obtain a formula for determining the absolute elongation (or shortening) of a section of a rod with length :

(3.13)

In formula (3.13) the product EA is called the rigidity of the beam in tension or compression, which is measured in kN, or MN.

This formula determines the absolute deformation if the longitudinal force is constant in the area. In the case where the longitudinal force is variable in the area, it is determined by the formula:

(3.14)

where N(x) is a function of the longitudinal force along the length of the section.

11. Transverse strain coefficient (Poisson's ratio

12.Determination of displacements during tension and compression. Hooke's law for a section of timber. Determination of displacements of beam sections

Let's determine the horizontal movement of the point A axis of the beam (Fig. 3.5) – u a: it is equal to the absolute deformation of part of the beam Ad, enclosed between the embedment and the section drawn through the point, i.e.

In turn, lengthening the section Ad consists of extensions of individual cargo sections 1, 2 and 3:

Longitudinal forces in the areas under consideration:

Hence,

Then

Similarly, you can determine the movement of any section of a beam and formulate the following rule:

moving any section jof a rod under tension-compression is determined as the sum of absolute deformations ncargo areas enclosed between the considered and fixed (fixed) sections, i.e.

(3.16)

The condition for the rigidity of the beam will be written in the following form:

, (3.17)

Where

– highest value displacement of the section, taken modulo from the displacement diagram; u – permissible value of displacement of the section for a given structure or its element, established in the standards.

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

For quantification basic properties of materials, such as

As a rule, the tension diagram is experimentally determined in coordinates  and  (Fig. 2.9). Characteristic points are marked on the diagram. Let's define them.

The highest stress to which a material follows Hooke's law is called limit of proportionality  P. Within the limits of Hooke's law, the tangent of the angle of inclination of the straight line  = f() to the  axis is determined by the value E.

The elastic properties of the material are maintained up to stress  U called elastic limit. Below the elastic limit  U is understood as the greatest stress up to which the material does not receive residual deformations, i.e. after complete unloading, the last point of the diagram coincides with the starting point 0.

Value  T called yield strength material. The yield strength is understood as the stress at which strain increases without a noticeable increase in load. If it is necessary to distinguish between the yield strength in tension and compression  T accordingly replaced by  TR and  TS. At voltages high  T plastic deformations develop in the body of the structure  P, which do not disappear when the load is removed.

The ratio of the maximum force that a sample can withstand to its initial cross-sectional area is called tensile strength, or tensile strength, and is denoted by  VR(with compression  Sun).

When performing practical calculations, the real diagram (Fig. 2.9) is simplified, and for this purpose various approximating diagrams are used. To solve problems taking into account elastically plastic properties of structural materials is most often used Prandtl diagram. According to this diagram, the stress changes from zero to the yield strength according to Hooke’s law  = E, and then as  increases,  =  T(Fig. 2.10).

The ability of materials to obtain residual deformations is called plasticity. In Fig. 2.9 presented a characteristic diagram for plastic materials.

Rice. 2.10 Fig. 2.11

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

1. What is body deformation called? How is Hooke's law formulated?

Vakhit Shavaliev

Deformations are any changes in the shape, size and volume of the body. Deformation determines the final result of the movement of body parts relative to each other.
Elastic deformations are deformations that completely disappear after the removal of external forces.
Plastic deformations are deformations that remain fully or partially after the action of external forces ceases.
Elastic forces are forces that arise in a body during its elastic deformation and are directed in the direction opposite to the displacement of particles during deformation.
Hooke's law
Small and short-term deformations with a sufficient degree of accuracy can be considered as elastic. For such deformations, Hooke’s law is valid:
The elastic force that arises during deformation of a body is directly proportional to the absolute elongation of the body and is directed in the direction opposite to the displacement of the particles of the body:
\
where F_x is the projection of force on the x-axis, k is the rigidity of the body, depending on the size of the body and the material from which it is made, the unit of rigidity in the SI system N/m.
http://ru.solverbook.com/spravochnik/mexanika/dinamika/deformacii-sily-uprugosti/

Varya Guseva

Deformation is a change in the shape or volume of a body. Types of deformation - stretching or compression (examples: stretching or squeezing an elastic band, accordion), bending (a board bent under a person, a sheet of paper bent), torsion (working with a screwdriver, squeezing out laundry by hand), shear (when a car brakes, the tires are deformed due to the friction force ) .
Hooke's law: The elastic force arising in a body during its deformation is directly proportional to the magnitude of this deformation
or
The elastic force that arises in a body during its deformation is directly proportional to the magnitude of this deformation.
Hooke's law formula: Fpr=kx

Hooke's law. Can it be expressed by the formula F= -khх or F= khх?

⚓ Otters ☸

Hooke's law is an equation of the theory of elasticity that relates stress and deformation of an elastic medium. Discovered in 1660 by the English scientist Robert Hooke. Since Hooke's law is written for small stresses and strains, it has the form of simple proportionality.

For a thin tensile rod, Hooke's law has the form:
Here F is the tension force of the rod, Δl is its elongation (compression), and k is called the elasticity coefficient (or rigidity). The minus in the equation indicates that the tension force is always directed in the direction opposite to the deformation.

The elasticity coefficient depends both on the properties of the material and on the dimensions of the rod. We can distinguish the dependence on the dimensions of the rod (cross-sectional area S and length L) explicitly by writing the elasticity coefficient as
The quantity E is called Young's modulus and depends only on the properties of the body.

If you enter the relative elongation
and normal stress in the cross section
then Hooke's law will be written as
In this form it is valid for any small volumes of matter.
[edit]
Generalized Hooke's law

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 Cijkl and contains 81 coefficients. Due to the symmetry of the Cijkl tensor, as well as the stress and strain tensors, only 21 constants are independent. Hooke's law looks like this:
For an isotropic material, the Cijkl tensor contains only two independent coefficients.

It should be borne in mind that Hooke's law is satisfied only for small deformations. When the proportionality limit is exceeded, the relationship between stress and strain becomes nonlinear. For many media, Hooke's law is not applicable even at small deformations.
[edit]

in short, you can do it this way or that, depending on what you want to indicate in the end: simply the modulus of the Hooke force or also the direction of this force. Strictly speaking, of course, -kx, since the Hooke force is directed against the positive increment in the coordinate of the end of the spring.

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 stretching 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.

Hooke's law usually called linear relationships between strain components and stress components.

Let's take an elementary rectangular parallelepiped with faces parallel to the coordinate axes, loaded with normal stress σ x, evenly distributed over two opposite faces (Fig. 1). At the same time σy = σ z = τ x y = τ x z = τ yz = 0.

Up to the limit of proportionality, the relative elongation is given by the formula

Where E— tensile modulus of elasticity. For steel E = 2*10 5 MPa, therefore, the deformations are very small and are measured as a percentage or 1 * 10 5 (in strain gauge instruments that measure deformations).

Extending an element in the axis direction X accompanied by its narrowing in the transverse direction, determined by the deformation components

Where μ - a constant called the lateral compression ratio or Poisson's ratio. For steel μ usually taken to be 0.25-0.3.

If the element in question is loaded simultaneously with normal stresses σx, σy, σ z, evenly distributed along its faces, then deformations are added

By superimposing the deformation components caused by each of the three stresses, we obtain the relations

These relationships are confirmed by numerous experiments. Applied overlay method or superpositions to find the total strains and stresses caused by several forces is legitimate as long as the strains and stresses are small and linearly dependent on the applied forces. In such cases, we neglect small changes in the dimensions of the deformed body and small movements of the points of application of external forces and base our calculations on the initial dimensions and initial shape of the body.

It should be noted that the smallness of the displacements does not necessarily mean that the relationships between forces and deformations are linear. So, for example, in a compressed force Q rod loaded additionally with shear force R, even with small deflection δ an additional point arises M = Qδ, which makes the problem nonlinear. In such cases, the total deflections are not linear functions of the forces and cannot be obtained by simple superposition.

It has been experimentally established that if shear stresses act along all faces of the element, then the distortion of the corresponding angle depends only on the corresponding components of the shear stress.

Constant G called the shear modulus of elasticity or shear modulus.

The general case of deformation of an element due to the action of three normal and three tangential stress components on it can be obtained using superposition: three shear strains, determined by relations (5.2b), are superimposed on three linear deformations determined by expressions (5.2a). Equations (5.2a) and (5.2b) determine the relationship between the components of strains and stresses and are called generalized Hooke's law. Let us now show that the shear modulus G expressed in terms of tensile modulus of elasticity E and Poisson's ratio μ . To do this, consider the special case when σ x = σ , σy = -σ And σ z = 0.

Let's cut out the element abcd planes parallel to the axis z and inclined at an angle of 45° to the axes X And at(Fig. 3). As follows from the equilibrium conditions of element 0 bс, normal stress σ v on all faces of the element abcd are zero and the shear stresses are equal

This state of tension is called pure shear. From equations (5.2a) it follows that

that is, the extension of the horizontal element is 0 c equal to the shortening of the vertical element 0 b: εy = -εx.

Angle between faces ab And bc changes, and the corresponding shear strain value γ can be found from triangle 0 bс:

It follows that