Hooke's law is formulated as follows: the elastic force that occurs when a body is deformed due to the application of external forces is proportional to its elongation. Deformation, in turn, is a change in the interatomic or intermolecular distance of a substance under the action of external forces. The elastic force is the force that tends to return these atoms or molecules to a state of equilibrium.

Formula 1 - Hooke's Law.

F - Force of elasticity.

k - rigidity of the body (Proportionality factor, which depends on the material of the body and its shape).

x - Deformation of the body (lengthening or compression of the body).

This law was discovered by Robert Hooke in 1660. He conducted an experiment which consisted in the fact that. A thin steel string was fixed at one end, and a different force was applied to the other end. Simply put, the string was suspended from the ceiling, and a load of various masses was applied to it.

Figure 1 - Stretching of a string under the action of gravity.

As a result of the experiment, Hooke found out that in small aisles, the dependence of the stretching of the body is linear with respect to the force of elasticity. That is, when a unit of force is applied, the body lengthens by one unit of length.

Figure 2 - Graph of the dependence of the elastic force on the elongation of the body.

Zero on the graph is the original length of the body. Everything on the right is an increase in body length. In this case, the elastic force has negative meaning. That is, she strives to return the body to its original state. Accordingly, it is directed opposite to the deforming force. Everything on the left is body compression. The force of elasticity is positive.

The stretching of the string of envy is not only from an external force, but also from the section of the string. A thin string will still somehow stretch from a small weight. But if you take a string of the same length, but let's say 1 m in diameter, it's hard to imagine how much weight it will take to stretch it.

To assess how a force acts on a body of a certain section, the concept of normal mechanical stress is introduced.

Formula 2 - normal mechanical stress.

S-Cross-sectional area.

This stress is ultimately proportional to the relative elongation of the body. Relative elongation is the ratio of the increment in the length of the body to its total length. And the coefficient of proportionality is called Young's modulus. Module because the value of body elongation is taken modulo, without taking into account the sign. It is not taken into account whether the body is shortened or lengthened. It is important to change its length.

Formula 3 - Young's modulus.

|e|- Relative elongation of the body.

s is the normal tension of the body.

Hooke's Law usually referred to as linear relationships between strain components and stress components.

Take an elementary rectangular parallelepiped with faces parallel to the coordinate axes, loaded with normal stress σ x, uniformly distributed over two opposite faces (Fig. 1). Wherein y = σz = τ x y = τ x z = τ yz = 0.

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

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

Extending an Element in the Axis Direction X is accompanied by its narrowing in the transverse direction, determined by the strain components

where μ is a constant called the transverse compression ratio or Poisson's ratio. For steel μ usually taken equal to 0.25-0.3.

If the element under consideration is simultaneously loaded with normal stresses σ x, y, σz, uniformly distributed over its faces, then deformations are added

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

These ratios are confirmed by numerous experiments. applied overlay method or superpositions to find the total strains and stresses caused by multiple 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 deformable body and small displacements of the points of application of external forces and base our calculations on the initial dimensions and initial form body.

It should be noted that the linearity of the relationships between forces and strains does not yet follow from the smallness of the displacements. So, for example, in compressed by forces Q rod loaded with an additional transverse force R, even with a small deflection δ there is an additional moment M = Qδ, which makes the problem non-linear. In such cases, full deflections are not linear functions efforts and cannot be obtained with a simple superimposition (superposition).

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

Constant G is called the shear modulus or shear modulus.

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

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 for the element 0 bс, normal stresses σ v on all faces of the element abcd are equal to zero, and shear stresses are equal

This stress state is called pure shift. Equations (5.2a) imply that

that is, the extension of the horizontal element 0 c equals the shortening of the vertical element 0 b: εy = -ε x.

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

Hence it follows that

TEST QUESTIONS

1) What is called deformation? What types of deformations do you know?

Deformation- change in the relative position of body particles associated with their movement. Deformation is the result of a change in interatomic distances and a rearrangement of blocks of atoms. Typically, deformation is accompanied by a change in the values ​​of interatomic forces, the measure of which is the elastic stress.

Types of deformations:

Tension-compression- in the resistance of materials - a type of longitudinal deformation of a rod or beam that occurs if a load is applied to it along its longitudinal axis (the resultant of the forces acting on it is normal to the cross section of the rod and passes through its center of mass).

Tension causes the rod to elongate (breakage and permanent deformation are also possible), compression causes the rod to shorten (buckling and buckling are possible).

bend- type of deformation, in which there is a curvature of the axes of straight bars or a change in the curvature of the axes of curved bars. Bending is associated with the occurrence of bending moments in the cross sections of the beam. Direct bending occurs when the bending moment in a given cross section of the beam acts in a plane passing through one of the main central axes of inertia of this section. In the case when the plane of action of the bending moment in a given cross section of the beam does not pass through any of the main axes of inertia of this section, it is called oblique.

If only a bending moment acts in the cross section of the beam during a straight or oblique bend, then there is a pure straight or a pure oblique bend, respectively. If a transverse force also acts in the cross section, then there is a transverse straight or transverse oblique bend.

Torsion- one of the types of body deformation. Occurs when a load is applied to a body in the form of a pair of forces (moment) in its transverse plane. In this case, only one internal force factor arises in the cross sections of the body - torque. Tension-compression springs and shafts work on torsion.

Types of deformation solid body. The deformation is elastic and plastic.

Deformation solid body may be the result of phase transformations associated with a change in volume, thermal expansion, magnetization (magnetostrictive effect), the appearance electric charge(piezoelectric effect) or the result of external forces.

A deformation is called elastic if it disappears after the removal of the load that caused it, and plastic if it does not disappear after the load is removed (at least completely). All real solids under deformation to a greater or lesser extent have plastic properties. Under certain conditions, the plastic properties of bodies can be neglected, as is done in the theory of elasticity. A solid body can be considered elastic with sufficient accuracy, that is, it does not show noticeable plastic deformations until the load exceeds a certain limit.

The nature of plastic deformation can be different depending on the temperature, the duration of the load, or the strain rate. With a constant load applied to the body, the deformation changes with time; this phenomenon is called creep. With increasing temperature, the creep rate increases. Relaxation and elastic aftereffect are particular cases of creep. One of the theories explaining the mechanism of plastic deformation is the theory of dislocations in crystals.

Derivation of Hooke's law for various types of deformation.

Net shift: Pure twist:

4) What is called the shear modulus and torsion modulus, what are their physical meaning?

Shear modulus or stiffness modulus (G or μ) characterizes the ability of a material to resist changing shape while maintaining its volume; it is defined as the ratio of shear stress to shear strain, defined as change right angle between planes on which shear stresses act). The shear modulus is one of the components of the viscosity phenomenon.

Shear modulus: Torsion modulus:

5) What is the mathematical expression of Hooke's law? What are the units of modulus and stress?

Measured in Pa - Hooke's law

Ministry of Education of the Autonomous Republic of Crimea

Taurida National University. Vernadsky

The study of physical law

HOOK'S LAW

Completed by: 1st year student

Faculty of Physics F-111

Potapov Evgeny

Simferopol-2010

Plan:

The relationship between what phenomena or quantities expresses the law.

The wording of the law

Mathematical expression of the law.

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

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

Experiments confirming the validity of a law formulated on the basis of a 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 expresses the law:

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

The wording of the law:

Hooke's law is an equation of the theory of elasticity that relates the 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 hardness). The minus in the equation indicates that the tension force is always directed in the direction opposite to the deformation.

If you enter a 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.

In the general case, stresses and strains 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: on the basis of experimental data or theoretically:

The law was discovered in 1660 by the English scientist Robert Hooke (Hooke) on the basis of observations and experiments. The discovery, as Hooke claimed in his essay "De potentia restitutiva", published in 1678, was made by him 18 years before that time, and in 1676 was placed in another of his books under the guise of an 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, and so on.

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

History is silent on 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 distance Δ l, then their product will be equal in absolute value to the force stretching the body (wire). This ratio will be fulfilled, however, not for all deformations, but for small ones. At large deformations, Hooke's law ceases to operate, the body is destroyed.

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

As follows from Hooke's law, the lengthening 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 to different values ​​of forces.

Literature.

1. Internet resources: - Wikipedia site (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. Grade 9

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

4. lectures on mechanics Ryabushkin D.S.

Types of deformations

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

elastic forces

During deformations of a solid body, its particles (atoms, molecules, ions) located at the nodes crystal lattice, are displaced from their equilibrium positions. This displacement is counteracted by the forces of interaction between the 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 the body during its elastic deformation and directed against the direction of displacement of the particles of the body caused by deformation are called elastic forces. Elastic forces act in any section of the deformed body, as well as in the place 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 the body to deform, 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 appearance of elastic forces during unilateral tension and compression of a solid body.

Hooke's law

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

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

Hooke's law for unilateral tension (compression) formulate as follows: the elastic force that occurs when a body is deformed is proportional to the elongation of this body.

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

We 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, the coordinate of which is x

It can be seen from the figure that the projection of the elastic force of the spring on the axis Ax 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. 20b shows a graph of Hooke's law. On the abscissa axis, the values ​​of the elongation x of the spring are plotted, and on the ordinate axis, the values ​​of the elastic force. The dependence of fx on x is linear, so the graph is a straight line passing through the origin.

Consider another experience.

Let one end of a thin steel wire be fixed on a bracket, and a load is suspended from the other end, the weight of which is the 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 modulus of force F, but also on the cross-sectional area of ​​the wire S.

Under the action of an external force applied to it, the wire is deformed and stretched. With not too much stretching, this deformation is elastic. In the elastically deformed wire there is an elastic force f y. According to Newton's third law, the elastic force is equal in absolute value and opposite in direction to the external force acting on the body, i.e.

f yn = -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). The normal stress s is equal to the ratio of the elastic modulus to the cross-sectional area of ​​the body:

s = f y /S (2.11)

Let the initial length of the unstretched wire be L 0 . After applying the force F, the wire stretched and its length became equal to L. The value DL \u003d L - L 0 is called absolute elongation of the wire. The value e = DL/L 0 (2.12) is called relative elongation of the body. 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 ways of writing Hooke's law for one-sided tension (compression). In this formula, the elongation is taken modulo, since it can be both positive and negative. The coefficient of proportionality E in Hooke's law is called the modulus of longitudinal 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. Therefore, Young's modulus is numerically equal to such a normal stress that should have arisen in the body with an increase in its length by 2 times. (if for such a large deformation Hooke's law was fulfilled). From formula (2.13) it is also seen that in SI Young's modulus is expressed in pascals (1 Pa = 1 N/m2).