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). Wherein 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 devices 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 σ x, 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 deformations, 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 equal to 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 = -ε x.
Angle between faces ab And bc changes, and the corresponding shear strain value γ can be found from triangle 0 bс:
It follows that
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
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 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.
When a rod is stretched and compressed, its length and cross-sectional dimensions change. If you mentally select from a rod in an undeformed state an element of length dx, then after deformation its length will be equal to dx ((Fig. 3.6). In this case, the absolute elongation in the direction of the axis Oh will be equal
and the relative linear deformation e x is determined by equality
Because the axis Oh coincides with the axis of the rod along which external loads act, let’s call the deformation e x longitudinal deformation, for which we will further omit the index. Deformations in directions perpendicular to the axis are called transverse deformations. If we denote by b characteristic size of the cross section (Fig. 3.6), then the transverse deformation is determined by the relation 
Relative linear deformations are dimensionless quantities. It has been established that transverse and longitudinal deformations during central tension and compression of the rod are related to each other by the relationship
The quantity v included in this equality is called Poisson's ratio or transverse strain coefficient. This coefficient is one of the main elastic constants of the material and characterizes its ability to undergo transverse deformations. For each material, it is determined from a tensile or compression experiment (see § 3.5) and is calculated using the formula
As follows from equality (3.6), longitudinal and transverse deformations always have opposite signs, which confirms the obvious fact that during tension the cross-sectional dimensions decrease, and during compression they increase.
Poisson's ratio is different for different materials. For isotropic materials it can take values ranging from 0 to 0.5. For example, for balsa wood Poisson's ratio is close to zero, and for rubber it is close to 0.5. For many metals at normal temperatures, the Poisson's ratio is in the range of 0.25+0.35.
As has been established in numerous experiments, for most structural materials at small deformations there is a linear relationship between stresses and strains
This law of proportionality was first established by the English scientist Robert Hooke and is called Hooke's law.
The constant included in Hooke's law E called the modulus of elasticity. The elastic modulus is the second main elastic constant of a material and characterizes its rigidity. Since deformations are dimensionless quantities, it follows from (3.7) that the elastic modulus has the dimension of stress.
In table Table 3.1 shows the values of the elastic modulus and Poisson's ratio for various materials.
When designing and calculating structures, along with calculating stresses, it is also necessary to determine the displacements of individual points and nodes of structures. Let's consider a method for calculating displacements during central tension and compression of rods.
Absolute elongation of element length dx(Fig. 3.6) according to formula (3.5) is equal to
Table 3.1
|
Name of material |
Modulus of elasticity, MPa |
Coefficient Poisson |
|
Carbon steel |
||
|
Aluminum alloys |
||
|
Titanium alloys |
||
|
(1.15-s-1.6) 10 5 |
||
|
along the grain |
(0,1 ^ 0,12) 10 5 |
|
|
across the grain |
(0,0005 + 0,01)-10 5 |
|
|
(0,097 + 0,408) -10 5 |
||
|
Brickwork |
(0,027 +0,03)-10 5 |
|
|
Fiberglass SVAM |
||
|
Textolite |
(0,07 + 0,13)-10 5 |
|
|
Rubber on rubber |
Integrating this expression over the range from 0 to x, we get
Where their) - axial displacement of an arbitrary section (Fig. 3.7), and C= u( 0) - axial displacement of the initial section x = 0. If this section is fixed, then u(0) = 0 and the displacement of an arbitrary section is equal to
The elongation or shortening of the rod is equal to the axial displacement of its free end (Fig. 3.7), the value of which is obtained from (3.8), taking x = 1:
Substituting the expression for deformation into formula (3.8)? from Hooke's law (3.7), we obtain
For a rod made of a material with a constant modulus of elasticity E axial movements are determined by the formula
The integral included in this equality can be calculated in two ways. The first method is to write the function analytically Oh) and subsequent integration. The second method is based on the fact that the integral under consideration is numerically equal to the area of the diagram a in the section . Introducing the designation
Let's consider special cases. For a rod stretched by a concentrated force R(rice. 3.3, a), longitudinal force./V is constant along the length and equal to R. Voltages a according to (3.4) are also constant and equal 
Then from (3.10) we obtain
From this formula it follows that if the stresses on a certain section of the rod are constant, then the displacements change according to a linear law. Substituting into the last formula x = 1, let's find the elongation of the rod:
Work E.F. called rigidity of the rod in tension and compression. The greater this value, the less the elongation or shortening of the rod.
Let's consider a rod under the action of a uniformly distributed load (Fig. 3.8). The longitudinal force in an arbitrary section located at a distance x from the fastening is equal to
By dividing N on F, we get the formula for stresses
Substituting this expression into (3.10) and integrating, we find
The greatest displacement, equal to the elongation of the entire rod, is obtained by substituting x = / in (3.13):
From formulas (3.12) and (3.13) it is clear that if the stresses linearly depend on x, then the displacements change according to the law of a square parabola. Diagrams N, about and And shown in Fig. 3.8.
General differential dependence connecting functions their) and a(x), can be obtained from relation (3.5). Substituting e from Hooke’s law (3.7) into this relation, we find
From this dependence follow, in particular, the patterns of changes in the function noted in the examples discussed above their).
In addition, it can be noted that if in any section the stresses a turn to zero, then in the diagram And there may be an extremum in this section.
As an example, let's build a diagram And for the rod shown in Fig. 3.2, putting E- 10 4 MPa. Calculating the area of a plot O for different areas, we find:
section x = 1 m:
section x = 3 m:
section x = 5 m:
On the upper section of the rod diagram And is a square parabola (Fig. 3.2, e). In this case, in the section x = 1 m there is an extremum. In the lower section, the nature of the diagram is linear.
The total elongation of the rod, which in this case is equal to
can be calculated using formulas (3.11) and (3.14). Since the lower section of the rod (see Fig. 3.2, A) stretched by force R ( its extension according to (3.11) is equal to
Action of force R ( is also transmitted to the upper section of the rod. In addition, it is compressed by force R 2 and is stretched by a uniformly distributed load q. In accordance with this, the change in its length is calculated by the formula
Summing up the values of A/ and A/ 2, we get the same result as given above.
In conclusion, it should be noted that, despite the small amount of displacement and elongation (shortening) of the rods during tension and compression, they cannot be neglected. The ability to calculate these quantities is important in many technological problems (for example, when installing structures), as well as for solving statically indeterminate problems.
8.2. Basic laws used in strength of materials
Statics relations. They are written in the form of the following equilibrium equations.
Hooke's law ( 1678): the greater the force, the greater the deformation, and, moreover, is directly proportional to the force. Physically, this means that all bodies are springs, but with great rigidity. When a beam is simply stretched by a longitudinal force N= F this law can be written as:
Here 
longitudinal force, l- beam length, A- its cross-sectional area, E- coefficient of elasticity of the first kind ( Young's modulus).
Taking into account the formulas for stresses and strains, Hooke’s law is written as follows: 
.
A similar relationship is observed in experiments between tangential stresses and shear angle:
.
G
calledshear modulus
, less often – elastic modulus of the second kind. Like any law, Hooke's law also has a limit of applicability. Voltage 
, up to which Hooke's law is valid, is called limit of proportionality(this is the most important characteristic in strength of materials).
Let's depict the dependence
from
graphically (Fig. 8.1). This picture is called stretch diagram
. After point B (i.e. at 
) this dependence ceases to be linear.
At 
after unloading, residual deformations appear in the body, therefore 
called elastic limit
.
When the voltage reaches the value σ = σ t, many metals begin to exhibit a property called fluidity. This means that even under constant load, the material continues to deform (that is, it behaves like a liquid). Graphically, this means that the diagram is parallel to the abscissa (section DL). The voltage σ t at which the material flows is called yield strength .
Some materials (St. 3 - construction steel) after a short flow begin to resist again. The resistance of the material continues up to a certain maximum value σ pr, then gradual destruction begins. The quantity σ pr is called tensile strength (synonym for steel: tensile strength, for concrete - cubic or prismatic strength). The following designations are also used:
=R b
A similar relationship is observed in experiments between shear stresses and shears.
3) Duhamel–Neumann law (linear temperature expansion):
In the presence of a temperature difference, bodies change their size, and in direct proportion to this temperature difference.
Let there be a temperature difference 
. Then this law looks like:
Here α - coefficient of linear thermal expansion, l - rod length, Δ l- its lengthening.
4) Law of Creep .
Research has shown that all materials are highly heterogeneous in small areas. The schematic structure of steel is shown in Fig. 8.2.
Some of the components have the properties of a liquid, so many materials under load receive additional elongation over time 
(Fig. 8.3.) (metals at high temperatures, concrete, wood, plastics - at normal temperatures). This phenomenon is called creep material.
The law for liquids is: the greater the force, the greater the speed of movement of the body in the liquid. If this relationship is linear (i.e. force is proportional to speed), then it can be written as:
E 
If we move on to relative forces and relative elongations, we get
Here the index " cr
"means that the part of the elongation that is caused by the creep of the material is considered. Mechanical characteristics 
called the viscosity coefficient.
Law of energy conservation.
Consider a loaded beam
Let us introduce the concept of moving a point, for example,
- vertical movement of point B;
- horizontal displacement of point C.
Powers 
while doing some work U.
Considering that the forces 
begin to increase gradually and assuming that they increase in proportion to displacements, we obtain:
.
According to the conservation law: no work disappears, it is spent on doing other work or turns into another energy (energy- this is the work that the body can do.).
Work of forces 
, is spent on overcoming the resistance of elastic forces arising in our body. To calculate this work, we take into account that the body can be considered to consist of small elastic particles. Let's consider one of them:
It is subject to tension from neighboring particles 
. The resultant stress will be 
Under the influence 
the particle will elongate. According to the definition, elongation is the elongation per unit length. Then:
Let's calculate the work dW, which the force does dN (here it is also taken into account that the forces dN begin to increase gradually and they increase proportionally to the movements):
For the whole body we get:
.
Job W which was committed 
, called elastic deformation energy.
According to the law of conservation of energy:
6)Principle possible movements .
This is one of the options for writing the law of conservation of energy.
Let the forces act on the beam F 1
,
F 2
,
…
. They cause points to move in the body 
and voltage 
. Let's give the body additional small possible movements
. In mechanics, a notation of the form 
means the phrase “possible value of the quantity A" These possible movements will cause the body additional possible deformations
. They will lead to the appearance of additional external forces and stresses 
,
δ.
Let us calculate the work of external forces on additional possible small displacements:
Here 
- additional movements of those points at which forces are applied F 1
,
F 2
,
…
Consider again a small element with a cross section dA and length dz (see Fig. 8.5. and 8.6.). According to the definition, additional elongation dz of this element is calculated by the formula:
dz= dz.
The tensile force of the element will be:
dN = (+δ) dA ≈ dA..
The work of internal forces on additional displacements is calculated for a small element as follows:
dW = dN dz = dA dz = dV
WITH 
summing up the deformation energy of all small elements we obtain the total deformation energy:
Law of energy conservation W = U gives:
.
This ratio is called principle of possible movements(it is also called principle of virtual movements). Similarly, we can consider the case when tangential stresses also act. Then we can obtain that to the deformation energy W the following term will be added:
Here is the shear stress, is the displacement of the small element. Then principle of possible movements will take the form:
Unlike the previous form of writing the law of conservation of energy, there is no assumption here that the forces begin to increase gradually, and they increase in proportion to the displacements
7) Poisson effect.
Let us consider the pattern of sample elongation:
The phenomenon of shortening a body element across the direction of elongation is called Poisson effect.
Let us find the longitudinal relative deformation.
The transverse relative deformation will be:
Poisson's ratio the quantity is called:
For isotropic materials (steel, cast iron, concrete) Poisson's ratio
This means that in the transverse direction the deformation less longitudinal
Note
: modern technologies can create composite materials with Poisson's ratio >1, that is, the transverse deformation will be greater than the longitudinal one. For example, this is the case for a material reinforced with rigid fibers at a low angle 
<<1
(см. рис.8.8.). Оказывается, что коэффициент
Пуассона при этом почти пропорционален
величине
, i.e. the less 
, the larger the Poisson's ratio.
Fig.8.8. Fig.8.9
Even more surprising is the material shown in (Fig. 8.9.), and for such reinforcement there is a paradoxical result - longitudinal elongation leads to an increase in the size of the body in the transverse direction.
8) Generalized Hooke's law.
Let's consider an element that stretches in the longitudinal and transverse directions. Let us find the deformation that occurs in these directions. 
Let's calculate the deformation 
arising from action 
:
Let's consider the deformation from the action 
, which arises as a result of the Poisson effect:
The overall deformation will be:
If valid and 
, then another shortening will be added in the direction of the x axis 
.
Hence: 
Likewise: 
These relations are called generalized Hooke's law.
It is interesting that when writing Hooke’s law, an assumption is made about the independence of elongation strains from shear strains (about independence from shear stresses, which is the same thing) and vice versa. Experiments well confirm these assumptions. Looking ahead, we note that strength, on the contrary, strongly depends on the combination of tangential and normal stresses.
Note: The above laws and assumptions are confirmed by numerous direct and indirect experiments, but, like all other laws, they have a limited scope of applicability.
Hooke's law was discovered in the 17th century by the Englishman Robert Hooke. This discovery about the stretching of a spring is one of the laws of elasticity theory and plays an important role in science and technology.
Definition and formula of Hooke's law
The formulation of this law is as follows: the elastic force that appears at the moment of deformation of a body is proportional to the elongation of the body and is directed opposite to the movement of particles of this body relative to other particles during deformation.
The mathematical notation of the law looks like this:
Rice. 1. Formula of Hooke's law
Where Fupr– accordingly, the elastic force, x– elongation of the body (the distance by which the original length of the body changes), and k– proportionality coefficient, called body rigidity. Force is measured in Newtons, and elongation of a body is measured in meters.
To reveal the physical meaning of stiffness, you need to substitute the unit in which elongation is measured in the formula for Hooke’s law - 1 m, having previously obtained an expression for k.
Rice. 2. Body stiffness formula
This formula shows that the stiffness of a body is numerically equal to the elastic force that occurs in the body (spring) when it is deformed by 1 m. It is known that the stiffness of a spring depends on its shape, size and the material from which the body is made.
Elastic force
Now that we know what formula expresses Hooke’s law, it is necessary to understand its basic value. The main quantity is the elastic force. It appears at a certain moment when the body begins to deform, for example, when a spring is compressed or stretched. It is directed in the opposite direction from gravity. When the elastic force and the force of gravity acting on the body become equal, the support and the body stop.
Deformation is an irreversible change that occurs in the size of the body and its shape. They are associated with the movement of particles relative to each other. If a person sits in a soft chair, then deformation will occur to the chair, that is, its characteristics will change. It comes in different types: bending, stretching, compression, shear, torsion.
Since the elastic force is related in origin to electromagnetic forces, you should know that it arises due to the fact that molecules and atoms - the smallest particles that make up all bodies - attract and repel each other. If the distance between the particles is very small, then they are affected by the repulsive force. If this distance is increased, then the force of attraction will act on them. Thus, the difference between attractive and repulsive forces manifests itself in elastic forces.
The elastic force includes the ground reaction force and body weight. The strength of the reaction is of particular interest. This is the force that acts on a body when it is placed on any surface. If the body is suspended, then the force acting on it is called the tension force of the thread.
Features of elastic forces
As we have already found out, the elastic force arises during deformation, and it is aimed at restoring the original shapes and sizes strictly perpendicular to the deformed surface. Elastic forces also have a number of features.
- they occur during deformation;
- they appear in two deformable bodies simultaneously;
- they are perpendicular to the surface in relation to which the body is deformed.
- they are opposite in direction to the displacement of body particles.
Application of the law in practice
Hooke's law is applied both in technical and high-tech devices, and in nature itself. For example, elastic forces are found in watch mechanisms, in shock absorbers in transport, in ropes, rubber bands, and even in human bones. The principle of Hooke's law underlies the dynamometer, a device used to measure force.