Generalized Hooke's law equation of motion. Hooke's law definition and formula
The stress and strain states considered above are components of a single physical entity - the stress-strain state at a point of the body.
When solving specific problems, it is necessary to take into account the physical relationships that exist between stresses and strains. In statically determinate problems, it is possible to find stresses without physical relationships, using only equilibrium equations. There is no such possibility in statically indeterminate problems.
The relationship between stresses and strains is usually established through experiments, and its complexity depends on the properties of the material. For isotropic materials widely used in practice, linear dependences are used, with the help of which it is possible to carry out calculations when stresses change over a fairly wide range.
Let us analyze the dependence between the components of the stressed and deformed states at a point of the body, using the principle of independence of the action of forces. To this end, we cut out an elementary parallelepiped from a solid body (Fig. 10.10).
Rice. 10.10.
Consider the case of action on the element of only shear stress t gu / (Fig. 10.10, but). In this case, the right angle changes only in planes parallel to the plane hu. Similarly, we can consider the angular displacements that arise from the action of shear stresses x yz and xzv . Assuming that the material is isotropic and there is a linear relationship between shear stresses and angular displacements, we arrive at the relations
where G- modulus of elasticity of the second kind.
Let us analyze the displacements caused by the action of normal stresses in the direction of the axis Oh(Fig. 10.10, b). The deformation caused by this stress in the direction of the Ox axis is equal to ct v /?, and in the direction of the other two axes, the displacements are determined using Poisson's ratio v according to the formula -vg v/?. Similarly, deformations in the direction of the axis Oh from and at and a 2 . Finally, by summing the deformations in all directions, we obtain
When the body temperature changes, the values a At, where At- change in body temperature; a is the coefficient of linear thermal expansion of the isotropic material. As for formulas (10.37), they will remain unchanged.
Relations (10.37) and (10.38) are called generalized Hooke's law for the case of a linearly elastic isotropic material.
When making calculations, the inverse relations are also useful:
Note that when deriving physical relationships, we implicitly assumed that the directions of the principal stresses and principal strains coincide with each other. This assumption is called conditions of coaxiality of stress and strain tensors.
In the case of anisotropic materials, whose properties differ in different directions, the coaxiality condition is not satisfied. For elastic anisotropic materials, the generalized Hooke's law is written in the following form:
Here a t -- constants of elasticity, which express the properties of the material. Let us introduce the notation
Then relations (10.40) can be represented in a vector-matrix form:
where (a) and (e) - vectors, respectively, stresses and strains; [BUT] matrix of elastic properties of the material.
For an isotropic linear elastic material of three constants E, G and v, as we established earlier, only two of them are independent. The matrix of elastic properties of such a material is as follows:
When writing the generalized Hooke's law for an anisotropic material (10.40), 36 constants were used. Let us establish how many of these quantities are independent. Consider two stress states (Fig. 10.11).
Rice. 10.11.
Element extension in direction at, due to the stress state of the first direction (Fig. 10.11, but), equals dAvl/= a 2 p x dy. Similarly, the elongation of the element in the first direction, due to the second stress state, is determined (Fig. 10.11, b): dA f/x = a x p y dx.
According to the principle of reciprocity of work
whence it follows that I |2 = a 21 .
Similarly, 14 more equalities can be obtained a:j= a jt ,i,j = 1, 2,..., 6, i*j. Material Compliance Matrix BUT is symmetrical. Thus, for anisotropic materials, out of 36 characteristics, only 21 are independent.
When analyzing composite materials, one has to deal with special cases of anisotropy. The most common case is orthotropic material, characterized by symmetry about three mutually perpendicular axes. An example of such anisotropy is wood. The elastic properties of an orthotropic medium are described by nine independent constants:
where by the symmetry property
The elastic constants of composite materials are in most cases determined experimentally.
- The notation of stresses and strains in the form of vector quantities is of a formal nature and is introduced for convenience.
As you know, physics studies all the laws of nature: from the simplest to the most general principles of natural science. Even in those areas where, it would seem, physics is not able to figure it out, it still plays a primary role, and every slightest law, every principle - nothing escapes it.
In contact with
It is physics that is the basis of the foundations, it is this that lies at the origins of all sciences.
Physics studies the interaction of all bodies, both paradoxically small and incredibly large. Modern physics is actively studying not just small, but hypothetical bodies, and even this sheds light on the essence of the universe.
Physics is divided into sections, this simplifies not only the science itself and its understanding, but also the methodology of study. Mechanics is concerned with the motion of bodies and the interaction of moving bodies, thermodynamics with thermal processes, and electrodynamics with electrical processes.
Why deformation should be studied by mechanics
Speaking of contractions or tensions, one should ask oneself the question: which branch of physics should study this process? With strong distortions, heat can be released, perhaps thermodynamics should deal with these processes? Sometimes, when liquids are compressed, it begins to boil, and when gases are compressed, liquids are formed? So what, the hydrodynamics should learn the deformation? Or molecular kinetic theory?
It all depends on the force of deformation, on its degree. If the deformable medium (a material that is compressed or stretched) allows, and the compression is small, it makes sense to consider this process as the movement of some points of the body relative to others.
And since the question is purely concerned, it means that mechanics will deal with this.
Hooke's law and the condition for its implementation
In 1660, the famous English scientist Robert Hooke discovered a phenomenon that can be used to mechanically describe the process of deformation.
In order to understand under what conditions Hooke's law is fulfilled, We restrict ourselves to two options:
- Wednesday;
- strength.
There are such media (for example, gases, liquids, especially viscous liquids close to solid states or, conversely, very fluid liquids) for which it is impossible to describe the process mechanically. And vice versa, there are such environments in which, with sufficiently large forces, the mechanics ceases to “work”.
Important! To the question: "Under what conditions is Hooke's law fulfilled?", one can give a definite answer: "For small deformations."
Hooke's law, definition: The deformation that occurs in a body is directly proportional to the force that causes that deformation.
Naturally, this definition implies that:
- compression or tension is small;
- the object is elastic;
- it consists of a material in which there are no non-linear processes as a result of compression or tension.
Hooke's law in mathematical form
Hooke's formulation, which we have given above, makes it possible to write it in the following form:
where is the change in the length of the body due to compression or tension, F is the force applied to the body and causing deformation (elastic force), k is the coefficient of elasticity, measured in N/m.
It should be remembered that Hooke's law valid only for small stretches.
We also note that it has the same form under tension and compression. Given that the force is a vector quantity and has a direction, then in the case of compression, the following formula will be more accurate:
But again, it all depends on where the axis will be directed, relative to which you are measuring.
What is the fundamental difference between compression and stretching? Nothing if it's insignificant.
The degree of applicability can be considered in the following form:
Let's take a look at the chart. As you can see, with small tensions (the first quarter of the coordinates), for a long time the force with the coordinate has a linear relationship (red straight line), but then the real dependence (dashed line) becomes nonlinear, and the law ceases to be fulfilled. In practice, this is reflected by such a strong stretch that the spring stops returning to its original position and loses its properties. With more stretch fracture occurs and the structure collapses material.
With small compressions (the third quarter of the coordinates), for a long time the force with the coordinate also has a linear relationship (red line), but then the real dependence (dashed line) becomes nonlinear, and everything again ceases to be true. In practice, this is reflected by such strong compression that heat starts to radiate and the spring loses its properties. With even greater compression, the coils of the spring “stick together” and it begins to deform vertically, and then completely melts.
As you can see, the formula expressing the law allows you to find the force, knowing the change in the length of the body, or, knowing the force of elasticity, measure the change in length:
Also, in some cases, you can find the coefficient of elasticity. To understand how this is done, consider an example task:
A dynamometer is connected to the spring. She was stretched, applying a force of 20, because of which she began to have a length of 1 meter. Then they let her go, waited until the vibrations stopped, and she returned to her normal state. In normal condition, its length was 87.5 centimeters. Let's try to find out what material the spring is made of.
Find the numerical value of the spring deformation:
From here we can express the value of the coefficient:
After looking at the table, we can find that this indicator corresponds to spring steel.
Trouble with the coefficient of elasticity
Physics, as you know, is a very precise science, moreover, it is so precise that it has created entire applied sciences that measure errors. As the standard of unwavering precision, she cannot afford to be clumsy.
Practice shows that the linear dependence we have considered is nothing more than Hooke's law for a thin and tensile rod. Only as an exception can it be used for springs, but even this is undesirable.
It turns out that the coefficient k is a variable, which depends not only on what material the body is made of, but also on the diameter and its linear dimensions.
For this reason, our conclusions require clarification and development, otherwise, the formula:
can not be called anything other than a relationship between three variables.
Young's modulus
Let's try to figure out the coefficient of elasticity. This parameter, as we found out, depends on three quantities:
- material (which suits us quite well);
- length L (which indicates its dependence on);
- area S.
Important! Thus, if we manage to somehow “separate” the length L and the area S from the coefficient, then we will get a coefficient that completely depends on the material.
What we know:
- the larger the cross-sectional area of the body, the greater the coefficient k, and the dependence is linear;
- the longer the body length, the smaller the coefficient k, and the dependence is inversely proportional.
So, we can write the coefficient of elasticity in this way:
where E is a new coefficient, which now exactly depends solely on the type of material.
Let us introduce the concept of “relative elongation”:
.
Output
We formulate Hooke's law for tension and compression: at low compressions, the normal stress is directly proportional to the relative elongation.
The coefficient E is called Young's modulus and depends solely on the material.
When a rod is stretched and compressed, its length and cross-sectional dimensions change. If we mentally select from the rod in the 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 to
and relative linear deformation e x is defined by the equality
Since the axis Oh coincides with the axis of the rod, along which external loads act, we call the deformation e x longitudinal deformation, for which the index will be omitted below. Deformations in directions perpendicular to the axis are called transverse deformations. If denoted 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 the transverse and longitudinal deformations during the central tension and compression of the rod are interconnected by the dependence
The quantity v included in this equality is called Poisson's ratio or coefficient of transverse deformation. This coefficient is one of the main constants of elasticity of the material and characterizes its ability to transverse deformations. For each material, it is determined from a tensile or compression test (see § 3.5) and is calculated by the formula
As follows from equality (3.6), longitudinal and transverse strains always have opposite signs, which is a confirmation of the obvious fact - when stretched, the dimensions of the cross section decrease, and when compressed, 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 cork wood, Poisson's ratio is close to zero, while for rubber it is close to 0.5. For many metals at normal temperatures, the value of Poisson's ratio is in the range of 0.25 + 0.35.
As established in numerous experiments, for most structural materials at small strains, 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 is called the modulus of elasticity. The modulus of elasticity is the second main constant of elasticity of a material and characterizes its rigidity. Since strains are dimensionless quantities, it follows from (3.7) that the modulus of elasticity has the dimension of stress.
In table. 3.1 shows the values of the modulus of elasticity and Poisson's ratio for various materials.
When designing and calculating structures, along with the calculation of stresses, it is also necessary to determine the displacements of individual points and nodes of structures. Consider a method for calculating displacements under central tension and compression of bars.
Absolute element extension length dx(Fig. 3.6) according to formula (3.5) is
Table 3.1
|
Material name |
Modulus of elasticity, MPa |
Coefficient Poisson |
|
Carbon steel |
||
|
aluminum alloys |
||
|
Titanium alloys |
||
|
(1.15-s-1.6) 10 5 |
||
|
along the fibers |
(0,1 ^ 0,12) 10 5 |
|
|
across the fibers |
(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 in the range from 0 to x, we get
where them) - axial displacement of an arbitrary section (Fig. 3.7), and C= and( 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
The elongation or shortening of the rod is equal to the axial displacement of its free end (Fig. 3.7), the value of which we obtain from (3.8), assuming x = 1:
Substituting in the formula (3.8) the expression for the deformation? from Hooke's law (3.7), we obtain
For a rod made of a material with a constant modulus of elasticity E axial displacements are determined by the formula
The integral included in this equality can be calculated in two ways. The first way is to analytically write the function Oh) and subsequent integration. The second method is based on the fact that the integral under consideration is numerically equal to the plot area a in the section. Introducing the notation
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 is equal to R. The stresses a according to (3.4) are also constant and equal to 
Then from (3.10) we obtain
It follows from this formula 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, find the elongation of the rod:
Work EF called stiffness of the rod in tension and compression. The larger this value, the smaller the elongation or shortening of the rod.
Consider a rod under the action of a uniformly distributed load (Fig. 3.8). The longitudinal force in an arbitrary section, spaced at a distance x from the fastening, is equal to
Dividing N on the F, we get the formula for stresses
Substituting this expression into (3.10) and integrating, we find
The largest displacement, equal to the elongation of the entire rod, is obtained by substituting x = / into (3.13):
From formulas (3.12) and (3.13) it can be seen that if the stresses depend linearly on x, then the displacements change according to the law of a square parabola. Plots N, oh and And shown in fig. 3.8.
General differential dependence linking functions them) 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 change in the function noted in the above examples them).
In addition, it can be noted that if in any section the stresses a vanish, then on 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 plot areas about for different areas, we find:
section x = 1 m:
section x = 3 m:
section x = 5 m:
On the upper section of the diagram bar And is a square parabola (Fig. 3.2, e). In this case, there is an extremum in the section x = 1 m. In the lower section, the character 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, but) stretched by force R ( its lengthening 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 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 above.
In conclusion, it should be noted that, despite the small value of displacements and elongations (shortenings) of rods under tension and compression, they cannot be neglected. The ability to calculate these quantities is important in many technological problems (for example, when assembling structures), as well as for solving statically indeterminate problems.
The action of external forces on a solid body leads to the appearance of stresses and strains at points in its volume. In this case, the stress state at a point, the relationship between stresses at different sites passing through this point, are determined by the equations of statics and do not depend on the physical properties of the material. The deformed state, the relationship between displacements and deformations are established using geometric or kinematic considerations and also do not depend on the properties of the material. In order to establish a relationship between stresses and strains, it is necessary to take into account the actual properties of the material and the loading conditions. Mathematical models describing the relationship between stresses and strains are developed on the basis of experimental data. These models should reflect the real properties of materials and loading conditions with a sufficient degree of accuracy.
The most common for structural materials are models of elasticity and plasticity. Elasticity is the property of a body to change shape and size under the action of external loads and restore its original configuration when the loads are removed. Mathematically, the property of elasticity is expressed in the establishment of a one-to-one functional relationship between the components of the stress tensor and the strain tensor. The property of elasticity reflects not only the properties of materials, but also the loading conditions. For most structural materials, the elasticity property manifests itself at moderate values of external forces, leading to small deformations, and at low loading rates, when energy losses due to temperature effects are negligible. A material is called linearly elastic if the components of the stress tensor and the strain tensor are connected by linear relations.
At high levels of loading, when significant deformations occur in the body, the material partially loses its elastic properties: when unloaded, its original dimensions and shape are not completely restored, and when external loads are completely removed, residual deformations are recorded. In this case the relationship between stresses and strains ceases to be unambiguous. This material property is called plasticity. The residual deformations accumulated in the process of plastic deformation are called plastic.
A high level of stress can cause destruction, i.e., the division of the body into parts. Solid bodies made of different materials are destroyed at different amounts of deformation. Fracture is brittle at small strains and occurs, as a rule, without noticeable plastic deformations. Such destruction is typical for cast iron, alloy steels, concrete, glass, ceramics and some other structural materials. For low-carbon steels, non-ferrous metals, plastics, a plastic type of fracture is characteristic in the presence of significant residual deformations. However, the division of materials according to the nature of their destruction into brittle and ductile is very conditional; it usually refers to some standard operating conditions. One and the same material can behave, depending on the conditions (temperature, nature of the load, manufacturing technology, etc.), as brittle or as ductile. For example, materials that are plastic at normal temperatures are destroyed as brittle at low temperatures. Therefore, it is more correct to speak not about brittle and plastic materials, but about the brittle or plastic state of the material.
Let the material be linearly elastic and isotropic. Let us consider an elementary volume under conditions of a uniaxial stress state (Fig. 1), so that the stress tensor has the form
Under such loading, there is an increase in dimensions in the direction of the axis Oh, characterized by linear deformation, which is proportional to the magnitude of the stress
Fig.1. Uniaxial stress state
This ratio is a mathematical notation Hooke's law establishing a proportional relationship between stress and the corresponding linear deformation in a uniaxial stress state. The coefficient of proportionality E is called the modulus of longitudinal elasticity or Young's modulus. It has the dimension of stresses.
Along with the increase in size in the direction of action; under the same stress, the dimensions decrease in two orthogonal directions (Fig. 1). The corresponding deformations will be denoted by and , and these deformations are negative for positive ones and are proportional to :
With the simultaneous action of stresses along three orthogonal axes, when there are no tangential stresses, the principle of superposition (superposition of solutions) is valid for a linear elastic material:
Taking into account formulas (1 - 4), we obtain
|
Tangential stresses cause angular deformations, and at small deformations they do not affect the change in linear dimensions, and therefore, linear deformations. Therefore, they are also valid in the case of an arbitrary stress state and express the so-called generalized Hooke's law.
The angular deformation is due to the shear stress , and the deformations and are due to the stresses and , respectively. Between the corresponding shear stresses and angular deformations for a linearly elastic isotropic body, there are proportional relationships
which express the law Hook on shift. The proportionality factor G is called shear module. It is significant that the normal stress does not affect the angular deformations, since in this case only the linear dimensions of the segments change, and not the angles between them (Fig. 1).
A linear relationship also exists between the average stress (2.18), which is proportional to the first invariant of the stress tensor, and the volumetric strain (2.32), which coincides with the first invariant of the strain tensor:
Fig.2. Planar shear strain
Corresponding aspect ratio TO called bulk modulus of elasticity.
Formulas (1 - 7) include the elastic characteristics of the material E, , G And TO, determining its elastic properties. However, these characteristics are not independent. For an isotropic material, two independent elastic characteristics are usually chosen as the elastic modulus E and Poisson's ratio. To express the shear modulus G across E And , Let us consider a plane shear deformation under the action of shear stresses (Fig. 2). To simplify the calculations, we use a square element with a side but. Calculate the principal stresses , . These stresses act on sites located at an angle to the original sites. From fig. 2 find the relationship between linear deformation in the direction of stress and angular deformation . The major diagonal of the rhombus characterizing the deformation is equal to
For small deformations
Given these ratios
Before deformation, this diagonal had the size . Then we will have
From the generalized Hooke's law (5) we obtain
Comparison of the obtained formula with the Hooke's law with shift (6) gives
As a result, we get
Comparing this expression with Hooke's volumetric law (7), we arrive at the result
Mechanical characteristics E, , G And TO are found after processing the experimental data of testing specimens for various types of loads. From the physical point of view, all these characteristics cannot be negative. In addition, it follows from the last expression that Poisson's ratio for an isotropic material does not exceed 1/2. Thus, we obtain the following restrictions for the elastic constants of an isotropic material:
Limit value leads to limit value , which corresponds to an incompressible material ( at ). In conclusion, we express the stresses in terms of deformations from the elasticity relations (5). We write the first of relations (5) in the form
Using equality (9), we will have
Similar relations can be derived for and . As a result, we get
|
Here relation (8) for the shear modulus is used. In addition, the designation
POTENTIAL ENERGY OF ELASTIC DEFORMATION
Consider first the elementary volume dV=dxdydz under conditions of uniaxial stress state (Fig. 1). Mentally fix the platform x=0(Fig. 3). A force acts on the opposite side .
This force does work in displacement. .
As the voltage increases from zero to the value
the corresponding deformation, by virtue of Hooke's law, also increases from zero to the value ,
and the work is proportional to the shaded one in Fig. 4 squares: 
. If we neglect the kinetic energy and losses associated with thermal, electromagnetic and other phenomena, then, by virtue of the law of conservation of energy, the work done will turn into potential energy accumulated during the deformation process: .
F= dU/dV called specific potential energy of deformation, which has the meaning of the potential energy accumulated in a unit volume of the body. In the case of a uniaxial stress state
8.2. Basic Laws Used in Strength of Materials
Relationships of statics. 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. With a simple tension of the beam by the longitudinal force N= F this law can be written as:
Here 
longitudinal force, l- bar length, BUT- 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 shear stresses and shear angle:
.
G
calledshear modulus
, less often - the elastic modulus of the second kind. Like any law, it has a limit of applicability and Hooke's law. Voltage 
, up to which Hooke's law is valid, is called limit of proportionality(this is the most important characteristic in sopromat).
Let's depict the dependence
from
graphically (Fig. 8.1). This painting is called stretch diagram
. After point B (i.e. at 
), this dependence is no longer linear.
At 
after unloading, residual deformations appear in the body, therefore 
called elastic limit
.
When the stress 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 (i.e. behaves like a liquid). Graphically, this means that the diagram is parallel to the abscissa (DL plot). The stress σ t at which the material flows is called yield strength .
Some materials (Art. 3 - building 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 value 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 dependence is observed in experiments between tangential stresses and shears.
3) Dugamel–Neumann law (linear thermal expansion):
In the presence of a temperature difference, the body changes its size, and is directly proportional to this temperature difference.
Let there be a temperature difference 
. Then this law takes the form:
Here α - coefficient of linear thermal expansion, l - rod length, Δ l- its lengthening.
4) law of creep .
Studies have shown that all materials are highly inhomogeneous in the small. The schematic structure of steel is shown in Fig. 8.2.
Some of the components have fluid properties, so many materials under load gain additional elongation over time. 
(fig.8.3.) (metals at high temperatures, concrete, wood, plastics - at ordinary temperatures). This phenomenon is called creep material.
For a liquid, the law is true: the greater the force, the greater the speed of the body in the fluid. If this relationship is linear (i.e. force is proportional to speed), then it can be written as:
E 
If we go over 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 characteristic 
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 offset of point C.
Forces 
while doing some work U.
Considering that the forces 
begin to increase gradually and assuming that they increase in proportion to displacements, we get:
.
According to the conservation law: no work disappears, it is spent on doing other work or goes into another energy (energy is the work that the body can do.
The work of forces 
, is spent on overcoming the resistance of the elastic forces that arise in our body. To calculate this work, we take into account that the body can be considered as consisting of small elastic particles. Let's consider one of them:
From the side of neighboring particles, a stress acts on it 
. The resultant stress will be 
Under the influence 
the particle is elongated. By definition, elongation is the elongation per unit length. Then:
Let's calculate the work dW that the force does dN (here it is also taken into account that the forces dN begin to increase gradually and they increase in proportion to displacements):
For the whole body we get:
.
Work W committed 
, called elastic deformation energy.
According to the law of conservation of energy:
6)Principle possible movements .
This is one of the ways to write the law of conservation of energy.
Let forces act on the beam F 1
,
F 2
,
…
. They cause points to move in the body 
and stress 
. Let's give the body additional small possible displacements
. In mechanics, the record of the form 
means the phrase "possible value of the quantity but". These possible movements will cause in 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 displacements of those points where 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
FROM 
summing the strain energy of all small elements, we obtain the total strain energy:
Law of energy conservation W = U gives:
.
This ratio is called the principle of possible movements(also called principle of virtual movements). Similarly, we can consider the case when shear stresses also act. Then it can be obtained that the strain energy W add the following term:
Here - shear stress, - shear of a 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.
Consider the elongation pattern of the sample:
The phenomenon of shortening of a body element across the direction of lengthening is called Poisson effect.
Let us find the longitudinal relative deformation.
The transverse relative deformation will be:
Poisson's ratio 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 a Poisson ratio > 1, that is, the transverse deformation will be greater than the longitudinal one. For example, this is the case for material reinforced with hard fibers at a low angle. 
<<1
(см. рис.8.8.). Оказывается, что коэффициент
Пуассона при этом почти пропорционален
величине
, i.e. the less 
, the greater 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, a paradoxical result takes place - longitudinal elongation leads to an increase in the size of the body in the transverse direction.
8) Generalized Hooke's law.
Consider an element that stretches in the longitudinal and transverse directions. Let us find the deformation arising in these directions. 
Calculate the deformation 
arising from the action 
:
Consider the deformation from the action 
, which results from the Poisson effect:
The total deformation will be:
If it works and 
, then add one more shortening in the direction of the x-axis 
.
Consequently: 
Similarly: 
These ratios are called generalized Hooke's law.
Interestingly, 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 the strength, on the contrary, strongly depends on the combination of shear 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 area of applicability.
