Basics of the conversion. Definition of tangent stresses. Mostzat.in.ua: voltage value of normal voltage at the point

The voltages are characterized by a numerical value and direction, i.e. the voltage is a vector inclined under a thorough angle to the section under consideration.

Suppose that at a point of any body cross section at a small area A acts the force F at some angle to the site (Fig. 63, a). Obeling this force f into area A, we will find the average voltage that occurs at the point M (Fig. 63, b):

True voltages at point m are determined when moving to the limit

Vector magnitude rcalled full voltage At point.

Full tension r It is possible to decompose into the components: according to the normal (perpendicular) to the site A and on the tangent of it (Fig, 63, B).

The voltage component according to the normal is called normal voltage at a given point of section and denote the Greek letter (sigma); The constituent component is called the tangent of voltage and denote the Greek letter (Tau).

The normal voltage directed from the section is considered a positive aimed at the cross section - negative.

Normal stresses occur when the particles, located on both sides of the section, seek to remove one from the other or to get close to the particle. Tangent stresses occur when the particles tend to move one relative to the other in the cross section plane.

The tangent stress can be decomposed on the coordinate axes into two components and (Fig.1.6, B). The first index indicates which axis is perpendicular to the cross section, the second - in parallel what axis acts voltage. If in the calculations, the direction of the tangent voltage does not matter, it is denoted without indices.

There is a dependency between the full voltage and its constituent

The voltage at which the material destruction occurs or noticeable plastic deformations occur, called the limit.

Task 4.1.1: A combination of stresses arising on the set of platforms passing through the point under consideration are called ...

2) full voltage;

3) normal voltage;

4) by tangent.

Decision:

1) The answer is correct. The stress state at the point is completely determined by the six components of the stress tensor: Σ X., σ y., σ Z., τ XY., τ yz., τ ZX.. Knowing these components, you can define voltages on any site passing through this point. A combination of stresses acting on a plurality of sites (sections) passing through this point is called a stressful state at the point.

2) The answer is incorrect! Ignorance of the determination of the full voltage at the point (the force per unit area of \u200b\u200bthe section).

3) The answer is incorrect! Recall that the projection of the vector of full voltage on the normal to the cross section is called normal voltage.

4) The answer is incorrect! An error is allowed in determining the term "tangent stress".
The projection of the full voltage vector on the axis lying in the cross section plane is called tangent.

Task 4.1.2: The playgrounds in the studied point of the intense body, on which tangent stresses are zero, called ...

1) oriented; 2) main platforms;

Decision:

1) The answer is incorrect! The term does not correspond to a given condition. Under the oriented areas are understood, which pass through the point at a predetermined direction.

2) The answer is correct.

At the rotation of the elementary volume 1, it is possible to find such a spatial orientation 2, in which tangent stresses on its edges will disappear and only normal voltages will remain (some of them may be zero). The sites (face), on which tangent stresses are zero, are called main sites.

3) The answer is incorrect! The term does not correspond to a given condition. Octahedrically referred to the grounds are placed to the main. Tangent stresses at octahedral sites are not equal to zero.

4) The answer is incorrect! We remind you that under the sections understand the sites carried out through the point in which the intense state is investigated.

Task 4.1.3: The main stresses for the intense state shown in the figure are equal to ... (voltage values \u200b\u200bare indicated in MPa).

1) σ 1 \u003d 150 MPa, σ 2 \u003d 50 MPa; 2) σ 1 \u003d 0 MPa, σ 2 \u003d 50 MPa, Σ 3 \u003d 150 MPa;

3) σ 1 \u003d 150 MPa, σ 2 \u003d 50 MPa, Σ 3 \u003d 0 MPa;

4) σ 1 \u003d 100 MPa, σ 2 \u003d 100 MPa, Σ 3 \u003d 0 MPa;

Decision:

1) The answer is incorrect! The value of the main voltage σ 3 \u003d 0 MPa is not specified.

2) The answer is incorrect! The designations of the main stresses do not comply with the rules of numbering.

3) The answer is correct. One edge of the element is free from tangent stresses. Therefore, this is the main platform, and the normal voltage (main voltage) on this site is also zero.
To determine the two other values \u200b\u200bof the main stresses, we use the formula
,
where the positive directions of voltages are shown in the figure.

For the example above we have ,, After transformations we find
In accordance with the rule of numbering the main stresses, we have,, i.e. Flat tense state.

4) The answer is incorrect! These are not the main stresses, but the specified values \u200b\u200bof normal voltages acting on the dedicated element.

Task 4.1.4: In the studied point of the intense body on the three main sites, the values \u200b\u200bof normal stresses are determined: the main stresses in this case are equal ...

1) σ 1 \u003d 150 MPa, σ 2 \u003d 50 MPa, σ 3 \u003d -100 MPa;

2) σ 1 \u003d 150 MPa, σ 2 \u003d -100 MPa, Σ 3 \u003d 50 MPa;

3) σ 1 \u003d 50 MPa, σ 2 \u003d -100 MPa, Σ 3 \u003d 150 MPa;

4) σ 1 \u003d -100 MPa, Σ 2 \u003d 50 MPa, Σ 3 \u003d 150 MPa;

Decision:

1) The answer is correct. The main stresses are assigned indexes 1, 2, 3 so that the condition is carried out. Hence,

2), 3), 4) the answer is incorrect! The main stresses are assigned indices 1, 2, 3 so that condition (in the algebraic sense) is satisfied.

Task 4.1.5: On the edges of the elementary volume (see Figure) defined voltage values \u200b\u200bin MPa. The angle between the positive axis direction x. And the external normal to the main site, on which the minimum main stress is valid, is equal to ...

1) ; 2) ; 3) ; 4) .

Decision:

1), 2), 4) The answer is incorrect! Apparently, the formula for determining the angle is incorrectly recorded. Proper entry:

3) The answer is correct.

The angle is determined by the formula
Substituting the numeric values \u200b\u200bof the voltages, we obtain the negative angle, lay the angle clockwise.

Task 4.1.6: The values \u200b\u200bof the main stresses are determined from the solution of the cubic equation the coefficients, called ...

1) intense state invariants; 2) elastic constant;

4) proportionate coefficients.

Decision:

1) The answer is correct. The roots of the equation are the main stresses - are determined by the nature of the intense state at the point and do not depend on the choice of the source coordinate system. Consequently, when turning the coordinate axes of coordinate coefficients



must remain unchanged. They are called invariants of intense state.

2) The answer is incorrect! Error in determining the term. Elastic constant characterize the properties of the material.

3) The answer is incorrect! Recall that the cosine guides are cosine of the angles that form normal with the axes of coordinates.

4) The answer is incorrect! The term does not comply with the condition of the issue

Through any point of the intensed body, it is usually possible to conduct _____________ mutually perpendicular platforms (s), on which tangent stresses will be zero.

			three
			two
			four
			six

Decision:

The figure shows the body loaded by external forces, and elemental volume with voltages on its faces. With a mental rotation of the elementary volume, it is possible to find such a spatial orientation in which tangent stresses on the edges will be zero. These faces will be the main platforms.

Topic: Stressful state at point. Main sites and main stresses
The main axes of the intense state are called ...

Decision:

The figure shows an elementary volume isolated in the vicinity of an arbitrary point of the loaded body. If, with a given orientation of the elementary volume, tangent stresses at its faces are zero, then the axis x., y., z. Called by the main axes of the intense state. When moving from one point to another direction, the main axes are generally changed.

Topic: Stressful state at point. Main sites and main stresses
Normal voltages operating on the main sites are called ...

Decision:
Three mutually perpendicular platforms on which there are no tangent stresses are called main sites. Normal voltages operating on the main sites are called main stresses. The maximum of the three main stresses is simultaneously the greatest full voltage acting on a plurality of sites passing through this point. The minimum of the three main stresses is the smallest of a plurality of complete voltages.

Topic: Stressful state at point. Main sites and main stresses

The intense state of the elementary volume shown in the figure is flat. The upper facet of the elementary volume is the main platform. The position of the two other main sites is determined by the angle

Decision:

The figure shows an elementary volume (top view). The direction of normal to the main platform is determined by the formula where - the angle between the positive direction of the axis x. And the normal to one of the main sites. For our case, substituting these values \u200b\u200bin the formula, get from where

Topic: Stressful state at point. Main sites and main stresses

The figure shows the rod, stretched forces F., and elemental volume isolated with edges parallel to the planes of the rod. When turning the elementary volume around the axis " u.»At an angle equal to 45 0, stressful condition ...

Decision:
In the figure, the elementary volume is highlighted by the main platforms. Main stresses: stressful state - linear. The type of stress state does not depend on the spatial orientation of the elementary volume and at any corner of the rotation remains linear.

4.2. Types of intense state

Task 4.2.1: Rod Round Diameter d. He is experiencing a pure bending and twist. Stressful state at the point IN Showing in the picture ...

1) ; 2) ; 3) ; 4) .

Decision:

1) The answer is incorrect! The torque causes the appearance of tangent stresses in the plane of the perpendicular axis of the rod.

2) The answer is incorrect! Direction of tangent voltage at point IN The cross section should correspond to the direction of torque in this section.

3) The answer is correct. Separate planes oriented along and across the axis of the rod, select the volumetric element. In the cross section of the rod in the sealing acts a bending moment M. And torque 2m. From bending moment M. At point IN Normal tensile tension occurs. Torque 2macting in the plane perpendicular to the axis of the rod, causes a tangent stress. The direction of tangent stress must be coordinated with the direction of torque. Therefore, the stressful state of the element in Figure 4 corresponds to the intense state at the point IN.

4) The answer is incorrect! From torque at point IN The cross section arises by the tangent stress. The direction of tangent stress must be coordinated with the direction of torque.

Task 4.2.2: The rod is experiencing stretching and pure bending. The intense state that occurs at a dangerous point is called ...

1) flat; 2) volume; 3) linear; 4) pure shift.

Decision:

1) The answer is incorrect! With a flat stressed state, one value of the main voltage is zero.

2) The answer is incorrect! At a dangerous point, only one main voltage is different from zero. With a voluminous stress state, three main stresses are different from zero.

3) The answer is correct. Dangerous points are located infinitely close to the top edge of the element. They have only stretching normal stresses from the longitudinal force and bending moment. Epures of voltage distribution from each internal power factor and the resulting step are shown in the figure.

Consequently, at a dangerous point there will be a linear intense state.

4) The answer is incorrect! With a pure shift, two main stresses are equal, but are opposed to the sign, and the third is zero.

Task 4.2.3: Stressful state "Clean shift" is shown in the picture ...

1) ; 2) ; 3) ; 4) .

Decision:

1) The answer is incorrect! The figure shows a flat stress state - a two-axis stretching.

2) The answer is incorrect! The element is under a flat intense state - a two-axis mixed stress state.

3) The answer is correct.

Pure shift is a stressful state when only tangent stresses apply on the edges of the selected elementary volume. If the elementary volume is rotated to an angle equal, then the tangent stresses at its edges (sites) will be zero, but the normal (main) voltages will appear. Thus, the pure shift can be realized by stretching and compression in two mutually perpendicular directions with voltages equal to an absolute value.
Consequently, the intense state "pure shift" is shown in Figure 3.

4) The answer is incorrect! This element is experiencing a linear intense state.

Task 4.2.4: The type of strenuous state shown in the figure is called ...

1) linear; 2) flat; 3) volume; 4) pure shift.

Decision:

1) The answer is correct. The type of stress state is determined depending on the values \u200b\u200bof the main stresses. In the example, one face is free from tangent stresses is the main playground. Normal voltage operating on the main site is called the main voltage. In this case, it is zero. Using the formula, we find two other main stresses. After the transformations we get ,. In accordance with the notation taken, we have ,. Two main stresses are zero. Consequently, the figure shows a linear intense state.

2) The answer is incorrect! With a flat stress state, one main voltage is zero. In this case, the two main stresses are zero.

3) The answer is incorrect! With a voluminous stress state in this case, the two main stresses are zero. Therefore, this intense state is not voluminous.

4) The answer is incorrect! With a pure shift ,. Calculations show that this case is incorrect.

Task 4.2.5: Stressful condition at values, called ...

1) volume; 2) pure shift; 3) flat; 4) linear.

Decision:

1) The answer is incorrect! With a voluminous stress state, all three main stresses are different from zero.

2) The answer is incorrect! With a pure shift, one value of the main voltage is zero, and the other two are equal in size, but are opposed to the sign.

3) The answer is correct. The type of stress state is determined by the values \u200b\u200bof the main stresses. In the case when all three main stresses are different from zero, we have a voluminous stress state. If one main voltage is zero - a flat stress state, and when two are zero - linear. Therefore, in this example there will be a flat intense state.

4) The answer is incorrect! With a linear stress state, only one main voltage is different from zero.

Task 4.2.6: On the edges of the elementary volume (see figure) the voltages specified in MPa. Stressful state at the point ...

1) linear; 2) flat (pure shift); 3) flat; 4) Volumetric.

Decision:

1) The answer is incorrect! Frontal edge of elementary volume is free from tangent stresses. This means that this line is the main platform and one of the three main stresses is equal to (-50 MPa). Two other major stresses determine the formula

2) The answer is incorrect! Recall that with a pure shift, one of the main stresses is zero. Two others are equal in absolute value and are opposite to the sign.

3) The answer is correct. The front edge of the elementary volume is free from tangent stresses. This means that it is the main platform and one of the three main stresses is equal to (-50 MPa). Two other major stresses Determine the formula

Delivering numeric values


Approaching the main stresses indexes, we have:

Thus, the stress state is flat (two-axis compression).

4) The answer is incorrect! Frontal edge of elementary volume is free from tangent stresses. This means that this line is the main platform and one of the three main stresses is equal to (-50 MPa). Two other major stresses can be determined by the formula
The calculation results will show which stress state is shown in the figure.

The intense state of the elementary volume shown in the figure is - ...

Decision:
The main stresses are the roots of the cubic equation
Where:



In our case, the cubic equation takes the view from where
Thus, the intense state of the elementary volume is linear (uniaxial stretching).

Subject: Types of intense State

A steel cube is inserted without a gap into a rigid clip (see Fig.). An evenly distributed intensity pressure operates on the upper edge of the cube r. The surfaces of the cube and the clip are absolutely smooth. The intense condition of the cube is shown in the picture ...

			in
			g.
			b.
			but

Decision:

The friction forces between the absolutely smooth surfaces of the cube and the clip are absent. Therefore, tangent stresses on the edges of the cube are zero, and all the faces are the main platforms. In the process of compressing the rib of the cube, directed along the axes x.and y., strive to lengthen. Extension along the axis y.it happens free. Extension along the axis x. It is impossible (prevents the tough clip). Due to the impossibility of elongation along the axis x.From the vertical planes of the closure on the cube there are efforts in the form of uniformly distributed over the area of \u200b\u200bloads with some intensity. Intensity r and should be considered as the main stresses. Thus, of the three main stresses one (on the front face of the cube). Therefore, the stress state of the cube is flat (Fig. in).

Subject: Types of intense State

The figure shows a rod working on a tension. Stressful state at the point TO is an - …

Decision:

At point TO cross-sectional is valid voltage from strength F.. The tangent of tangent of torque from the torque is shown in Figure 1. In the angular points therefore the intense state at the point TO - linear (uniaxial stretching, Fig. 2).

Subject: Types of intense State

The intense state of the elementary volume is - ...

Decision:

The upper facet of the elementary volume is the main platform, so one main voltage is the two other main stresses by calculating the formula
In this case (see Fig.) Substituting in the formula, we get
Assigning the corresponding indexes to the main stresses, we get
Stressful condition - volumetric.

Subject: Types of intense State

On the body acts uniformly distributed over the surface pressure r(See Fig.). The intense state of the elementary volume is - ...

Decision:

If the body acts uniformly distributed over the surface pressure r(See Fig.), The stressful state at any point of the body volumetric (three-axis compression). In this case, with any spatial orientation of the elementary volume.

The voltage is vector and as any vector can be represented normal (relative to the site) and tangential components (Fig. 2.3). The normal component of the voltage vector will denote by the tangent. Experimental studies found that the influence of normal and tangent stresses on the strength of the material is different, and therefore will continue to be necessary to separately consider the components of the stress vector.

Fig. 2.3. Normal and tangent stress in the site

Fig. 2.4. Tangent voltage with bolt cut

When stretching the bolt (see Fig. 2.2) In cross section, normal voltage is valid

When operating a bolt on a slice (Fig. 2.4) in the Sechenya P, an effort should arise, balancing force.

From equilibrium conditions it follows that

In fact, the last ratio determines some mean voltage in a section, which sometimes use for approximate strength estimates. In fig. 2.4 shows the type of bolt after exposure to considerable effort. The destruction of the bolt began, and one half of it was shifted relative to the other: a shift or cut was deformed.

Examples of stress determination in structural elements.

We will analyze the simplest examples in which the assumption of the uniform distribution of stresses can be considered almost acceptable. In such cases, the magnitudes of the stresses are determined by the method of sections from the equations of the statics (equations of equilibrium).

Top of a thin-walled round shaft.

The thin-walled round shaft (tube) transmits torque (for example, from the aviation motor to the air screw). It is required to determine the voltages in the cross section of the shaft (Fig. 2.5, a). We carry out the plane of the section P perpendicular to the axis of the shaft and consider the equilibrium of the cut-off part (Fig. 2.5, b).

Fig. 2.5. True thin-walled round shaft

From the condition of the axial symmetry, given the low wall thickness, it can be assumed that the voltages in all points of the cross section are the same.

Strictly speaking, such an assumption is true only with a very small wall thickness, but in practical calculations it is used if the wall thickness

where is the average section radius.

The external forces applied to the cut-off part of the shaft are reduced only to the torque, and therefore normal stresses in cross section must be absent. Torque is balanced by tangent stresses, the moment of which is equal

Of the last ratio, we find the tangent stress in the sections of the shaft:

Voltages in a thin-walled cylindrical vessel (pipe).

In the thin-walled cylindrical vessel, the pressure is applied (Fig. 2.6, a).

We carry out the cross section with the plane P, perpendicular to the axis of the cylindrical shell, and consider the equilibrium of the cut-off part. The pressure acting on the vessel cover creates amplifies

This force is balanced by the shell arising in the cross section, and the intensity of the specified forces - the voltage will be equal to

The shell thickness 5 is assumed to be small compared to the average radius, the voltages are considered evenly distributed in all points of the cross section (Fig. 2.6, b).

However, not only voltages in the longitudinal direction, but also the circumferential (or annular) stresses in the perpendicular direction act on the pipe material. To identify them, we allocate two cross sections the ring of length I (Fig. 2.7), and then we will carry out the diametrical section separating the half of the ring.

In fig. 2.7, and show voltages on the surfaces of the cross section. Pressure is valid for the inside surface of the pipe

Fig. 2.8. Crack in a cylindrical shell under the action of destructive internal pressure

Voltage The intensity of the internal forces at the point of the body is called, that is, the voltage is an inner force entering the unit area. By nature, the voltage is arising on the inner surfaces of the contact of the parts of the body. The voltage, as well as the intensity of the external surface load, is expressed in units of force related to unit area: Pa \u003d N / m 2 (MPa \u003d 10 6 N / m 2, kgf / cm 2 \u003d 98 066 Pa ≈ 10 5 Pa, TC / m 2, etc.).

We highlight a small platform ΔA.. The inner force acting on it is denoted by Δ \\ VEC (R). Complete average voltage on this site \\ VEC (P) \u003d Δ \\ VEC (R) / ΔA. Find the limit of this relationship at Δa \\ to 0. This will be complete voltage on this site (point) of the body.

\\ TextStyle \\ Vec (P) \u003d \\ LIM _ (\\ Delta A \\ To 0) (\\ Delta \\ VEC (R) \\ Over \\ Delta A)

The total voltage \\ VEC P, as well as the equal internal forces applied on the elementary platform, is a vector value and can be decomposed into two components: perpendicular to the site under consideration - normal voltage Σ n and tangent to the site - tangent voltage \\ Tau_n. Here n. - Normal to the selected area.

The tangent stress, in turn, can be decomposed into two components parallel to the coordinate axes x, Y.associated with cross section - \\ Tau_ (NX), \\ Tau_ (NY). In the title of the tangent voltage, the first index indicates normal to the site, the second index is the direction of the tangent voltage.

$$ \\ VEC (P) \u003d \\ LEFT [\\ Matrix (\\ Sigma _n \\\\ \\ Tau _ (NX) \\\\ \\ Tau _ (NX)) \\ Right] $$

Note that in the future we will deal mainly not with full voltage \\ VEC P, but with its constituent σ_x, \\ tau _ (xy), \\ tau _ (xz). In the general case, two types of stresses may occur at the site: normal σ and tangent τ .

Tensor stresses

When analyzing stresses in the vicinity of the considered point, an infinitely small volumetric element is distinguished (parallelepiped with sides dX, DY, DZ) For each face of which there are, in general, three stresses, for example, for the edge, perpendicular axis X (x) - σ_x, \\ tau _ (xy), \\ tau _ (xz)

Voltage components in three perpendicular element faces form the voltage system described by a special matrix - tensor stress

$$ T _ \\ Sigma \u003d \\ Left [\\ Matrix (
\\ sigma _x \\ tau _ (yx) \\ tau _ (zx) \\\\
\\ Tau _ (xy) \\ sigma _y \\ tau _ (zy) \\\\ \\ tau _ (xz) \\ tau _ (yz) \\ sigma _z
) \\ RIGHT] $$

Here the first column represents the components of stresses on the courts,
Normal to the X axis, the second and third to the axis y and z, respectively.

When turning the axes of coordinates that coincide with the standards to the harms of the allocated
Element, voltage components are changed. Rotating the selected element around the coordinate axes, you can find this position of the element, in which all tangent stresses on the edges of the element are zero.

The playground on which tangent stresses are zero, called main platform .

Normal voltage on the main platform is called main voltage

Normal to the main site is called the main axis of stress .

At each point, you can spend three mutually perpendicular main sites.

When the coordinate axes turn the components of stresses, but the stress-strain state of the body (VAT) is changed.

Domestic efforts have the result of bringing the internal forces attached to the elementary section of the cross section. Voltage is a measure that characterizes the distribution of internal forces in cross section.

Suppose we know the tension in each elementary platform. Then you can write:

Longitudinal effort on the site dA: dn \u003d Σ z Da
Transverse force along the x axis: dQ X \u003d \\ Tau (ZX) DA
Transverse force along the Y axis: dQ Y \u003d \\ Tau (ZY) DA
Elementary moments around the axes x, y, z: $$ \\ begin (array) (LCR) DM _x \u003d Σ _z Da \\ Cdot Y \\\\ Dm _y \u003d Σ _z Da \\ Cdot X \\\\ Dm _z \u003d Dm _k \u003d \\ Tau _ (ZY) DA \\ CDOT X - \\ Tau _ (ZX) DA \\ CDOT Y \\ End (Array) $$

After performing integration on the cross-sectional area, we obtain:

That is, each internally effort is the total result of the voltage action throughout the transverse cross section of the body.

Stressful and deformed state of the elastic body. Communication between stresses and deformations

The concept of body voltage at this point. Normal and tangent stresses

Internal power factors arising when loading an elastic body characterize a state of a particular body cross section, but do not answer the question of which the cross-sectional point is the most loaded, or, as they say, dangerous point. Therefore, it is necessary to enter into consideration some additional value that characterizes the state of the body at this point.

If the body to which the external forces is applied is in equilibrium, then in any cross section there are internal forces of resistance. Denote by an internal force acting on the elementary platform, and the normal to this site is then the value

(3.1)

called full voltage.

In general, the total voltage does not coincide in the direction of the standard to the elementary platform, therefore it is more convenient to operate with the components along the coordinate axes -

If the external normal coincides with any coordinate axis, for example, with axis H., the components of the voltage will take a view of the component turns out to be perpendicular to the cross section and is called normal tension, and the components will lie in the cross section plane and are called tangent stresses.

To easily distinguish normal and tangent stresses usually apply other notation: - normal voltage - tangent.

We highlight from the body that under the action of external forces is infinitely small parallelepiped, the face of which is parallel to the coordinate planes, and the ribs have a length. On each face of such an elementary parallelepiped, there are three components of stresses parallel to the coordinate axes. In total, we obtain 18 components of stresses.

Normal voltages are referred to in the form where the index denotes the normal to the corresponding face (i.e., it can take values). Tangent stresses are related; Here, the first index corresponds to normal to the platform on which this tangent voltage acts, and the second indicates the axis parallel to which this voltage is directed (Fig. 1.1).

Fig.3.1. Normal and tangent stresses

For these stresses, the following is taken rule of signs. Normal tension It is considered positive when tensile, or that the same thing when it coincides with the direction of external normal to the site on which it acts. Tanner tension It is considered positive if on the site, the normal to which coincides with the direction of the coordinate axis parallel to it, it is directed towards the corresponding voltage of the positive coordinate axis.

The components of the stresses are the functions of the three coordinates. For example, normal voltage at the point with coordinates can be denoted

At the point, which is considered to be considered at an infinitely small distance, the voltage with an accuracy of an infinitely small first order can be decomposed into a series of Taylor:

For sites that are parallel to the plane only changes the coordinate h.and increment therefore on the verge of parallelepiped, which coincides with the plane normal voltage will, and on a parallel face, distinguished on an infinitely small distance, - The stresses on the other parallel edges of the parallelepiped are associated in the same way. Consequently, from the 18 components of the voltage unknown are only nine.

The law is proved in the theory of elasticity party tangent stressesAccording to which there are components of tangent stresses in two mutually perpendicular venues, perpendicular to the intersection lines of these sites are equal to each other:

It can be shown that voltages (3.3) do not simply characterize the intense state of the body at this point, but determine it is uniquely. The combination of these stresses forms a symmetric matrix, which is called tensor stress:

(3.4)

Since every point will be your stress tensor, then there is field Stress tensors.

When the tensor is multiplied by a scalar value, a new tensor will be obtained, all the components of which are many more components of the original tensor.