Relationship between transverse and longitudinal deformation. Longitudinal and transverse deformations

The ratio of the absolute elongation of the rod to its original length is called relative elongation (- epsilon) or longitudinal deformation. Longitudinal deformation is a dimensionless quantity. Dimensionless deformation formula:

In tension, the longitudinal deformation is considered positive, and in compression, negative.
The transverse dimensions of the rod as a result of deformation also change, while they decrease during tension, and increase during compression. If the material is isotropic, then its transverse deformations are equal to each other:
.
It has been experimentally established that during tension (compression) within the limits of elastic deformations, the ratio of transverse to longitudinal deformation is a constant value for a given material. The modulus of the ratio of transverse to longitudinal strain, called Poisson's ratio or transverse strain ratio, is calculated by the formula:

For different materials, Poisson's ratio varies within. For example, for cork, for rubber, for steel, for gold.

Hooke's law
The elastic force that occurs in the body when it is deformed is directly proportional to the magnitude of this deformation
For a thin tensile rod, Hooke's law has the form:

Here is the force that stretches (compresses) the rod, is the absolute elongation (compression) of the rod, and is the coefficient of elasticity (or stiffness).
The coefficient of elasticity depends both on the properties of the material and on the dimensions of the rod. It is possible to distinguish the dependence on the dimensions of the rod (cross-sectional area and length) explicitly by writing the coefficient of elasticity as

The value is called the modulus of elasticity of the first kind or Young's modulus and is a mechanical characteristic of the material.
If you enter a relative elongation

And the normal stress in the cross section

Then Hooke's law in relative units will be written as

In this form, it is valid for any small volumes of material.
Also, when calculating straight rods, Hooke's law is used in relative form

Young's modulus
Young's modulus (modulus of elasticity) is a physical quantity that characterizes the properties of a material to resist tension / compression during elastic deformation.
Young's modulus is calculated as follows:

Where:
E - modulus of elasticity,
F - strength,
S is the area of the surface over which the action of the force is distributed,
l is the length of the deformable rod,
x is the modulus of change in the length of the rod as a result of elastic deformation (measured in the same units as the length l).
Through Young's modulus, the velocity of propagation of a longitudinal wave in a thin rod is calculated:

Where is the density of the substance.
Poisson's ratio
Poisson's ratio (denoted as or) is the absolute value of the ratio of the transverse to longitudinal relative deformation of a material sample. This coefficient does not depend on the size of the body, but on the nature of the material from which the sample is made.
The equation
,
where
- Poisson's ratio;
- deformation in the transverse direction (negative in axial tension, positive in axial compression);
- longitudinal deformation (positive in axial tension, negative in axial compression).

Laws of R. Hooke and S. Poisson

Let us consider the deformations of the rod shown in fig. 2.2.

Rice. 2.2 Longitudinal and transverse tensile strains

Denote by the absolute elongation of the rod. When stretched, this is a positive value. Through - absolute transverse deformation. When stretched, this is a negative value. Signs and accordingly change during compression.

Relations

(epsilon) or , (2.2)

called relative elongation. It is positive in tension.

Relations

Or , (2.3)

called relative transverse strain. It is negative when stretched.

R. Hooke in 1660 discovered the law, which read: "What is the elongation, such is the force." In modern writing, R. Hooke's law is written as follows:

that is, the stress is proportional to the relative strain. Here, E. Young's modulus of elasticity of the first kind is a physical constant within the limits of R. Hooke's law. It is different for different materials. For example, for steel it is 2 10 6 kgf / cm 2 (2 10 5 MPa), for wood - 1 10 5 kgf / cm 2 (1 10 4 MPa), for rubber - 100 kgf / cm 2 ( 10 MPa), etc.

Taking into account that , and , we get

where is the longitudinal force on the power section;

- the length of the power section;

– tensile-compressive stiffness.

That is, the absolute deformation is proportional to the longitudinal force acting on the power section, the length of this section, and inversely proportional to the tensile-compression stiffness.

When calculating by the action of external loads

where is the external longitudinal force;

is the length of the section of the rod on which it acts. In this case, the principle of independence of action of forces* is applied.

S. Poisson proved that the ratio is a constant value, different for different materials, that is

or , (2.7)

where is the S. Poisson ratio. This is, generally speaking, a negative value. In reference books, its value is given "modulo". For example, for steel it is 0.25 ... 0.33, for cast iron - 0.23 ... 0.27, for rubber - 0.5, for cork - 0, that is. However, for wood it can be more than 0.5.

Experimental study of the processes of deformation and

Destruction of tensioned and compressed rods

Russian scientist V.V. Kirpichev proved that the deformations of geometrically similar samples are similar if the forces acting on them are similarly located, and that the results of testing a small sample can be used to judge the mechanical characteristics of the material. In this case, of course, the scale factor is taken into account, for which a scale factor determined experimentally is introduced.

Mild Steel Tension Chart

Tests are carried out on discontinuous machines with simultaneous recording of the fracture diagram in coordinates - force, - absolute deformation (Fig. 2.3, a). Then the experiment is recalculated in order to construct a conditional diagram in coordinates (Fig. 2.3, b).

According to the diagram (Fig. 2.3, a), the following can be traced:

- Hooke's law is valid up to the point;

- from point to point, deformations remain elastic, but Hooke's law is no longer valid;

- from point to point, deformations grow without increasing load. Here, the cement skeleton of the ferrite grains of the metal is destroyed, and the load is transferred to these grains. Chernov–Luders shear lines appear (at an angle of 45° to the sample axis);

- from point to point - the stage of secondary hardening of the metal. At the point, the load reaches its maximum, and then a narrowing appears in the weakened section of the sample - the “neck”;

- at the point - the sample is destroyed.

Rice. 2.3 Fracture diagrams of steel in tension and compression

Diagrams allow you to get the following basic mechanical characteristics of steel:

- proportionality limit - the highest stress up to which Hooke's law is valid (2100 ... 2200 kgf / cm 2 or 210 ... 220 MPa);

- elastic limit - the highest stress at which the deformations still remain elastic (2300 kgf / cm 2 or 230 MPa);

- yield strength - stress at which deformations grow without increasing load (2400 kgf / cm 2 or 240 MPa);

- strength limit - stress corresponding to the highest load withstood by the sample during the experiment (3800 ... 4700 kgf / cm 2 or 380 ... 470 MPa);

Stresses and strains in tension and compression are interconnected by a linear relationship, which is called Hooke's law , named after the English physicist R. Hooke (1653-1703), who established this law.
Hooke's law can be formulated as follows: normal stress is directly proportional to relative elongation or shortening .

Mathematically, this dependence is written as follows:

σ = Eε.

Here E - coefficient of proportionality, which characterizes the rigidity of the material of the beam, i.e. its ability to resist deformation; he is called modulus of elasticity , or modulus of elasticity of the first kind .
The modulus of elasticity, like stress, is expressed in terms of pascals (Pa) .

Values E for various materials are established experimentally and experimentally, and their value can be found in the relevant reference books.
So, for steel E \u003d (1.96 ... 2.16) x 105 MPa, for copper E \u003d (1.00 ... 1.30) x 105 MPa, etc.

It should be noted that Hooke's law is valid only within certain loading limits.
If we substitute the previously obtained values of relative elongation and stress into the formula of Hooke's law: ε = ∆l / l ,σ = N / A , then you can get the following dependency:

Δl \u003d N l / (E A).

The product of the modulus of elasticity and the cross-sectional area E × BUT , standing in the denominator, is called the stiffness of the section in tension and compression; it simultaneously characterizes the physical and mechanical properties of the material of the beam and the geometric dimensions of the cross section of this beam.

The above formula can be read as follows: the absolute elongation or shortening of a beam is directly proportional to the longitudinal force and length of the beam, and inversely proportional to the rigidity of the beam section.
Expression E A / l called stiffness of the beam in tension and compression .

The above formulas of Hooke's law are valid only for bars and their sections having a constant cross section, made of the same material and with a constant force. For a beam that has several sections that differ in material, cross-sectional dimensions, longitudinal force, the change in the length of the entire beam is determined as the algebraic sum of the extensions or shortenings of individual sections:

Δl = Σ (Δl i)

Deformation

Deformation(English) deformation) is a change in the shape and size of a body (or part of a body) under the influence of external forces, with changes in temperature, humidity, phase transformations and other influences that cause a change in the position of body particles. With increasing stress, the deformation can end in destruction. The ability of materials to resist deformation and destruction under the influence of various types of loads is characterized by the mechanical properties of these materials.

On the appearance of one or another type of deformation the nature of the stresses applied to the body has a great influence. Alone deformation processes are associated with the predominant action of the tangential component of the stress, others - with the action of its normal component.

Types of deformation

By the nature of the load applied to the body types of deformation subdivided as follows:

Tensile deformation;
compression deformation;
Shear (or shear) deformation;
Torsional deformation;
Bending deformation.

TO the simplest types of deformation include: tensile strain, compressive strain, shear strain. The following types of deformation are also distinguished: deformation of all-round compression, torsion, bending, which are various combinations of the simplest types of deformation (shear, compression, tension), since the force applied to the body subjected to deformation is usually not perpendicular to its surface, but is directed at an angle , which causes both normal and shear stresses. By studying the types of deformation engaged in such sciences as solid state physics, materials science, crystallography.

In solids, in particular metals, they emit two main types of deformations- elastic and plastic deformation, the physical nature of which is different.

A shear is a type of deformation when only shear forces occur in cross sections.. Such a stressed state corresponds to the action on the rod of two equal oppositely directed and infinitely close transverse forces (Fig. 2.13, a, b) causing a shear along a plane located between the forces.

Rice. 2.13. Shear strain and stress

The cut is preceded by deformation - the distortion of the right angle between two mutually perpendicular lines. At the same time, on the faces of the selected element (Fig. 2.13, in) shear stresses occur. The amount of offset of the faces is called absolute shift. The value of the absolute shift depends on the distance h between planes of force F. The shear deformation is more fully characterized by the angle by which the right angles of the element change - relative shift:

. (2.27)

Using the previously considered method of sections, it is easy to verify that only shear forces arise on the side faces of the selected element Q=F, which are the resultant shear stresses:

Taking into account that shear stresses are distributed uniformly over the cross section BUT, their value is determined by the ratio:

. (2.29)

It has been experimentally established that within the limits of elastic deformations, the magnitude of shear stresses is proportional to the relative shear (Hooke's law in shear):

where G is the modulus of elasticity in shear (modulus of elasticity of the second kind).

There is a relationship between the moduli of longitudinal elasticity and shear

where is Poisson's ratio.

Approximate values of the modulus of elasticity in shear, MPa: steel - 0.8·10 5 ; cast iron - 0.45 10 5; copper - 0.4 10 4; aluminum - 0.26 10 5; rubber - 4.

2.4.1.1. Shear strength calculations

Pure shear in real structures is extremely difficult to implement, since due to the deformation of the connected elements, an additional bending of the rod occurs, even with a relatively small distance between the planes of action of forces. However, in a number of designs, the normal stresses in the cross sections are small and can be neglected. In this case, the condition of the strength reliability of the part has the form:

, (2.31)

where - allowable shear stress, which is usually assigned depending on the magnitude of the allowable tensile stress:

– for plastic materials under static load =(0.5…0.6) ;

- for fragile ones - \u003d (0.7 ... 1.0) .

2.4.1.2. Shear stiffness calculations

They are reduced to limiting elastic deformations. By solving expression (2.27)–(2.30) together, the magnitude of the absolute shift is determined:

, (2.32)

where is the shear stiffness.

Torsion

2.4.2.1. Plotting Torques

2.4.2.2. Torsional deformations

2.4.2.4. Geometric characteristics of sections

2.4.2.5. Torsional Strength and Stiffness Calculations

Torsion is a type of deformation when a single force factor arises in cross sections - torque.

Torsional deformation occurs when the beam is loaded by pairs of forces, the planes of action of which are perpendicular to its longitudinal axis.

2.4.2.1. Plotting Torques

To determine the stresses and deformations of the beam, a torque diagram is built showing the distribution of torques along the length of the beam. Applying the method of sections and considering any part in equilibrium, it becomes obvious that the moment of internal elastic forces (torque) must balance the action of external (rotating) moments on the considered part of the beam. It is customary to consider the moment positive if the observer looks at the section under consideration from the side of the outer normal and sees the torque T directed counter-clockwise. In the opposite direction, the moment is assigned a minus sign.

For example, the equilibrium condition for the left side of the beam has the form (Fig. 2.14):

- in section A-A:

- in section B-B:

The boundaries of the sections in the construction of the diagram are the planes of action of torques.

Rice. 2.14. Calculation scheme of a bar (shaft) in torsion

2.4.2.2. Torsional deformations

If a grid is applied to the side surface of a rod of circular cross section (Fig. 2.15, but) from equidistant circles and generators, and apply pairs of forces with moments to the free ends T in planes perpendicular to the axis of the rod, then with a small deformation (Fig. 2.15, b) can be found:

Rice. 2.15. Diagram of torsion deformation

· generatrices of the cylinder turn into large pitch helical lines;

· the squares formed by the grid turn into rhombuses, i.e. there is a shift of cross sections;

sections, round and flat before deformation, retain their shape after deformation;

The distance between the cross sections remains virtually unchanged;

· there is a rotation of one section relative to another by a certain angle.

Based on these observations, the theory of bar torsion is based on the following assumptions:

cross-sections of the beam, flat and normal to its axis before deformation, remain flat and normal to the axis after deformation;

Equidistant cross-sections rotate relative to each other at equal angles;

· the radii of cross-sections do not bend during deformation;

Only tangential stresses occur in cross sections. Normal stresses are small. The length of the beam can be considered unchanged;

· the material of the bar during deformation obeys Hooke's law in shear: .

In accordance with these hypotheses, the torsion of a rod with a circular cross section is represented as the result of shifts caused by the mutual rotation of the sections.

On a rod of circular cross section with a radius r, sealed at one end and loaded with torque T at the other end (Fig. 2.16, but), denote on the lateral surface the generatrix AD, which under the action of the moment will take the position AD 1. On distance Z from the termination, select an element with a length dZ. As a result of torsion, the left end of this element will turn by an angle , and the right end by an angle (). Formative sun element will take position B 1 From 1, deviating from the initial position by an angle . Due to the smallness of this angle

The ratio represents the angle of twist per unit length of the rod and is called relative angle of twist. Then

Rice. 2.16. Design scheme for determining stresses
during torsion of a rod of circular cross section

Taking into account (2.33), Hooke's law in torsion can be described by the expression:

. (2.34)

By virtue of the hypothesis that the radii of circular cross sections are not curved, shear shear stresses in the vicinity of any point of the body located at a distance from the center (Fig. 2.16, b) are equal to the product

those. proportional to its distance from the axis.

The value of the relative angle of twist by the formula (2.35) can be found from the condition that the elementary circumferential force () on an elementary area of size dA, located at a distance from the axis of the beam, creates an elementary moment relative to the axis (Fig. 2.16, b):

The sum of elementary moments acting over the entire cross section BUT, is equal to the torque M Z. Considering that:

The integral is a purely geometric characteristic and is called polar moment of inertia of the section.

Under the action of tensile forces along the axis of the beam, its length increases, and the transverse dimensions decrease. Under the action of compressive forces, the opposite occurs. On fig. 6 shows a beam stretched by two forces P. As a result of tension, the beam lengthened by Δ l, which is called absolute elongation, and get absolute transverse constriction Δa .

The ratio of the magnitude of absolute elongation and shortening to the original length or width of the beam is called relative deformation. In this case, the relative deformation is called longitudinal deformation, but - relative transverse deformation. The ratio of relative transverse strain to relative longitudinal strain is called Poisson's ratio: (3.1)

Poisson's ratio for each material as an elastic constant is determined empirically and is within: ; for steel.

Within the limits of elastic deformations, it is established that the normal stress is directly proportional to the relative longitudinal deformation. This dependency is called Hooke's law:

, (3.2)

where E is the coefficient of proportionality, called modulus of normal elasticity.

Let, as a result of deformation, the initial length of the rod l will become equal. l 1. Changing the length

is called the absolute elongation of the bar.

In tension, the longitudinal deformation is considered positive, and in compression, negative.

The transverse dimensions of the rod as a result of deformation also change, while they decrease during tension, and increase during compression. If the material is isotropic, then its transverse deformations are equal to each other:

It has been experimentally established that during tension (compression) within the limits of elastic deformations, the ratio of transverse to longitudinal deformation is a constant value for a given material. The modulus of the ratio of transverse to longitudinal strain, called Poisson's ratio or transverse strain ratio, is calculated by the formula:

For different materials, Poisson's ratio varies within . For example, for cork, for rubber, for steel, for gold.

Longitudinal and transverse deformations. Poisson's ratio. Hooke's law

Poisson's ratio for each material as an elastic constant is determined empirically and is within: ; for steel.

Within the limits of elastic deformations, it is established that the normal stress is directly proportional to the relative longitudinal deformation. This dependency is called Hooke's law:

, (3.2)

where E is the coefficient of proportionality, called modulus of normal elasticity.

If we substitute the expression into the formula of Hooke's law and , then we get the formula for determining the elongation or shortening in tension and compression:

, (3.3)

where is the product EF is called tensile and compressive stiffness.

Longitudinal and transverse deformations. Hooke's law

Have an idea about longitudinal and transverse deformations and their relationship.

Know Hooke's law, dependencies and formulas for calculating stresses and displacements.

To be able to carry out calculations on the strength and stiffness of statically determinate bars in tension and compression.

Tensile and Compressive Deformations

Consider the deformation of the beam under the action of the longitudinal force F(Fig. 4.13).

The initial dimensions of the beam: - initial length, - initial width. The beam is extended by the amount Δl; Δ1- absolute elongation. When stretched, the transverse dimensions decrease, Δ but- absolute narrowing; ∆1 > 0; Δ but 0.

In the resistance of materials, it is customary to calculate deformations in relative units: fig.4.13

- relative extension;

Relative contraction.

There is a relationship between longitudinal and transverse strains ε'=με, where μ is the coefficient of transverse strain, or Poisson's ratio, is a characteristic of the plasticity of the material.

Encyclopedia of Mechanical Engineering XXL

Equipment, materials science, mechanics and.

Longitudinal deformation in tension (compression)

It has been experimentally established that the ratio of transverse strain ej. to longitudinal deformation e under tension (compression) up to the limit of proportionality for a given material is a constant value. Denoting the absolute value of this ratio (X), we get

Experiments have established that the relative transverse strain eo in tension (compression) is a certain part of the longitudinal strain e, i.e.

The ratio of transverse to longitudinal strain in tension (compression), taken as an absolute value.

In the previous chapters of the strength of materials, simple types of beam deformation were considered - tension (compression), shear, torsion, direct bending, characterized by the fact that in the cross sections of the beam there is only one internal force factor during tension (compression) - longitudinal force, during shear - transverse force, in torsion - torque, in pure straight bending - bending moment in a plane passing through one of the main central axes of the beam cross section. With direct transverse bending, two internal force factors arise - a bending moment and a transverse force, but this type of beam deformation is referred to as simple, since the combined effect of these force factors is not taken into account in strength calculations.

When stretched (compressed), the transverse dimensions also change. The ratio of the relative transverse strain e to the relative longitudinal strain e is a physical constant of the material and is called Poisson's ratio V = e/e.

When stretching (compressing) the beam, its longitudinal and transverse dimensions receive changes characterized by deformations of the longitudinal prod (bg) and transverse (e, e). which are related by the relation

As experience shows, when the beam is stretched (compressed), its volume changes slightly with an increase in the length of the beam by the value Ar, each side of its section decreases by We will call the relative longitudinal deformation the value

Longitudinal and transverse elastic deformations that occur during tension or compression are related to each other by the dependence

So, consider a beam of isotropic material. The hypothesis of flat sections establishes such a geometry of deformations in tension and compression that all longitudinal fibers of the beam have the same deformation x, regardless of their position in the cross section F, i.e.

An experimental study of volumetric deformations was carried out under tension and compression of fiberglass samples with simultaneous registration on a K-12-21 oscilloscope of changes in the longitudinal and transverse deformations of the material and the force under loading (on a testing machine TsD-10). The test until reaching the maximum load was carried out at almost constant loading speeds, which was ensured by a special regulator that the machine is equipped with.

As experiments show, the ratio of the transverse strain b to the longitudinal strain e in tension or compression for a given material within the application of Hooke's law is a constant value. This ratio, taken in absolute value, is called the transverse strain ratio or Poisson's ratio.

Here /p(szh) - longitudinal deformation in tension (compression) /u - transverse deformation in bending I - length of the deformable beam P - area of its cross section / - moment of inertia of the cross-sectional area of the sample relative to the neutral axis - polar moment of inertia P - applied force -torsion moment - coefficient, uchi-

The deformation of the rod during tension or compression consists in changing its length and cross section. Relative longitudinal and transverse deformations are determined, respectively, by the formulas

The ratio of the height of the side plates (tank walls) to the width in batteries of significant dimensions is usually more than two, which makes it possible to calculate the tank walls using the formulas for cylindrical bending of the plates. The tank lid is not rigidly fastened to the walls and cannot prevent their buckling. Neglecting the influence of the bottom, it is possible to reduce the calculation of the tank under the action of horizontal forces to the calculation of a closed statically indeterminate frame-strip separated from the tank by two horizontal sections. The modulus of normal elasticity of glass-reinforced plastic is relatively small; therefore, structures made of this material are sensitive to buckling. The strength limits of fiberglass in tension, compression and bending are different. A comparison of the calculated stresses with the limiting ones should be made for the deformation that is predominant.

Let us introduce the notation used in the algorithm, the values with indices 1,1-1 refer to the current and previous iterations at the time stage m - Am, m and 2 - respectively, the rate of longitudinal (axial) deformation in tension (i > > 0) and compression (2 deformations are related by the relation

Relationships (4.21) and (4.31) were tested on a large number of materials and under various loading conditions. The tests were carried out in tension-compression at a frequency of about one cycle per minute and one cycle per 10 minutes over a wide range of temperatures. Both longitudinal and transverse strain gauges were used to measure strains. At the same time, solid (cylindrical and corset) and tubular samples were tested from boiler steel 22k (at temperatures of 20-450 C and asymmetries - 1, -0.9 -0.7 and -0.3, in addition, the samples were welded and with notch), heat-resistant steel TS (at temperatures of 20-550 ° C and asymmetries -1 -0.9 -0.7 and -0.3), heat-resistant nickel alloy EI-437B (at 700 ° C), steel 16GNMA, ChSN , Kh18N10T, steel 45, aluminum alloy AD-33 (with asymmetries -1 0 -b0.5), etc. All materials were tested as delivered.

The coefficient of proportionality E, linking both normal stress and longitudinal deformation, is called the modulus of elasticity in tension-compression of the material. This coefficient has other names, the modulus of elasticity of the 1st kind, Young's modulus. Elastic modulus E is one of the most important physical constants characterizing the ability of a material to resist elastic deformation. The larger this value, the less the beam is stretched or compressed when the same force P is applied.

If we assume that in Fig. 2-20, and the shaft O is the driving one, and the shafts O1 and O2 are driven, then when the disconnector is turned off, the thrust LL1 and L1L2 will work in compression, and when turned on, in tension. As long as the distances between the axes of the shafts O, 0 and O2 are small (up to 2000 mm), the difference between the deformation of the rod in tension and compression (longitudinal bending) does not affect the operation of the synchronous transmission. In a disconnector for 150 kV, the distance between the poles is 2800 mm, for 330 kV - 3500 mm, for 750 kV - 10,000 mm. With such large distances between the centers of the shafts and significant loads that they must transmit, they say /> d. This length is chosen for reasons of greater stability, since a long sample, in addition to compression, may experience buckling deformation, which will be discussed in the second part of the course. Samples of building materials are made in the form of a cube with dimensions of 100 X YuO X YuO or 150 X X 150 X 150 mm. During the compression test, the cylindrical sample assumes an initially barrel-shaped shape. If it is made of a plastic material, then further loading leads to flattening of the sample; if the material is brittle, then the sample suddenly cracks.

At any point of the beam under consideration, there is the same stress state and, therefore, the linear deformations (see 1.5) are the same for all its currents. Therefore, the value can be defined as the ratio of the absolute elongation A / to the original length of the beam /, i.e. e, = A / / /. Linear deformation during tension or compression of the beams is usually called relative elongation (or relative longitudinal deformation) and is denoted e.

See pages where the term is mentioned Longitudinal deformation in tension (compression) : Technical Handbook of the Railwayman Volume 2 (1951) - [ c.11 ]

Longitudinal and transverse deformations in tension - compression. Hooke's law

When tensile loads are applied to the rod, its initial length / increases (Fig. 2.8). Let us denote the length increment by A/. The ratio of the increase in the length of the rod to its original length is called elongation or longitudinal deformation and is denoted by g:

Relative elongation is a dimensionless value, in some cases it is customary to express it as a percentage:

When stretched, the dimensions of the rod change not only in the longitudinal direction, but also in the transverse direction - the rod narrows.

Rice. 2.8. Tensile deformation of the rod

Change ratio A but cross-sectional size to its original size is called relative transverse narrowing or transverse deformation.

It has been experimentally established that there is a relationship between the longitudinal and transverse deformations

where p is called Poisson's ratio and are constant for a given material.

Poisson's ratio is, as can be seen from the above formula, the ratio of transverse to longitudinal deformation:

For various materials, Poisson's ratio values range from 0 to 0.5.

On average, for metals and alloys, Poisson's ratio is approximately 0.3 (Table 2.1).

The value of Poisson's ratio

When compressed, the picture is reversed, i.e. in the transverse direction, the initial dimensions decrease, and in the transverse direction, they increase.

Numerous experiments show that up to certain loading limits for most materials, the stresses that arise during tension or compression of the rod are in a certain dependence on the longitudinal deformation. This dependency is called Hooke's law, which can be formulated as follows.

Within known loading limits, there is a directly proportional relationship between the longitudinal deformation and the corresponding normal stress

Proportionality factor E called modulus of longitudinal elasticity. It has the same dimension as the voltage, i.e. measured in Pa, MPa.

The modulus of longitudinal elasticity is a physical constant of a given material, which characterizes the ability of a material to resist elastic deformations. For a given material, the modulus of elasticity varies within narrow limits. So, for steel of different grades E=(1.9. 2.15) 10 5 MPa.

For the most commonly used materials, the modulus of elasticity has the following values in MPa (Table 2.2).

The value of the modulus of elasticity for the most commonly used materials

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