Poisson's ratio. Hooke's law. Longitudinal and transverse deformations. Hooke's law Hooke's law for absolute deformation

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

is called the absolute elongation of the rod.

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

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

The transverse dimensions of the rod also change as a result of deformation; when stretched, they decrease, and when compressed, they increase. If the material is isotropic, then its transverse deformations are equal:

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

For various 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

When tensile forces act along the axis of the beam, its length increases and its transverse dimensions decrease. When compressive forces act, the opposite phenomenon occurs. In Fig. Figure 6 shows a beam stretched by two forces P. As a result of tension, the beam lengthened by an amount Δ l, which is called absolute elongation, and we get absolute transverse contraction Δа .

The ratio of the 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, A - 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 experimentally and is within the limits: ; for steel.

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

, (3.2)

Where E- proportionality coefficient, called modulus of normal elasticity.

If we substitute the expression and , then we get a formula for determining elongation or shortening during tension and compression:

, (3.3)

where is the product EF called tensile and compressive stiffness.

Longitudinal and transverse deformations. Hooke's law

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

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

Be able to carry out calculations of the strength and stiffness of statically determined beams in tension and compression.

Tensile and compressive strains

Let us consider the deformation of a beam under the action of a longitudinal force F(Fig. 4.13).

Initial dimensions of the timber: - initial length, - initial width. The beam is lengthened by an amount Δl; Δ1- absolute elongation. When stretched, the transverse dimensions decrease, Δ A- absolute narrowing; Δ1 > 0; Δ A 0.

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

— relative elongation;

Relative narrowing.

There is a relationship between longitudinal and transverse deformations ε′=με, where μ is the transverse deformation coefficient, or Poisson’s ratio, a characteristic of the plasticity of the material.

Encyclopedia of Mechanical Engineering XXL

Equipment, materials science, mechanics, etc.

Longitudinal deformation in tension (compression)

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

Experiments have established that the relative transverse deformation eo during tension (compression) constitutes a certain part of the longitudinal deformation e, i.e.

The ratio of transverse to longitudinal deformation during tension (compression), taken in absolute value.

In previous chapters, material strengths were discussed simple types beam deformations - tension (compression), shear, torsion, straight bending, characterized by the fact that in the cross sections of the beam only one internal force factor arises during tension (compression) - longitudinal force, during shear - transverse force, during torsion - torque, with clean straight bend- bending moment in a plane passing through one of the main central axes cross section timber. With direct transverse bending two internal force factors arise - a bending moment and a transverse force, but this type of beam deformation is classified as simple, since when calculating the strength, the joint influence of these force factors is not taken into account.

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

When a beam is stretched (compressed), its longitudinal and transverse dimensions undergo changes characterized by longitudinal (bg) and transverse (e, e) deformations. which are related by the relation

As experience shows, when a beam is stretched (compressed), its volume changes slightly as the length of the beam increases by the value Ar, each side of its cross-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 relationship

So, let's consider a beam made of isotropic material. Hypothesis flat sections establishes such a geometry of deformations during 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 during tension and compression of fiberglass samples while simultaneously recording on a K-12-21 oscilloscope changes in longitudinal, transverse deformations of the material and force during loading (on a TsD-10 testing machine). Test to reach maximum load was carried out at almost constant loading speeds, which was ensured by a special regulator with which the machine was equipped.

As experiments show, the ratio of transverse deformation b to longitudinal deformation e during tension or compression for a given material, within the limits of 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 /р(сж) - longitudinal deformation during tension (compression) /u - transverse deformation during bending I - length of the deformed beam P - its cross-sectional area / - moment of inertia of the cross-sectional area of the sample relative to the neutral axis - polar moment of inertia P - applied force - torsional moment - coefficient, teaching -

The deformation of a rod during tension or compression consists of a change in 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 attached to the walls and cannot prevent them from bulging. Neglecting the influence of the bottom, it is possible to reduce the calculation of a tank under the action of horizontal forces on it to the calculation of a closed statically indeterminate strip frame separated from the tank by two horizontal sections. The normal elastic modulus of fiberglass is relatively small, so structures made from this material are sensitive to longitudinal bending. 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; quantities with indices 1,1-1 refer to the current and previous iteration at the time stage t - At, t and 2 - respectively, the rate of longitudinal (axial) deformation during tension (i > > 0) and compression (2 deformations are related by the relation

Dependencies (4.21) and (4.31) were checked for large number materials and different conditions loading. Tests were carried out under tension-compression with a frequency of about one cycle per minute and one cycle per 10 minutes over a wide temperature range. Both longitudinal and transverse strain gauges were used for strain measurements. At the same time, solid (cylindrical and corset) and tubular samples from boiler steel 22k were tested (at temperatures of 20-450 C and asymmetries - 1, -0.9 -0.7 and -0.3, in addition, welded samples and notch), heat-resistant steel TS (at temperatures 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 , Х18Н10Т, steel 45, aluminum alloy AD-33 (with asymmetries -1 0 -b0.5), etc. All materials were tested as delivered.

The proportionality coefficient E, which relates normal stress and longitudinal strain, is called the elastic modulus of the material in tension-compression. This coefficient also has other names: elastic modulus of the 1st kind, Young’s modulus. The elastic modulus E is one of the most important physical constants characterizing the ability of a material to resist elastic deformation. The greater this value, the less the beam stretches or contracts when the same force P is applied.

If we assume that in Fig. 2-20, and shaft O is driving, and shafts O1 and O2 are driven, then when the traction disconnector is turned off, LL1 and L1L2 will work in compression, and when turned on, they will work 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 during tension and compression (longitudinal bending) does not affect the operation of the synchronous transmission. In a 150 kV disconnector, the distance between the poles is 2800 mm, in a 330 kV disconnector - 3500 mm, in a 750 kV disconnector - 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 greater stability, since a long sample, in addition to compression, may experience deformation longitudinal bending, which will be discussed in the second part of the course. Samples from building materials are manufactured in the shape of a cube with dimensions 100 X 100 X 150 mm or 150 X X 150 X 150 mm. During a compression test, the cylindrical sample initially takes on a barrel-shaped shape. If it is made from plastic material, then further loading leads to flattening of the sample; if the material is brittle, then the sample suddenly cracks.

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

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

Longitudinal and transverse deformations during tension and compression. Hooke's law

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

Relative elongation is a dimensionless quantity, in some cases it is usually expressed 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 A cross-sectional size to its original size is called relative transverse contraction or transverse deformation'.

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

where p is called Poisson's ratio and are a constant value 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).

Poisson's ratio value

During compression, the opposite picture occurs, i.e. in the transverse direction the original dimensions decrease, and in the transverse direction they increase.

Numerous experiments show that, up to certain loading limits for most materials, the stresses arising during tension or compression of a rod are in a certain dependence on the longitudinal deformation. This dependence 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 voltage, i.e. measured in Pa, MPa.

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

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

The elastic modulus value for the most commonly used materials

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In tension, the longitudinal strain is considered positive, and in compression, it is considered negative.
The transverse dimensions of the rod also change as a result of deformation; when stretched, they decrease, and when compressed, they increase. If the material is isotropic, then its transverse deformations are equal:
.
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 limits. For example, for cork, for rubber, for steel, for gold.

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

Here, is the force with which the rod is stretched (compressed), is the absolute elongation (compression) of the rod, and is the coefficient of elasticity (or rigidity).
The elasticity coefficient depends both on the properties of the material and on the dimensions of the rod. We can distinguish the dependence on the dimensions of the rod (cross-sectional area and length) explicitly by writing the elasticity coefficient as

The quantity is called the elastic modulus of the first kind or Young's modulus and is a mechanical characteristic of the material.
If you enter the 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, the notation of Hooke’s law in relative form is used

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

Where:
E - elastic modulus,
F - strength,
S is the surface area over which 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).
Using Young's modulus, the speed 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.
Equation
,
Where
- Poisson's ratio;
- deformation in the transverse direction (negative for axial tension, positive for axial compression);
- longitudinal deformation (positive for axial tension, negative for axial compression).

Let us consider a straight beam of constant cross-section with length l, embedded at one end and loaded at the other end with a tensile force P (Fig. 2.9, a). Under the influence of force P, the beam elongates by a certain amount?l, which is called complete, or absolute, elongation (absolute longitudinal deformation).

At any points of the beam under consideration there is an identical state of stress, and, therefore, the linear deformations for all its points are the same. Therefore, the value can be defined as the ratio of the absolute elongation?l to the initial length of the beam l, i.e. . Linear deformation during tension or compression of beams is usually called relative elongation, or relative longitudinal deformation, and is designated

Hence,

Relative longitudinal strain is measured in abstract units. Let us agree to consider the elongation strain to be positive (Fig. 2.9, a), and the compression strain to be negative (Fig. 2.9, b).

The greater the magnitude of the force stretching the beam, the greater, other things being equal, the elongation of the beam; how larger area cross-section of the beam, the less elongation of the beam. Bars made from different materials elongate differently. For cases where the stresses in the beam do not exceed the proportionality limit, the following relationship has been established by experience:

Here N is the longitudinal force in the cross sections of the beam;

F - cross-sectional area of the beam;

E - coefficient depending on physical properties material.

Considering that the normal stress in the cross section of the beam we obtain

The absolute elongation of a beam is expressed by the formula

those. absolute longitudinal deformation is directly proportional to the longitudinal force.

For the first time, the law of direct proportionality between forces and deformations was formulated by R. Hooke (in 1660).

A more general formulation is the following formulation of Hooke's law: the relative longitudinal strain is directly proportional to the normal stress. In this formulation, Hooke's law is used not only in the study of tension and compression of beams, but also in other sections of the course.

The value E included in the formulas is called the longitudinal elastic modulus (abbreviated as the elastic modulus). This value is a physical constant of the material, characterizing its rigidity. How more value E, the less, other things being equal, is the longitudinal deformation.

The product EF is called the cross-sectional stiffness of the beam in tension and compression.

If the transverse size of the beam before applying compressive forces P to it is designated b, and after the application of these forces b +?b (Fig. 9.2), then the value?b will indicate the absolute transverse deformation of the beam. The ratio is the relative transverse strain.

Experience shows that at stresses not exceeding the elastic limit, the relative transverse strain is directly proportional to the relative longitudinal strain e, but has the opposite sign:

The proportionality coefficient in formula (2.16) depends on the material of the beam. It is called the transverse deformation ratio, or Poisson's ratio, and is the ratio of transverse deformation to longitudinal deformation, taken in absolute value, i.e.

Poisson's ratio, along with the elastic modulus E, characterizes the elastic properties of the material.

The value of Poisson's ratio is determined experimentally. For various materials it has values from zero (for cork) to a value close to 0.50 (for rubber and paraffin). For steel, Poisson's ratio is 0.25-0.30; for a number of other metals (cast iron, zinc, bronze, copper) it has values from 0.23 to 0.36.

Table 2.1 Elastic modulus values.

Table 2.2 Transverse strain coefficient values (Poisson's ratio)

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

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

Be able to carry out calculations of the strength and stiffness of statically determined beams in tension and compression.

Tensile and compressive strains

Let us consider the deformation of a beam under the action of a longitudinal force F (Fig. 21.1).

In the strength of materials, it is customary to calculate deformations in relative units:

There is a relationship between longitudinal and transverse deformations

Where μ - coefficient of transverse deformation, or Poisson's ratio, - characteristic of the plasticity of the material.

Hooke's law

Within the limits of elastic deformations, deformations are directly proportional to the load:

- coefficient. IN modern form:

Let's get a dependency

Where E- modulus of elasticity, characterizes the rigidity of the material.

Within elastic limits, normal stresses are proportional to elongation.

Meaning E for steels within (2 – 2.1) 10 5 MPa. All other things being equal, the stiffer the material, the less it deforms:

Formulas for calculating the displacements of beam cross sections under tension and compression

We use well-known formulas.

Elongation

As a result, we obtain the relationship between the load, the dimensions of the beam and the resulting deformation:

Δl- absolute elongation, mm;

σ - normal stress, MPa;

l- initial length, mm;

E - elastic modulus of the material, MPa;

N- longitudinal force, N;

A - cross-sectional area, mm 2;

Work AE called section rigidity.

Conclusions

1. The absolute elongation of a beam is directly proportional to the magnitude of the longitudinal force in the section, the length of the beam and inversely proportional to the cross-sectional area and elastic modulus.

2. The relationship between longitudinal and transverse deformations depends on the properties of the material, the relationship is determined Poisson's ratio, called transverse deformation coefficient.

Poisson's ratio: steel μ from 0.25 to 0.3; at the traffic jam μ = 0; near rubber μ = 0,5.

3. Transverse deformations are less than longitudinal ones and rarely affect the performance of the part; if necessary, the transverse deformation is calculated using the longitudinal one.

Where Δа- transverse narrowing, mm;

and about- initial transverse size, mm.

4. Hooke's law is satisfied in the elastic deformation zone, which is determined during tensile tests using a tensile diagram (Fig. 21.2).

During operation, plastic deformations should not occur; elastic deformations are small compared to the geometric dimensions of the body. The main calculations in the strength of materials are carried out in the zone of elastic deformations, where Hooke's law operates.

In the diagram (Fig. 21.2), Hooke’s law operates from the point 0 to the point 1 .

5. Determining the deformation of a beam under load and comparing it with the permissible one (which does not impair the performance of the beam) is called rigidity calculation.

Examples of problem solving

Example 1. The loading diagram and dimensions of the beam before deformation are given (Fig. 21.3). The beam is pinched, determine the movement of the free end.

Solution

1. The beam is stepped, so diagrams of longitudinal forces and normal stresses should be constructed.

We divide the beam into loading areas, determine the longitudinal forces, and build a diagram of the longitudinal forces.

2. We determine the values of normal stresses along sections, taking into account changes in the cross-sectional area.

We build a diagram of normal stresses.

3. At each section we determine the absolute elongation. We summarize the results algebraically.

Note. Beam pinched occurs in the patch unknown reaction in the support, so we start the calculation with free end (right).

1. Two loading sections:

section 1:

stretched;

section 2:

Three voltage sections:

Example 2. For a given stepped beam (Fig. 2.9, A) construct diagrams of longitudinal forces and normal stresses along its length, and also determine the displacements of the free end and section WITH, where the force is applied R 2. Modulus of longitudinal elasticity of the material E= 2.1 10 5 N/"mm 3.

Solution

1. The given beam has five sections /, //, III, IV, V(Fig. 2.9, A). The diagram of longitudinal forces is shown in Fig. 2.9, b.

2. Let’s calculate the stresses in the cross sections of each section:

for the first

for the second

for the third

for the fourth

for the fifth

The normal stress diagram is shown in Fig. 2.9, V.

3. Let's move on to determining the displacements of cross sections. The movement of the free end of the beam is defined as the algebraic sum of the lengthening (shortening) of all its sections:

Substituting numeric values, we get

4. The displacement of section C, in which the force P 2 is applied, is defined as the algebraic sum of the lengthening (shortening) of sections ///, IV, V:

Substituting the values from the previous calculation, we get

Thus, the free right end of the beam moves to the right, and the section where the force is applied R 2, - to the left.

5. The displacement values calculated above can be obtained in another way, using the principle of independence of the action of forces, i.e., determining the displacements from the action of each force R 1; R 2; R 3 separately and summing up the results. We recommend that the student do this independently.

Example 3. Determine what stress occurs in a steel rod of length l= 200 mm, if after applying tensile forces to it its length becomes l 1 = 200.2 mm. E = 2.1*10 6 N/mm 2.

Solution

Absolute elongation of the rod

Longitudinal deformation of the rod

According to Hooke's law

Example 4. Wall bracket (Fig. 2.10, A) consists of a steel rod AB and a wooden strut BC. Rod cross-sectional area F 1 = 1 cm 2, cross-sectional area of the strut F 2 = 25 cm 2. Determine the horizontal and vertical displacements of point B if a load is suspended in it Q= 20 kN. Modules of longitudinal elasticity of steel E st = 2.1*10 5 N/mm 2, wood E d = 1.0*10 4 N/mm 2.

Solution

1. To determine the longitudinal forces in the rods AB and BC, we cut out node B. Assuming that the rods AB and BC are stretched, we direct the forces N 1 and N 2 arising in them from the node (Fig. 2.10, 6 ). We compose the equilibrium equations:

Effort N 2 turned out with a minus sign. This indicates that the initial assumption about the direction of the force is incorrect - in fact, this rod is compressed.

2. Calculate the elongation of the steel rod Δl 1 and shortening the strut Δl 2:

Traction AB lengthens by Δl 1= 2.2 mm; strut Sun shortened by Δl 1= 7.4 mm.

3. To determine the movement of a point IN Let's mentally separate the rods in this hinge and mark their new lengths. New point position IN will be determined if the deformed rods AB 1 And B 2 C bring them together by rotating them around the points A And WITH(Fig. 2.10, V). Points B 1 And B 2 in this case they will move along arcs, which, due to their smallness, can be replaced by straight segments V 1 V" And V 2 V", respectively perpendicular to AB 1 And SV 2. The intersection of these perpendiculars (point IN") gives the new position of point (hinge) B.

4. In Fig. 2.10, G the displacement diagram of point B is shown on a larger scale.

5. Horizontal movement of a point IN

Vertical

where the component segments are determined from Fig. 2.10, g;

Substituting numerical values, we finally get

When calculating displacements, the absolute values of the lengthening (shortening) of the rods are substituted into the formulas.

Security questions and tasks

1. A steel rod 1.5 m long is stretched by 3 mm under load. What is the relative elongation? What is relative contraction? ( μ = 0,25.)

2. What characterizes the transverse deformation coefficient?

3. State Hooke's law in modern form for tension and compression.

4. What characterizes the elastic modulus of a material? What is the unit of elastic modulus?

5. Write down the formulas for determining the elongation of the beam. What characterizes the work AE and what is it called?

6. How is the absolute elongation of a stepped beam loaded with several forces determined?

7. Answer the test questions.