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 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 various materials Poisson's ratio varies within. 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. It is possible to distinguish the dependence on the size of the rod (area cross section and length) explicitly, writing the elasticity coefficient as
The quantity is called the elastic modulus of the first kind or Young’s modulus and is mechanical characteristics 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, 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 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
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. Denoting the absolute value of this ratio (X, we obtain
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 there is only one internal force factor during tension (compression) - longitudinal force, with shear - shear force, with torsion - torque, with pure straight bend- bending moment in a plane passing through one of the main central axes of the cross section of the beam. 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. The hypothesis of plane sections establishes such a geometry of deformations during tension and compression that all longitudinal fibers of a 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). The test until the maximum load was reached 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 formulas for the 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
Dependences (4.21) and (4.31) were checked on a large number of materials and at 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 in the delivered condition.
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 larger 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 reasons of greater stability, since a long sample, in addition to compression, may experience longitudinal bending deformation, 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, a cylindrical sample initially takes barrel-shaped. 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 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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Let us consider the deformations that occur during tension and compression of the rods. When stretched, the length of the rod increases and the transverse dimensions decrease. When compressed, on the contrary, the length of the rod decreases and the transverse dimensions increase. In Fig. 2.7 the dotted line shows the deformed view of a stretched rod.
ℓ – length of the rod before applying the load;
ℓ 1 – length of the rod after applying the load;
b – cross dimension before applying load;
b 1 – transverse size after application of load.
Absolute longitudinal strain ∆ℓ = ℓ 1 – ℓ.
Absolute transverse strain ∆b = b 1 – b.
The value of the relative linear deformation ε can be defined as the ratio of the absolute elongation ∆ℓ to the initial length of the beam ℓ
Transverse deformations are found similarly
When stretched, the transverse dimensions decrease: ε > 0, ε′< 0; при сжатии: ε < 0, ε′ >0. Experience shows that during elastic deformations, the transverse deformation is always directly proportional to the longitudinal one.
ε′ = – νε. (2.7)
The proportionality coefficient ν is called Poisson's ratio or transverse strain ratio. It represents the absolute value of the ratio of transverse to longitudinal deformation during axial tension
Named after the French scientist who first proposed it at the beginning of the 19th century. Poisson's ratio is a constant value for a material within the limits of elastic deformations (i.e. deformations that disappear after the load is removed). For various materials, Poisson's ratio varies within the range 0 ≤ ν ≤ 0.5: for steel ν = 0.28…0.32; for rubber ν = 0.5; for a plug ν = 0.
There is a relationship between stress and elastic deformation known as Hooke's law:
σ = Eε. (2.9)
The proportionality coefficient E between stress and strain is called the normal elastic modulus or Young's modulus. The dimension E is the same as that of voltage. Just like ν, E is the elastic constant of the material. The greater the value of E, the less, other things being equal, the longitudinal deformation. For steel E = (2...2.2)10 5 MPa or E = (2...2.2)10 4 kN/cm 2.
Substituting into formula (2.9) the value of σ according to formula (2.2) and ε according to formula (2.5), we obtain an expression for the absolute deformation
The product EF is called the rigidity of the timber in tension and compression.
Formulas (2.9) and (2.10) are different forms of writing Hooke’s law, proposed in the middle of the 17th century. Modern form recordings of this fundamental law of physics appeared much later - at the beginning of the 19th century.
Formula (2.10) is valid only within those areas where the force N and stiffness EF are constant. For a stepped rod and a rod loaded with several forces, the elongations are calculated in sections with constant N and F and the results are summed algebraically
If these quantities change according to a continuous law, ∆ℓ is calculated by the formula
In a number of cases, to ensure the normal operation of machines and structures, the dimensions of their parts must be chosen so that, in addition to the strength condition, the rigidity condition is ensured
where ∆ℓ – change in part dimensions;
[∆ℓ] – the permissible value of this change.
We emphasize that the calculation of rigidity always complements the calculation of strength.
2.4. Calculation of a rod taking into account its own weight
The simplest example of a problem about stretching a rod with parameters that vary along its length is the problem about stretching a prismatic rod under the influence of its own weight (Fig. 2.8a). The longitudinal force N x in the cross section of this beam (at a distance x from its lower end) is equal to the force of gravity of the underlying part of the beam (Fig. 2.8, b), i.e.
N x = γFx, (2.14)
where γ is the volumetric weight of the rod material.
The longitudinal force and stress vary linearly, reaching a maximum in the embedment. The axial displacement of an arbitrary section is equal to the elongation of the upper part of the beam. Therefore, it must be determined using formula (2.12), integration is carried out from the current value x to x = ℓ:
We obtained an expression for an arbitrary section of the rod
At x = ℓ the displacement is greatest, it is equal to the elongation of the rod
Figure 2.8, c, d, e shows graphs of N x, σ x and u x
Multiply the numerator and denominator of formula (2.17) by F and get:
The expression γFℓ is equal to the own weight of the rod G. Therefore
Formula (2.18) can be immediately obtained from (2.10), if we remember that the resultant of the own weight G must be applied at the center of gravity of the rod and therefore it causes elongation of only the upper half of the rod (Fig. 2.8, a).
If the rods, in addition to their own weight, are also loaded with concentrated longitudinal forces, then stresses and deformations are determined based on the principle of independence of the action of forces separately from concentrated forces and from their own weight, after which the results are added up.
The principle of independent action of forces follows from the linear deformability of elastic bodies. Its essence lies in the fact that any value (stress, displacement, deformation) from the action of a group of forces can be obtained as the sum of values found from each force separately.
Changing the size, volume and possibly shape of the body, with external influence on it is called deformation in physics. A body deforms when stretched, compressed, and/or when its temperature changes.
Deformation occurs when different parts of the body undergo different movements. So, for example, if a rubber cord is pulled by the ends, then its different parts will move relative to each other, and the cord will be deformed (stretched, lengthened). During deformation, the distances between atoms or molecules of bodies change, which is why elastic forces arise.
Let a straight beam, long and having a constant cross-section, be fixed at one end. The other end is stretched by applying force (Fig. 1). In this case, the body lengthens by an amount called absolute elongation (or absolute longitudinal deformation).
At any point of the body under consideration there is an identical state of stress. Linear deformation () during tension and compression of such objects is called relative elongation (relative longitudinal deformation):
Relative longitudinal strain
Relative longitudinal deformation is a dimensionless quantity. As a rule, the relative elongation is much less than unity ().
Elongational strain is usually considered positive and compressive strain negative.
If the stress in the beam does not exceed a certain limit, the following relationship has been experimentally established:
where is the longitudinal force in the cross sections of the beam; S is the cross-sectional area of the beam; E - elastic modulus (Young's modulus) - a physical quantity, a characteristic of the rigidity of the material. Taking into account that the normal stress in the cross section ():
The absolute elongation of a beam can be expressed as:
Expression (5) is a mathematical representation of R. Hooke’s law, which reflects the direct relationship between force and deformation under small loads.
In the following formulation, Hooke's law is used not only when considering tension (compression) of a beam: The relative longitudinal deformation is directly proportional normal voltage.
Relative shear strain
During shear, the relative deformation is characterized using the formula:
where is the relative shift; - absolute shift of layers parallel to each other; h is the distance between layers; - shear angle.
Hooke's law for shift is written as:
where G is the shear modulus, F is the shear-causing force parallel to the shearing layers of the body.
Examples of problem solving
EXAMPLE 1
| Exercise | What is the relative elongation of a steel rod if its upper end is fixed motionless (Fig. 2)? Cross-sectional area of the rod. A mass of kg is attached to the lower end of the rod. Consider that the own mass of the rod is much less than the mass of the load. |
| Solution | The force that causes the rod to stretch is equal to the gravitational force of the load that is located at the lower end of the rod. This force acts along the axis of the rod. We find the relative elongation of the rod as:
Where . Before carrying out the calculation, you should find the Young's modulus for steel in reference books. Pa. |
| Answer |
EXAMPLE 2
| Exercise | The lower base of a metal parallelepiped with a base in the form of a square with side a and height h is fixed motionless. A force F acts on the upper base parallel to the base (Fig. 3). What is the relative shear strain ()? Consider the shear modulus (G) to be known. |
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. We will 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; The larger the cross-sectional area 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 is a coefficient depending on the physical properties of the 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. The greater the value of E, the less, other things being equal, 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 denoted by 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)