Young's modulus (elasticity). Elastic and strength characteristics of materials
As a manuscript
Ministry of Education of the Russian Federation
Volgograd State Academy of Architecture and Civil Engineering
Department of Physics
YOUNG'S MODULUS MEASUREMENT
rod bending method
Guidelines for laboratory work No. 5
Volgograd 2010
UDC 539.4(076.5)
Measurement of Young's modulus by the rod bending method: Method. directions to laboratory work/ Comp. , ; VolgGASA. Volgograd, 2003, 16 p.
The aim of the work is to study elastic deformations, verify Hooke's law and determine the Young's modulus of a metal rod by the bending method. The definitions of the basic concepts of the theory of elasticity are given, the microscopic mechanisms of elastic and plastic deformations are explained, tabular data on the elastic and strength properties of solids are given. The technique of measurements is stated, the order of performance of work and the analysis of experimental data is described. Tasks for UIRS are formulated. Safety rules are given and control questions are given.
For students of all specialties in the discipline "Physics".
Il. 6. Tab. 3. Bibliography. 8 titles
© Volgograd State
Academy of Architecture and Civil Engineering, 2003
© Compilation,
C spruce work . Study of elastic deformations, verification of Hooke's law and
determination of Young's modulus of metal by the method of rod bending.
Instruments and accessories : installation for measuring the deflection of metal samples in the form of rods, samples for research, a set of weights, a caliper, a micrometer.
1. Theoretical introduction
1.1. Deformations, types of deformations
Unlike gases, which have neither their own shape nor their own volume, unlike liquids, which do not have their own shape, but have their own volume, solids have both their own volume and their own shape. Under the influence of external mechanical forces and for other reasons (for example, when heated, under the influence of electric or magnetic fields), solids change both their volume and their shape, i.e. deformed.
When a solid body is deformed, its particles are displaced from their original equilibrium positions to new ones. This displacement is prevented by the forces of interaction between the particles: in the deformed body, elastic forces arise that balance the external forces that caused the deformation.
By the nature of the emerging forces allocate elastic And plastic deformations. If the forces acting on a rigid body are small enough, so that after the elimination of these forces, both the volume of the body and its shape are restored (i.e., the deformation disappears), then the deformations are called elastic. In this case, the particles of the solid body return to their original equilibrium positions. With sufficiently large external forces or their long-term action, an irreversible rearrangement of the crystal lattice occurs, and deformations do not completely disappear after the removal of external forces. Such deformations are called plastic.
By the nature of geometric distortions There are two main types of deformations: deformation sprains (compression) and deformation shear(Fig. 1). Any other deformation, for example, bending, torsion, can be represented as a combination of these two main types of deformation.
By the nature of the distribution of deformations in the volume of the body, homogeneous and inhomogeneous deformations are distinguished. The deformation is called homogeneous, if all the elementary cubes, from which the body can be mentally composed, are deformed in the same way. The simplest elementary deformations are elongation and shear. Change in body length as a result of its stretching (or compression) from its original value l 0 to l, equal to , is called absolute tensile strain(D l > 0) or compression(D l < 0). Relative elongation on 
the quantity e = D l/l 0.
During deformation of a uniform shear, only the shape changes, while the volume of the body remains unchanged (Fig. 1, b). Each horizontal layer is shifted relative to its adjacent layers. During shearing, any straight line that was perpendicular to the sheared layers before deformation will rotate through some angle . The value is called relative shift. The angle is small, so they assume .
, Where is the resultant of the forces acting on the surface element https://pandia.ru/text/78/101/images/image009_97.gif" width="87" height="25">, (1)
where is the force applied along the normal to the section of the rod body (Fig. 1, A).
Tangential stress, arising from a uniform shear, can be calculated similarly:
- tangential force parallel to the shear plane (Fig. 1, b).
The voltage is called true if the change in area is taken into account S under deformation, and conditional, if S is the area of the undeformed body.
1.2. Hooke's Law
For small elastic deformations performed Hooke's law: stresses arising in an elastically deformed body are directly proportional to the magnitude of the relative deformation. For elastic deformations of tension (compression) and shear, Hooke's law is expressed by the equations:
Where E And G– characteristics of the elastic properties of the substance. Proportionality factor E between normal stress s n and relative strain (compression) e is called the modulus of elasticity or Young's modulus. Proportionality factor G between tangential stress st and relative shear https://pandia.ru/text/78/101/images/image015_66.gif" width="64" height="19">, (4)
Where K is the coefficient of all-round compression (modulus of volumetric deformation).
Formulas (3) express the so-called elementary Hooke's law, which determines the relationship between stress and strain in the same direction (the direction of the applied force). However, deformations can also occur in directions that do not coincide with the direction of the force. For example, when the sample is stretched (Fig. 1, A) is not only elongated, but also compressed in the transverse direction. The transverse strain in tension or compression is characterized by Poisson's ratio n, equal to the ratio transverse to longitudinal strain in the elastic region (see Table 1). The generalized Hooke's law, written taking into account possible deformations in three directions, has the form:
https://pandia.ru/text/78/101/images/image017_60.gif" width="173" height="29">, (5)
,
where indices x, y And z designate the directions of the coordinate axes along which the corresponding stresses and relative strains of tension (compression) are calculated. And similarly generalized Hooke's law for shift:
https://pandia.ru/text/78/101/images/image022_40.gif" width="193" height="51">. (7)
1.3. Stretch Chart
A typical dependence of normal stress on relative strain in unilateral tension (tension diagram) is shown in fig. 2. Point B on the diagram separates the areas of elastic and plastic deformations, the point C corresponds to the beginning of the destruction of the body.
https://pandia.ru/text/78/101/images/image024_43.gif" width="13" height="16 src="> and remains, but when fully unloaded, the body retains residual deformation OR. In materials where plastic deformations are strongly developed, there is a yield region BB¢ , where the increase in body size occurs at a constant stress. This stage of material loading can be replaced by a section B¢ C non-linear relationship between https://pandia.ru/text/78/101/images/image025_39.gif" width="16" height="16">. Then the point B¢ identified with the yield strength. Usually clear boundaries between plots BB¢ And B¢ C no, and the yield strength is determined conditionally. Conditional yield point(s0.2) is the stress, after loading to which and subsequent unloading, the residual deformation is 0.2% of the original length, i.e. = 0.002 (for comparison: the conditional elastic limit is the stress, after which the residual deformation is less than 0.05 % original length). Yield area BB¢ observed not for all materials, but only for ductile, with a viscous nature of fracture. In brittle materials, the elastic limit coincides with the tensile strength; the destruction of such materials that occurs without visible plastic deformation is called brittle.
Tensile strength(temporal resistance 628 " style="width:471.3pt;border-collapse:collapse">
Material
E, GPa
Shear modulus
G, GPa
Coefficient
Poisson
tensile strength
tensile strength
for compression
Tensile strength
V bend, MPa
(17–17,5)∙103
Aluminum
Wood
plexiglass
titanium alloys
High strength steels
In brittle fracture https://pandia.ru/text/78/101/images/image025_39.gif" width="16" height="16">> B, the deformation is concentrated on one section of the sample, where the cross section decreases, forming called neck. A crack occurs in the neck perpendicular to the tension axis, which grows in this direction until the sample is completely destroyed. In this case, B characterizes the resistance of the material to plastic deformation, and not to destruction..gif" width="16 height=16" height="16 ">0.2), Young's modulus E are the basic parameters included in the GOST for the supply of structural materials, in the acceptance test certificates; they are included in the calculations of strength and resource.
1.3. Microscopic deformation mechanisms
The elastic properties of bodies depend on their structure, the nature of the relative position and movement of the particles (atoms, molecules) that make up them. The mutual arrangement and movement of particles is determined by the forces of interaction between them. Atoms and ions of a crystal experience from neighboring particles the action as forces of attraction f pr, and repulsive forces f from, the values of which depend on the distance between the particles. By their origin, these are forces of an electrostatic nature, the directions of force vectors f at f are opposite, the potential energy of attraction is negative, and the potential energy of repulsion is positive. In this case, the repulsive forces decrease faster with increasing distance than the attractive forces. Therefore, the dependences of the total potential energy W sweat and resultant force f cut from distance r have the form shown in Fig. 3. For some distance between particles r 0, called equilibrium, the potential energy is minimal (Fig. 3, A), and the resulting force vanishes (Fig. 3, b).
When the body is compressed by external forces, the distance between the particles becomes smaller r 0, and repulsive forces arise in the body that prevent its compression. When a body is stretched, the distances between its particles exceed r 0, resulting in attractive forces that prevent stretching. Thus, when the particles deviate from the equilibrium position in any direction, forces arise that tend to return them to the equilibrium state.
With steady elastic deformation, the resultant of internal elastic forces in any section of the body balances the external forces acting on the body. Therefore, under elastic deformation, the magnitude of internal forces can be determined from the magnitude of external forces applied to the body. After the elimination of external forces, the internal forces will return the particles to their equilibrium positions, and the deformations will disappear. However, this will take place only at small deformations, when the environment of the moving particles remains unchanged. In this case, the forces of their interaction are proportional to the deviation of the particle from the equilibrium position ( r – r 0), which corresponds to Hooke's law on the site cd crooked f(r) (Fig. 3, b).
At sufficiently large displacements, particles of a deformable body from the previous equilibrium positions fall into neighboring ones, previously occupied by other particles, which also move to new equilibrium positions. With the disappearance of external forces, new equilibrium positions are preserved, therefore, residual deformations take place. Such is the mechanism of the occurrence of plastic deformations, which is usually realized during the shifts of atoms - the sliding of atomic planes or during their reorientation (twinning).
It is wrong to think that plastic shear deformations are formed by displacement of one part of the crystal relative to another. If this were the case, then the shear strength of the crystals would be 100–1000 times greater than the real one that takes place in reality. The nature of shear formation is associated with the imperfection of the crystal structure of solids, with the formation and movement of defects. Structural defects are divided into point (zero-dimensional), linear (one-dimensional), surface (two-dimensional) and volumetric (three-dimensional) defects according to geometric features.
Point defects localized at individual points of the crystal include vacancies(vacant sites of the crystal lattice), atoms in interstices And impurity atoms at sites or interstices.
Linear defects are those in which the violation of the regularity of the structure of the crystal lattice is concentrated near certain lines. The lines separating the region of shear deformations from the undeformed region are called dislocations. Distinguish regional And screw dislocations(Fig. 4, a, b). Edge dislocation OO" (in Fig. 4, A it is indicated by the symbol) arose when a part of the crystal was shifted by one interatomic distance and represents the edge of an extra half-plane. Edge dislocation is perpendicular to the shear vector, screw dislocation OO" parallel to the shift vector (Fig. 4, b).
The dislocation, causing elastic distortion of the lattice, creates a force field around itself, which is characterized at each point by a certain tangent (st) and normal (s n) stresses. When another dislocation enters this field, forces arise that tend to bring the dislocations closer together or push them away from each other. The strength of the material depends on the density and mobility of dislocations.
Humidity" href="/text/category/vlazhnostmz/" rel="bookmark">humidity and temperature of the environment, vibrocompaction methods). Strengthening technologies are developed depending on the type and purpose of concrete (heavy, light, hydraulic, road, heat-resistant, etc. . P.). Reinforced concrete structures reinforced with prestressing. Stressed concrete is created by heating the reinforcement, leading to its thermal expansion, and subsequent cooling after the completion of the concrete hardening process. The resulting compressive deformations of the reinforcement create compressive stresses in the concrete. During the operation of the structure in the conditions of its tension, the existing internal stresses are directed against external forces, which significantly increases the tensile strength. In a similar way, the flexural strength is increased by creating internal moments of forces inside the structure that are opposite to the external moments of forces that occur in the operating mode.
2. Measurement technique
The aim of the work is to determine the Young's modulus based on the study of elastic bending deformation. Bending deformation is experienced by the details of many structures. A beam or slab lying on supports sags both under the action of its own weight and under the action of an applied load. F(Fig. 5). The bending test scheme (Fig. 5) is provided by GOST for determining the bending strength limits. The same scheme in present work is used to determine Young's modulus.
https://pandia.ru/text/78/101/images/image030_33.gif" width="56" height="21">. (8)
Measuring https://pandia.ru/text/78/101/images/image031_31.gif" width="15" height="20 src=">/ F and calculate Young's modulus using the formula
Where l- length, b- width, h is the thickness of the rod, k is the coefficient of elasticity in bending, determined from (8).
To justify formula (9), we consider a fragment of a rod experiencing bending deformations (Fig. 6, A). At balance, the force F is balanced by the resultant of elastic forces F t directed tangentially to the deformable layers (Fig. 6, A, b). On the other hand, the resultant of the elastic forces is perpendicular to the section of the rod and creates normal stresses.
When bending on the convex side, the body experiences tensile strain, and on the concave side, compressive strain. Inside the bent rod there is neutral layer, in which there are no compressive or tensile deformations. Since the neutral layer does not change the length, the length of the line O 1O 2 belonging to the neutral layer is equal to dx = r d a , Where r is the radius of curvature of the neutral layer, d a is the angle between the planes of the section of the rod.
Line AB, lying below the neutral layer at a distance z, experiences tensile strain. Its length is 
. Accordingly, the absolute and relative elongations are equal:
https://pandia.ru/text/78/101/images/image037_26.gif" width="136" height="48 src=">.
From Hooke's law for stretching we get
https://pandia.ru/text/78/101/images/image039_26.gif" width="85" height="25">, and its moment is . The total moment of force is found by integration:
https://pandia.ru/text/78/101/images/image042_21.gif" width="99" height="31 src="> (m4 unit) is a measure of the resistance of a section of a body to bending deformation, in contrast to the physical concepts of the moment of inertia of a rigid body https://pandia.ru/text/78/101/images/image044_20.gif" width="172" height="60 src=">,
whence formulas (8) and (9) follow.
In standard strength tests, the applied load is increased until the body breaks, fixing the force F = fm at which the rod breaks. Bending strength is calculated by the formula
https://pandia.ru/text/78/101/images/image046_20.gif" width="65" height="25 src=">.gif" width="168" height="55">, (12 )
where D Ei= E Wed - Ei, Student's coefficient a find according to the Student's table at W= 0.95 and n= 5. In accordance with the error, round the result and present it as E = (E cf ± D E) Pa. Compare the results obtained with the table. Formulate conclusions on the work, including a commentary on the feasibility of Hooke's law and evaluation of the results obtained.
table 2
Dimensions of the investigated rod
Material (steel, brass…) |
||||
width, mm | thickness, mm | |||
Table 3
Young's modulus results
ni 1 mm | ni 2 mm | ni 3 mm | ni sr, mm |
(n0 Wed - ni Wed) | E, | ( E)2, |
|||||
E exp = ( E Wed E) 1011 Pa |
Safety
· The steel rod is not fixed on the supports. Place the weights carefully to prevent the rod and weights from falling.
· Do not leave the unit switched on.
Tasks for educational and research work
1. Study of the elastic properties of various building materials.
2. Study of deviations from Hooke's law for rods made of plastic, organic glass, and other plastic materials.
3. Estimation of microscopic parameters of interatomic interactions.
4. Estimation of the theoretical strength of solids with an ideal crystal lattice, comparison with experimental values. Modern theories of destruction.
When completing assignments, use additional literature.
Control questions
1. Types of deformations. Hooke's law for elastic deformations: uniaxial and all-round tension (compression). Hooke's law for shear deformations.
2. physical meaning Young's modulus, shear modulus, Poisson's ratio, the relationship between these quantities. Generalized Hooke's law.
3. Microscopic mechanism of deformation of solids. Show on the graphs the dependence of the potential energy and the interaction force on the distance between the atoms, the area of feasibility of Hooke's law.
4. Diagram of stretching. Limits of elasticity, yield strength, strength.
5. The main mechanism of destruction of solids. The role of defects. Types of defects. Methods for increasing the strength of materials.
6. Task. Find the relative elongation of a vertically suspended steel cable under the influence of its own weight of 100 kg. Cross-sectional area S = 5 cm2.
7. Task. To two opposite faces of a steel bar with a cross section S= 10 cm2 forces applied F 1 = F 2 = 10 kg. Determine the amount of relative shift.
8. Task. According to the values of Young's modulus obtained in the work, to estimate what is the largest load that a wire with a diameter of d= 1 mm without going beyond the elastic limit? Estimate also the range of values of the applied forces corresponding to the yield region. For calculations, use the value of Young's modulus obtained in your work, and the data in Table. 1.
9. Task. For prestressing structures, two methods are used: mechanical tension and thermal expansion of reinforcement, in which it is necessary to create a stress s0, which is 90% of the yield strength. Determine the required elongation of the steel bar for the required stress s0. Calculate what force must be applied to the steel reinforcement bar for this, or by how many degrees should it be heated? With thermal expansion, the relative elongation is directly proportional to the temperature increment e = a D T, where a = 1.2 10–5 deg–1. Rod length l 0 = 2.5 m, diameter 10 mm, Young's modulus of steel E= 210 GPa, yield strength st = 260 MPa.
Bibliographic list
1. Physics course. M.: Higher. school, 1999.
2. Short Course Physics: Proc. allowance for universities. M.: Higher. school, 2000.
3. Physics course / , . M.: Higher. school, 1999.
4. Yavorsky B. M. Handbook of physics for students of technical colleges and engineers. - 2nd ed. correct and additional / , . M.: Higher. school, 1999.
5. Solid state physics / , M.: Vyssh. school, 2000. Ch. 2–4.
6. Solid state physics. M.: Higher. Shk., 1975, pp. 56–88.
7. Construction Materials and products. M.: Higher. school, 1983. §1.3, §6, 7.
8. Thermophysical properties of materials: Educational research work on the course of physics / Comp. , ; VolgISI. Volgograd. 1983, pp. 6–8.
9. Gorchakov materials: Proc. For universities. / , . Moscow: Stroyizdat, 1986.– 688 p.
10. Physical quantities: Handbook /, etc.; Ed. , . Moscow: Energoizdat, 1991.1232 p.
Young's modulus is also called the elastic stiffness constant or simply stiffness.
* Given for heavy, high-strength concrete (for light concrete sv = 5–15 MPa).
** Given for road concrete.
Application:
Measurement of the modulus of elasticity, shear modulus and Poisson's ratio (transverse strain) in non-dispersive isotropic structural materials.
General information:
Defined as the ratio of stress (force per unit area) to compressive strain.
Defined as the ratio of shear stress to shear strain.
Poisson's ratio the ratio of relative transverse compression to relative longitudinal tension.
These basic properties of materials are necessarily taken into account in production and in various scientific research, and are determined using the measured values of the speed of sound and the density of the material. The speed of sound propagation is easily calculated by pulse-echo ultrasonic testing using appropriate equipment. The procedure below is valid for any homogeneous, isotropic, non-dispersive material (speed of sound does not change with frequency). This includes the most common metals, industrial ceramics and glass, provided that the cross-sectional dimensions are not close to the wavelength of the control frequency. Rigid plastics such as polystyrene and acrylic can also be measured, although they have a high ultrasonic attenuation coefficient.
Rubber cannot be measured ultrasonically due to its high dispersion and non-linear elastic properties. Soft plastics likewise show a high degree of shear wave attenuation, and cannot usually be measured. In the case of anisotropic materials, the elasticity varies with direction, as does the propagation velocity of compressional waves and/or shear waves. Generating the full elastic modulus matrix in anisotropic samples usually requires six series of ultrasonic measurements. The porosity or graininess of a material can affect the accuracy of the elastic modulus measurement because it causes the speed of sound to fluctuate based on grain size and orientation or pore size and distribution, regardless of the elasticity of the material.
Equipment:
To measure the speed of sound in elasticity calculations, a 38DL PLUS or 45MG precision thickness gauge with single-element probe software, or flaw detectors with a sound speed measurement function (for example, the EPOCH series) are usually used. Model 5072PR or 5077PR generators/receivers, in combination with an oscilloscope or signal sampler, can also be used to measure propagation time. This test will require two transducers suitable for pulse-echo measurement of the speed of sound in a material with compressional and shear waves. Among the most used probes are the M112 or V112 broadband P-wave transducer (10 MHz) and the V156 normal-incidence transducer (5 MHz). They are suitable for measuring the most common metals and fired ceramic samples. Special transducers are required to measure very thick and very thin materials or samples with high attenuation. In some cases, a shadow testing method (through sounding method) is used using two transducers located on the same axis, on opposite sides of the product being tested. An Olympus specialist should be consulted when selecting a transducer or setting up an instrument.
The test piece can be of any shape that allows for a pulse-echo measurement of the transit time of ultrasound through the material. Typically, this is a 12.5 mm thick sample with even, parallel surfaces, and is wider or larger than the diameter of the transducer being used. Extreme care must be taken when measuring narrow specimens due to possible edge effects that can affect the measured transit time. When using very thin samples, resolution will be limited due to small fluctuations in the time it takes the pulse to travel through the short path. We recommend taking samples with a minimum thickness of 5 mm, but preferably thicker. In all cases, the thickness of the test piece must be precisely known.
Procedure:
Measure the compressional and shear velocity of the test piece using the appropriate probe and instrument settings. To measure shear wave velocity, a special high viscosity couplant, such as SWC, will be required. The 38DL PLUS and 45MG thickness gauges can directly measure the speed of sound in a material based on an entered sample thickness, while the EPOCH Series flaw detectors measure the speed of sound during sound speed calibration. In both cases, follow the recommended sound velocity measurement procedure provided in the instrument's instruction manual. If using a transmitter/receiver, record the round-trip time of the signal through an area of known thickness using P- and S-wave converters, and calculate:
If necessary, convert the sound speed units to inches/s or cm/s. (Time is usually measured in microseconds; to get measurements in inches/s or cm/s, multiply in/µs or cm/µs by 10 6 .) The resulting speeds of sound can be used in the following formulas.
Note: If the speed of sound is expressed in cm/s and the density is expressed in g/cm 3 , the modulus of elasticity will be expressed in dynes/cm 2 . If you are using imperial units (in/s and psi) to calculate the modulus of elasticity in psi. inch (PSI), do not confuse the pound (a unit of force) with the pound (a unit of mass). Since the modulus of elasticity is expressed as a force per unit area, when calculating in imperial units, it is necessary to multiply the result of the above formula by the mass/force conversion factor (1 / acceleration free fall) to get the elasticity value in psi. inch. If original calculations are in metric units, use a conversion factor of 1 psi = 6.89 x 10 4 dynes/cm 2 . You can also enter the speed of sound in inches/s and the density in g/cm 3 and then divide by a conversion factor of 1.07 x 10 4 to get elasticity in PSI.
To determine the shear modulus, multiply the square of the shear wave velocity by the density.
Again, use the units cm/s and g/cm3 to get the modulus of elasticity in dynes/cm2 or imperial units (in/s and lb/in3) and multiply the result by the mass/force conversion factor.
Bibliography
For more information on the measurement of elastic modulus using the ultrasonic method, see the sources below:
1. Moore, P. (ed.), non-destructive testing handbook, Volume 7, American Society for Nondestructive Testing, 2007, pp. 319-321.
2. Krautkramer, J., H. Krautkramer, Ultrasonic Testing of Materials, Berlin, Heidelberg, New York 1990 (Fourth Edition), pp. 13-14, 533-534.
Before using any material in construction work, you should familiarize yourself with its physical characteristics in order to know how to handle it, what mechanical effect will be acceptable for it, and so on. One of important characteristics, which is very often paid attention to, is the modulus of elasticity.
Below we consider the concept itself, as well as this value in relation to one of the most popular in construction and repair work material - steel. These indicators will also be considered for other materials, for the sake of an example.
Modulus of elasticity - what is it?
The modulus of elasticity of a material is called set of physical quantities, which characterize the ability of a solid body deform elastically when a force is applied to it. It is expressed by the letter E. So it will be mentioned in all the tables that will go further in the article.
It cannot be argued that there is only one way to determine the value of elasticity. Various approaches to the study of this quantity led to the fact that there are several different approaches at once. Below are three main ways to calculate the indicators of this characteristic for different materials:
Table of indicators of elasticity of materials
Before proceeding directly to this steel characteristic, let's first consider, as an example, and additional information, a table containing data on this value in relation to other materials. Data is measured in MPa.
As you can see from the table above, this value is different for different materials, moreover, the indicators differ if one or another option for calculating this indicator is taken into account. Everyone is free to choose exactly the option of studying indicators that suits him best. It may be preferable to consider Young's modulus, since it is more often used specifically to characterize a particular material in this regard.
After we briefly got acquainted with the data of this characteristic of other materials, we will proceed directly to the characteristic of steel separately.
To start let's look at dry numbers and derive various indicators of this characteristic for different types steels and steel structures:
- Modulus of elasticity (E) for casting, hot-rolled reinforcement from steel grades referred to as St.3 and St. 5 equals 2.1*106 kg/cm^2.
- For such steels as 25G2S and 30KhG2S, this value is 2 * 106 kg / cm ^ 2.
- For wire of a periodic profile and cold-drawn round wire, there is such a value of elasticity equal to 1.8 * 106 kg / cm ^ 2. For cold-flattened reinforcement, the indicators are similar.
- For strands and bundles of high-strength wire, the value is 2 10 6 kg / cm ^ 2
- For steel spiral ropes and ropes with a metal core, the value is 1.5·10 4 kg/cm^2, while for ropes with an organic core, this value does not exceed 1.3·10 6 kg/cm^2.
- The shear modulus (G) for rolled steel is 8.4·10 6 kg/cm^2.
- And finally, Poisson's ratio for steel is equal to 0.3
These are general data given for types of steel and steel products. Each value was calculated according to all physical rules and taking into account all the available relationships that are used to derive the values of this characteristic.
Below will be all general information about this characteristic of steel. Values will be given as n about Young's modulus, and in shear modulus, both in one unit of measure (MPa) and in others (kg/cm2, newton*m2).
Steel and several different grades
The values of the elasticity indices of steel differ, since there are multiple modules, which are calculated and calculated differently. One can notice the fact that, in principle, the indicators do not differ much, which testifies in favor of different studies of elasticity. various materials. But it is not worth going deep into all calculations, formulas and values, since it is enough to choose a certain value of elasticity in order to be guided by it in the future.
By the way, if you do not express all the values by numerical ratios, but take it immediately and calculate it completely, then this characteristic of the steel will be equal to: Е=200000 MPa or Е=2,039,000 kg/cm^2.
This information will help you understand the very concept of the modulus of elasticity, as well as get acquainted with the main values \u200b\u200bof this characteristic for steel, steel products, as well as for several other materials.
It should be remembered that the elastic modulus indicators are different for different steel alloys and for different steel structures that contain other compounds in their composition. But even in such conditions, one can notice the fact that the indicators do not differ much. The value of the modulus of elasticity of steel practically depends on the structure. as well as carbon content. The method of hot or cold processing of steel also cannot greatly affect this indicator.
Goal of the work: experimental determination of the elastic moduli of plates made of various materials using the bending method.
Instruments and accessories: installation "Young's Modulus", plates, a set of weights weighing 0.05 kg, 0.1 kg and 0.15 kg.
Elements of the theory and method of experiment
In various elements of structures and machines, only longitudinal forces often occur, which cause tensile or compressive deformation in them.
The 17th-century English scientist Robert Hooke discovered a fundamental pattern between forces and the displacements they cause, establishing a directly proportional relationship between the elongation of a sample and the tensile force.
Thomas Young, an English scientist of the 19th century, first expressed the idea that for each material there is a constant value that characterizes its ability to resist external loads. The concept of this quantity, which he called the "modulus of elasticity" (later "Young's modulus"), was formulated in 1807 in the work "Natural Philosophy".
The modulus of elasticity characterizes the most important property structural material- rigidity - and is a fundamental concept, without which not a single engineering calculation of structural elements and structures can do. On fig. 1 shows a rod with a straight axis under the action of longitudinal forces N, where
σ – normal voltage,
A is the cross-sectional area of the rod.
Rice. 1. Longitudinal and transverse deformations of the rod
Under the action of longitudinal forces, the rod is deformed. If it is stretched, then its length increases and becomes equal to L+∆ L, Where ∆ L is the absolute longitudinal deformation (elongation) of the bar. Its transverse dimensions decrease and take on the values H–∆ H And B–∆ B, Where ∆ H And ∆ B are the absolute transverse deformations of the bar.
The ratio of the absolute longitudinal deformation of the rod to its original length is called the relative longitudinal deformation:
The ratio of the absolute transverse deformation of the rod to its original transverse dimension is called the relative transverse deformation:
Here, the “+” sign for the deformation and the “–” sign for the deformations and are set because, when stretched, the longitudinal dimensions of the rod increase, and the transverse ones decrease.
The last step in the formation of Hooke's law in its modern form was made by the French mathematician Cauchy, who in 1822 introduced the concepts of "stress" and "strain" into the scientific literature, and the French scientist Navier, who in 1826 defined the modulus of elasticity as the ratio of load per unit area of the cross section, to the relative elongation produced by it
Where E– Young's modulus (modulus of elasticity of the first kind).
Thus Hooke's law is practical use as a formula
Elastic modulus E is the physical constant of the material and is determined experimentally. Its value is expressed in the same units as the stresses σ, i.e., in pascals (Pa), since ε is a dimensionless quantity. The elastic modulus of most materials has large numerical values and is usually expressed in gigapascals (GPa).
The absolute value of the ratio of relative transverse strain and relative longitudinal strain in tension or compression in the domain of Hooke's law is called Poisson's ratio
This is a dimensionless coefficient that characterizes the properties of the material and is determined experimentally. It bears the name of the French scientist who first introduced it into the theory.
After an external load is applied to the body, its points move. Usually, the values of elastic displacements are considered small in comparison with geometric dimensions deformable bodies. Consider these displacements using the example of a cantilever beam with a length L with one-sided external termination shown in fig. 2. A concentrated force is applied to the free end of the beam F, which causes deformations of its points. The deflection of the beam in the current section is denoted δ . Select the volume element of the beam with the length Dz located at a distance Z from the fixed end.
Rice. 2. Bending of the cantilever beam
The deformed state in the current section of the beam is described by the radius of curvature or the curvature of its curved axis.
It is known that the equation of the bent axis of the beam is:
Where I X - axial moment of inertia of the beam section relative to the axis Ox. Work EI X is called the section stiffness in bending about the corresponding axis.
On fig. 3 shows an arbitrary section, which is a flat geometric figure, whose area A. Let us allocate an elementary area on it DA.
Let's define the moment of inertia rectangular section about axes C X and C Y passing through its center, as shown in Fig. 4.
Divide the area of the rectangle into elementary rectangles with dimensions B And Dy, whose area is . Substituting the value into expression (9) and integrating, we get:
Similarly
Consider a beam with a length L, mounted on two supports and loaded, as shown in Fig. 5.
The solution of differential equation (8) can be obtained by successive integration. When the external load is placed symmetrically with respect to the supports, as shown in fig. 5, then the solution of this equation will take the form:
Therefore, Young's modulus is defined by the formula
Taking into account expression (10), we obtain
Therefore, having determined the load F and the value of deflection δ for a beam (plate) with a length L with cross-sectional dimensions B And H, according to the formula (14) it is possible to calculate the Young's modulus of the material from which it is made.
Description of the experimental setup
A schematic representation of the installation "Young's Modulus" is shown in fig. 6.
The "Young's Module" installation consists of a base 1 on which a rack 2 is fixed. On the rack there is a bracket 3 with two prismatic supports 4. The test sample 5 (plate) is mounted on the supports. Using the sample loading device 7, which is a bracket with a prismatic support, a stacked weight 6 and a clock indicator 8 are attached to the sample.
Work order
1. Place one of the test plates on prismatic supports 4.
2. Install the hour indicator 8 so that its tip touches the plate.
3. Hang the device bracket 7 in the middle of the plate.
4. Attach a weight to the bracket M1 =0.1 kg.
5. On the indicator scale 8 determine the value of the plate deflection δ 1 .
6. Remove the load.
7. Hang a weight on the bracket M2 =0.15 kg.
8. On the indicator scale 8 determine the value of the deflection of the plate δ 2 .
Where G- acceleration of gravity.
10. The value of the deflection of the plate is determined as
11. Find Young's modulus using formula (14), where L\u003d 0.114 m - distance between prisms (plate length); B\u003d 0.012 m - width of the plate section; H\u003d 0.0008 m - plate thickness; δ - the value of the deflection of the plate, m.
12. Do the above steps with the second plate.
13. Repeat steps for both springs. 1-12 two more times.
The material of the studied samples is spring steel and bronze.
Explain the obtained results of the elastic moduli of the plates, compare them with reference data.
The procedure for estimating errors
Assume that the error in estimating the value of Young's modulus according to formula (14) is determined by the error in measuring the length of the plate L(systematic error) and deflection estimation error d (systematic + random errors).
Record the results of direct measurements of the specified parameters:
A) L=< L> ± D L, Where D L= D L Sist;
B) d =< D > ± Dd , Where , .
Record the results of indirect measurements:
E=<Е> ± D E, Where , , , , .
Questions and tasks for self-control
1. What is the difference between normal stress and shear stress?
2. What formulas are used to determine the absolute and relative deformations?
3. What value is called the modulus of elasticity of the first kind?
4. How is Poisson's ratio determined?
5. What is called the bending stiffness of the section?
6. What is the difference between the formulas for the axial moment of inertia of the section relative to the axes Ox And Oy?
7. What formula expresses the deflection of a two-bearing beam?
Ministry of Education and Science of the Russian Federation State educational institution higher professional education
œKuzbass State Technical University
Department of Strength of Materials
DETERMINATION OF THE ELASTIC MODULUS OF THE FIRST KIND
AND POISSON RATIO
Guidelines for laboratory work on the discipline "Strength of materials for students of technical specialties"
Compiled by I. A. Panachev M. Yu. Nasonov
Approved at the meeting of the department Minutes No. 8 dated 01/31/2011 Recommended for printing by the educational and methodological commission of specialty 150202 Minutes No. 6 dated 03/02/2011 An electronic copy is in the library of KuzSTU
Kemerovo 2011
The purpose of the work: experimental determination of the "elastic" constants of the material - steel VST3
– modulus of longitudinal elasticity (modulus of elasticity of the first kind, Young's modulus);
– transverse strain coefficient (Poisson's ratio).
” 1. Modulus of longitudinal elasticity (modulus of elasticity of the first kind, Young's modulus) - definition and use
item 1. Designation
The modulus of longitudinal elasticity is indicated by the Latin letter - "E".
P. 2. Semantic definition
E - this is a characteristic of the rigidity (elasticity) of a material, showing its ability to resist longitudinal deformation (tension, compression) and bending.
Item 3. Properties of E
1. E is "elastic" material constant, the application of which is valid only within the limits of linear elastic deformations of the material, i.e., within the limits of the Hooke's law (Fig. 1).
Area of action | |||
Hooke's law | |||
E = tgα | |||
Rice. Fig. 1. Tensile diagram of steel Vst3 A-B - section of the linear relationship between strains - ε
and stresses - σ (section of Hooke's law); В-С - section of non-linear dependence between deformations
and stresses
2. E relates strains and stresses in the formula of Hooke's law in tension (compression) and is graphically estimated as follows E = tg (see Fig. 1).
3. Material with a large numerical value E is more rigid and requires more effort to deform it.
4. Most materials correspond to a certain constant (constant) value E .
5. The values of E for basic materials are given in the manuals on the strength of materials and the manuals of the machine builder, and in the absence of data in the manuals, they are determined experimentally.
P. 4. Use of E
E used in the strength of materials in the evaluation of strength
performance, rigidity and stability of structural elements:
1) when calculating strength in the process of determining experimentally stresses from measured strains
≤ [σ]; (1) 2) when calculating stiffness in the process of theoretical determination
strain reduction
3) when calculating stability in the process of solving all types of problems.
P. 5. Numerical definition
E numerically equal to the voltage that could arise
V beam with its elastic tension by 100% (2 times).
E - the characteristic is conditional, because when determining it, it is conditionally considered that any material is capable of elastically deforming, increasing in length an infinite number of times, although it is known
- no more than 2% (except for rubber, rubber).
The basis of 100% is adopted for the convenience of using E in the formulas of Hooke's law.
E practically determined by stretching the sample by a fraction of a percent and increasing the resulting stress by the corresponding number of times.
Example 1: when the sample is stretched by \u003d 1%, the stresses arising in the sample are, for example, 1000 MPa (10,000 kg / cm2), then the modulus of elasticity will be equal to
E \u003d 100 \u003d 100,000 MPa (1,000,000 kg / cm2). Example 2: \u003d 0.1% \u003d 100 MPa (1,000 kg / cm2)
E \u003d 1000 \u003d 100,000 MPa (1,000,000 kg / cm2).
P. 6. Units E
E has the dimension: [kN/cm 2] or [MPa].
P. 7. Examples of the numerical value of E
The modulus of elasticity E for different materials is
2.1 104 kN/cm2 | 2.1 105 MPa | 2,100,000 kg/cm2 |
||
1.15 104 kN/cm2 | 1.15 105 MPa | 1 150 000 kg/cm2 |
||
1.0 104 kN/cm2 | 1.0 105 MPa | 1,000,000 kg/cm2 |
||
aluminum - 0.7 104 kN/cm2 | 0.7 105 MPa | 700,000 kg/cm2 |
||
0.15 104 kN/cm2 | 0.15 105 MPa = | 150,000 kg/cm2 |
||
rubber - | 0.00008 104 kN/cm2 = 0.0008 105 MPa = 80 kg/cm2. |
|||
From the data in the list, one can draw a conclusion about the ratio of the stiffness of the materials (the stiffness of the material proportionally depends on the modulus of elasticity). For example, steel is 2 times stiffer than copper, therefore, when considering samples of the same type made of steel and copper, in order to stretch them to the same length within the boundaries of elastic deformations, a load must be applied to the steel sample twice as large as compared to copper.
” 2. Transverse strain ratio (Poisson’s ratio) –
definition and use
item 1. Designation
Poisson's ratio is denoted by the Greek letter "" (mu).
P. 2. Semantic definition
- elastic mechanical characteristic of the material, characterizing the ability of the material to deform in transverse
in the longitudinal direction under the longitudinal application of the load, since when the sample is stretched, along with its longitudinal elongation, its transverse narrowing also takes place (Fig. 2).
Rice. 2. Longitudinal and transverse deformation of the specimen under tension
From fig. 2 it follows that the absolute deformations of the sample
l = l1 – l0 , | b \u003d b 1 -b 0, |
where l and b are the absolute elongation and absolute narrowing of the
l 0 and l 1 | samples (absolute deformations); | ||
are the initial and final length of the sample; |
|||
b 0 and b 1 | are the initial and final width of the sample. |
||
If we accept that l 1 l 0 | L, and b1 b0 = b, | then relatively |
|
nye deformations of the sample will be equal to: | |||
L/l | " = b/b, | ||
- relative longitudinal and relative pop- |
|||
river deformation of the sample (relative elongation |
|||
and relative contraction). | |||
numerically is equal to the ratio relative narrowing of the sample to its relative elongation during its longitudinal deformation, i.e., the ratio between the relative transverse and longitudinal deformations. This relationship is expressed
formula | |||||||||||||||
item 3. Properties
1. Each material corresponds to a certain constant value (constant).
2. For most materials, the numerical value is given in the manuals on the strength of materials and manuals of the machine builder, otherwise it is determined experimentally.
4. Use
It is used in the resistance of materials as a coefficient in the formula of the generalized Hooke's law (2) and connects the elastic moduli of the first and second kind, which will be discussed below.
P. 5. Units of measurement
is a dimensionless quantity (b/c).
P. 6. Limits of change
Generally speaking, for the known studied isotropic (having the same elastic properties in all directions) materials, the range of variation of the Poisson's ratio = 0 0.5.
item 7. Examples of a Numeric Value
Poisson's ratio - for various kinds material-
cork tree - 0.
” 3. Description of the test equipment
IN a tensile testing machine is used to stretch the sample in the lab R-5 (Fig. 3).
Rice. Fig. 3. Scheme of R-5 tensile testing machine: 1 – handle; 2 - nut; 3 - screw;
9 - force meter; 10 - strain gauges
The installation during the experiment works as follows. The rotation of the handle /1/ is transmitted through the gearbox to the nut /2/, which causes the vertical movement of the screw /3/. This leads to stretching of the sample /6/ fixed in the grips /4/ and /5/. The force in the sample is created by a system of levers /7/ and a pendulum /8/. The amount of effort is fixed on the scale of the force meter /9/. To determine the absolute longitudinal and transverse strains, lever-type strain gauges (Guggenberger strain gauge) /10/.P
Rice. 4. Lever strain gauge (Guggenberger strain gauge): a - general form; b - simplified scheme;
l bt - base of the strain gauge; l bt - change in the base of the strain gauge; 1 - sample; 2 - screw; 3 - mounting clamp;
Price 4 - measuring one small scale; division 5 of the scale - index tensiometer arrow; - C tenz is equal to 0.0016 - hinge; mm (0.00017 - fixed cm / div.). support; 8 - movable support
The strain gauge can only measure the deformation of the area where it is located, i.e. the area called " strain gauge base", but cannot measure the absolute strains of the entire sample, unless of course the length of the sample is equal to the base of the strain gauge.
Due to the fact that the measurements in the experiment will be made by strain gauges with dimensions (bases) much smaller than the dimensions of the test sample, the length and width of the measured section of the sample will be limited by the bases of the longitudinal and transverse strain gauges.
E and are the characteristics of the material, not the sample, therefore, E and obtained by measuring the deformations of a section of the sample will be the same as when measuring the deformations of the entire sample.
item 3. Location of strain gauges and measuring sections on the sample
In laboratory work, to improve the accuracy of the results obtained, the values of E and will be determined by two participants
stacks of the test sample located on its opposite faces (Fig. 5).
I section |
II section
Rice. 5. Scheme of the location of the studied sections of the sample and strain gauges on the sample
1, 2 – longitudinal strain gauges; 3, 4 – transverse strain gauges; (dashed line shows strain gauges on the invisible face of the sample)
This arrangement of strain gauges is due to the fact that in the process of stretching the sample, the lines of action of tensile forces P do not always coincide with the longitudinal axis of the sample, i.e., there is an eccentricity (displacement of the line of action of forces P from the longitudinal axis). The average readings of strain gauges taken from two sections of the sample will give a true picture.
item 4. Remarks
1. The application to the sample of an additional load equal to the loading stage should give each time the same increment of its length. This is due to the fact that the stretching of the sample in this laboratory work is carried out only within the limits of the elastic properties of the material, within the limits of the Hooke's law, which is linear dependence between load and deformation. This provision allows the experiment to be carried out repeatedly, using as a basis a constant additional load equal to the loading stage - P, with a uniform increase in the total load. To bring the experimental setup into operation
state used preliminary load stage
niya - P 0.
2. F arr - the cross-sectional area of the test sample is determined in accordance with fig. 6.
h = 0.3 cm
a = 8 cm
” 3. Working formulas for determining the modulus of longitudinal elasticity - E and Poisson's ratio -
In laboratory work, the desired characteristics are determined taking into account the stepwise method of force increment and the equality of the dimensions of the test sections to the bases of the longitudinal and transverse strain gauges:
1) E is determined from formula (3) - Hooke's law (type II) -
l Nl;
P lbt | |||||
l btF arr |
|||||
where P | is the increment of the force applied to the sample (step |
||||
l bt | loading); | ||||
– base of the longitudinal strain gauge; |
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l bt - change in the base of the longitudinal strain gauge; F arr - cross-sectional area of the sample.
