It would seem that the results obtained in the previous paragraphs solve the problem of testing a compressed rod for stability; All that remains is to choose the safety factor. However, this is not the case. A closer study of the numerical values ​​obtained using Euler’s formula shows that it gives correct results only within certain limits.

Figure 1 shows the dependence of the magnitude of critical stresses calculated at various values ​​of flexibility for steel 3, usually used in metal structures. This dependence is represented by a hyperbolic curve, the so-called “Eulerian hyperbola”:

When using this curve, we must remember that the formula it represents was obtained by integrating the differential equation of the curved axis, i.e., under the assumption that the stresses in the rod at the moment of loss of stability do not exceed the proportionality limit.

Fig.1. Hyperbolic dependence of the critical stress on the flexibility of the rod

Consequently, we do not have the right to use the values ​​of critical stresses calculated using Euler’s formula if they are obtained above this limit for a given material. In other words, Euler’s formula is applicable only if the following conditions are met:

If we express flexibility from this inequality, then the condition for the applicability of Euler’s formulas will take a different form:

Substituting the corresponding values ​​of the modulus of elasticity and the limit of proportionality for a given material, we find the smallest value of flexibility at which it is still possible to use Euler’s formula. For steel 3, the proportionality limit can be taken equal to , therefore, for rods made of this material, you can use the Euler formula only with flexibility

i.e. greater than 100%

For steel 5 at Euler's formula is applicable under flexibility; for cast iron at , for pine at etc. If we draw a horizontal line in Fig. 1 with an ordinate equal to , then it will cut the Euler hyperbola into two parts; You can only use the lower part of the graph, which relates to relatively thin and long rods, the loss of stability of which occurs at stresses lying no higher than the proportionality limit.

The theoretical solution obtained by Euler turned out to be applicable in practice only for a very limited category of rods, namely, thin and long rods with great flexibility. Meanwhile, rods with low flexibility are very often found in structures. Attempts to use Euler's formula to calculate critical stresses and check stability at low flexibility sometimes led to very serious disasters, and experiments on compression of rods show that at critical stresses greater than the proportionality limit, the actual critical forces are significantly lower than those determined by Euler's formula.

Thus, it is necessary to find a way to calculate the critical stresses also for those cases where they exceed the proportionality limit of the materials, for example, for mild steel rods with slenderness from 0 to 100.

It should immediately be noted that at present the most important source for establishing critical stresses beyond the limit of proportionality, i.e., at low and medium flexibility, are the results of experiments. There are attempts to theoretically solve this problem, but they rather point the way to further research than provide grounds for practical calculations.

First of all, it is necessary to select rods with low flexibility, from 0 to approximately 30 x 40; their length is relatively small in relation to the cross-sectional dimensions. For example, for a rod of circular cross-section, flexibility 20 corresponds to a length-to-diameter ratio of 5. For such rods, it is difficult to talk about the phenomenon of loss of stability of the rectilinear shape of the entire rod as a whole in the sense that this is the case for thin and long rods.

These short rods will fail mainly due to the fact that the compressive stresses in them will reach the yield point (for ductile materials) or the strength limit (for brittle materials). Therefore, for short rods, up to a flexibility of approximately 3040, the critical stresses “will be equal to, or slightly lower (due to some curvature of the rod axis that is still observed), respectively, either (steel) or (cast iron, wood).

Thus, we have two limiting cases of operation of compressed rods: short rods, which lose their load-carrying capacity mainly due to the destruction of the material from compression, and long rods, for which the loss of load-carrying capacity is caused by a violation of the stability of the rectilinear shape of the rod. A quantitative change in the ratio of the length and transverse dimensions of the rod changes the entire nature of the destruction phenomenon. What remains common is the suddenness of the onset of a critical state in the sense of a sudden sharp increase in deformations.

In compressed rods of great flexibility, for which Euler’s formula is applicable, after reaching the force R critical value, a sharp increase in deformations is usually observed. Up to this point, deflections tend to increase with increasing load, but remain insignificant. Theoretically, one would expect the rod to remain straight until the critical force; However, a number of circumstances that are inevitable in practice - the initial curvature of the rod, some eccentricity in the application of the load, local overstresses, heterogeneity of the material - cause small deflections even at compressive forces less than critical.

The dependence of shortening on stress during compression of short rods has a similar character; we have the same suddenness of growth of deformations at a certain magnitude of stress (when ).

It now remains for us to consider the behavior of compressed rods at average values ​​of flexibility, for example, for steel rods with flexibility from 40 to 100; Engineers most often encounter similar values ​​of flexibility in practice.

By the nature of destruction, these rods approach the category of thin and long rods; they lose their linear shape and are destroyed by significant lateral buckling. In experiments with them, one can note the presence of a clearly expressed critical force in the “Eulerian” sense; critical stresses are obtained above the proportionality limit and below the yield strength for ductile and tensile strength for brittle materials.

However, the loss of rectilinear shape and the reduction in critical stresses compared to short bars for these “medium” flexibility bars are associated with the same phenomena of deterioration in material strength that cause loss of load-carrying capacity in short bars. Here, both the influence of length, which reduces the value of critical stresses, and the influence of a significant increase in material deformations at stresses beyond the proportionality limit are combined.

Experimental determination of the critical forces for compressed rods has been carried out repeatedly both here and abroad. Particularly extensive experimental material was collected by Prof. F. Yasinsky, who compiled a table of critical (“breaking”) stresses c. depending on flexibility for a number of materials and laid the foundation for modern methods of calculating compressed rods for stability.

Based on the experimental material obtained, we can assume that at critical stresses less than the proportionality limit, all experiments confirm Euler’s formula for any material.

For rods of medium and low flexibility, various empirical formulas have been proposed, showing that the critical stresses for such flexibility change according to a law close to linear:

Where A And b coefficients depending on the material, a flexibility of the rod. For cast iron, Yasinsky received: A = 338,7MPa, b = 1,483 MPa. For steel 3 with flexibility from = 40 to = 100 coefficients A And b may be accepted: A = 336 MPa; b = 1,47MPa. For wood (pine): A = 29,3 MPa; b = 0,194 MPa.

Sometimes empirical formulas are convenient, giving for the inelastic region the change in critical stresses according to the law of a square parabola; these include the formula

Here at = 0 is considered for ductile and brittle materials; coefficient A, selected from the condition of smooth conjugation with the Euler hyperbola, has the value:

for steel with yield strength = 280 MPa a = 0,009 MPa

Given the data given here, a complete graph of critical stresses (depending on flexibility) for any material can be constructed. Figure 2 shows such a graph for building steel with a yield strength and the limit of proportionality .

Fig.2. Complete critical stress chart for structural steel.

The graph consists of three parts: an Euler hyperbola at, an inclined straight line at and a horizontal, or slightly inclined, straight line at . Similar graphs can be constructed by combining Euler’s formula with experimental results for other materials.

Checking compressed rods for stability.

It was previously noted that for compressed bars two checks must be made:

for strength

for sustainability

Where

To establish the permissible stress for stability, we now only have to select the safety factor k.

In practice, this coefficient varies for steel from 1.8 to 3.0. The safety factor for stability is selected higher than the safety factor for strength, equal to 1.5 × 1.6 for steel.

This is explained by the presence of a number of circumstances that are inevitable in practice (initial curvature, eccentricity of action, loads, material heterogeneity, etc.) and have almost no effect on the operation of the structure under other types of deformation (torsion, bending, tension).

For compressed rods, due to the possibility of loss of stability, these circumstances can greatly reduce the load-carrying capacity of the rod. For cast iron, the safety factor ranges from 5.0 to 5.5, for wood from 2.8 to 3.2.

To establish a connection between the permissible stress for stability and the permissible stress for strength, let us take their ratio:

Designating

here is the reduction factor for the main permissible stress for compressed rods.

Having a graph of the dependence on for a given material, knowing or and choosing safety factors for strength and stability, you can create tables of coefficient values ​​as a function of flexibility. Such data is provided in our technical specifications for the design of structures; they are tabulated.

Tensile strength

A certain threshold value for a specific material, exceeding which will lead to the destruction of the object under the influence of mechanical stress. The main types of strength limits: static, dynamic, compressive and tensile. For example, the tensile strength is the limit value of a constant (static limit) or variable (dynamic limit) mechanical stress, exceeding which will rupture (or unacceptably deform) the product. Unit of measurement - Pascal [Pa], N/mm² = [MPa].

Yield strength (σ t)

The amount of mechanical stress at which the deformation continues to increase without increasing the load; used for calculating permissible stresses in plastic materials.

After passing the yield point, irreversible changes are observed in the metal structure: the crystal lattice is rearranged, and significant plastic deformations appear. At the same time, self-strengthening of the metal occurs and after the yield point, the deformation increases with increasing tensile force.

This parameter is often defined as “the stress at which plastic deformation begins to develop,” thus identifying the limits of yield and elasticity. However, it should be understood that these are two different parameters. The yield strength values ​​exceed the elastic limit by approximately 5%.

Endurance limit or fatigue limit (σ R)

The ability of a material to withstand loads that cause cyclic stress. This strength parameter is defined as the maximum stress in a cycle at which fatigue failure of the product does not occur after an indefinitely large number of cyclic loads (the basic number of cycles for steel is Nb = 10 7). The coefficient R (σ R) is taken to be equal to the cycle asymmetry coefficient. Therefore, the fatigue limit of the material in the case of symmetrical loading cycles is denoted as σ -1, and in the case of pulsating ones - as σ 0.

Note that fatigue tests of products are very long and labor-intensive; they involve the analysis of large volumes of experimental data with an arbitrary number of cycles and a significant scatter of values. Therefore, special empirical formulas are most often used that connect the endurance limit with other strength parameters of the material. The most convenient parameter is considered to be the tensile strength.

For steels, the bending endurance limit is usually half the tensile strength: For high-strength steels, you can take:

For ordinary steels during torsion under conditions of cyclically changing stresses, the following can be accepted:

The above ratios should be used with caution, because they were obtained under specific loading conditions, i.e. during bending and torsion. However, when tested in tension-compression, the endurance limit becomes approximately 10-20% less than in bending.

Proportional limit (σ)

The maximum stress value for a particular material at which Hooke’s law still applies, i.e. The deformation of the body is directly proportional to the applied load (force). Please note that for many materials, reaching (but not exceeding!) the elastic limit leads to reversible (elastic) deformations, which, however, are no longer directly proportional to stress. In this case, such deformations may be somewhat “lag” relative to the increase or decrease in load.

Diagram of the deformation of a metal sample under tension in the coordinates elongation (Є) - stress (σ).

1: Absolute elasticity limit.

2: Limit of proportionality.

3: Elastic limit.

Today, there are several methods for testing material samples. At the same time, one of the simplest and most revealing tests are tensile (tensile) tests, which make it possible to determine the proportionality limit, yield strength, elastic modulus and other important characteristics of the material. Since the most important characteristic of the stressed state of a material is deformation, determining the deformation value for known dimensions of the sample and the loads acting on the sample makes it possible to establish the above characteristics of the material.

Here the question may arise: why can’t we simply determine the resistance of a material? The fact is that absolutely elastic materials, which collapse only after overcoming a certain limit - resistance, exist only in theory. In reality, most materials have both elastic and plastic properties; we will consider what these properties are below using the example of metals.

Tensile tests of metals are carried out in accordance with GOST 1497-84. For this purpose, standard samples are used. The test procedure looks something like this: a static load is applied to the sample, and the absolute elongation of the sample is determined Δl, then the load increases by a certain step value and the absolute elongation of the sample is again determined, and so on. Based on the data obtained, a graph of elongation versus load is constructed. This graph is called a stress diagram.

Figure 318.1. Stress diagram for a steel sample.

In this diagram we see 5 characteristic points:

1. Limit of proportionality R p(point A)

Normal stresses in the cross section of the sample when the proportionality limit is reached will be equal to:

σ p = P p /F o (318.2.1)

The proportionality limit limits the area of ​​elastic deformations on the diagram. In this section, the deformations are directly proportional to the stresses, which is expressed by Hooke’s law:

R p = kΔl (318.2.2)

where k is the stiffness coefficient:

k = EF/l (318.2.3)

where l is the length of the sample, F is the cross-sectional area, E is Young’s modulus.

Elastic moduli

The main characteristics of the elastic properties of materials are Young's modulus E (modulus of elasticity of the first kind, modulus of elasticity in tension), modulus of elasticity of the second kind G (modulus of elasticity in shear) and Poisson's ratio μ (transverse deformation coefficient).

Young's modulus E shows the ratio of normal stresses to relative strains within the limits of proportionality

Young's modulus is also determined empirically when testing standard tensile samples. Since the normal stresses in the material are equal to the force divided by the initial cross-sectional area:

σ = Р/F о (318.3.1), (317.2)

and relative elongation ε - the ratio of absolute deformation to the initial length

ε pr = Δl/l o (318.3.2)

then Young’s modulus according to Hooke’s law can be expressed as follows

E = σ/ε pr = Pl o /F o Δl = tg α (318.3.3)

Figure 318.2. Stress diagrams of some metal alloys

Poisson's ratio μ shows the ratio of transverse to longitudinal strains

Under the influence of loads, not only does the length of the sample increase, but also the area of ​​the cross-section under consideration decreases (if we assume that the volume of material in the region of elastic deformation remains constant, then an increase in the length of the sample leads to a decrease in the cross-sectional area). For a sample having a circular cross-section, the change in cross-sectional area can be expressed as follows:

ε pop = Δd/d o (318.3.4)

Then Poisson's ratio can be expressed by the following equation:

μ = ε pop /ε pr (318.3.5)

Shear modulus G shows the ratio of shear stresses T to the shear angle

The shear modulus G can be determined experimentally by testing specimens for torsion.

During angular deformations, the section under consideration does not move linearly, but at a certain angle - the shift angle γ to the initial section. Since the shear stress is equal to the force divided by the area in the plane in which the force acts:

T= Р/F (318.3.6)

and the tangent of the angle of inclination can be expressed as the ratio of the absolute deformation Δl to the distance h from the place where the absolute deformation was recorded to the point relative to which the rotation was made:

tgγ = Δl/h (318.3.7)

then at small values ​​of the shear angle the shear modulus can be expressed by the following equation:

G= T/γ = Ph/FΔl (318.3.8)

Young's modulus, shear modulus and Poisson's ratio are related to each other by the following relationship:

E = 2(1 + μ)G (318.3.9)

The values ​​of the constants E, G and µ are given in table 318.1

Table 318.1. Approximate values ​​of the elastic characteristics of some materials

Note: Elastic moduli are constant values, however, manufacturing technologies for various building materials change and more accurate values ​​of elastic moduli should be clarified according to currently valid regulatory documents. The modulus of elasticity of concrete depends on the class of concrete and therefore is not given here.

Elastic characteristics are determined for various materials within the limits of elastic deformations limited on the stress diagram by point A. Meanwhile, several more points can be identified on the stress diagram:

2. Elastic limit Р у

Normal stresses in the cross section of the sample when the elastic limit is reached will be equal to:

σ y = Р y /F o (318.2.4)

The elastic limit limits the area in which the appearing plastic deformations are within a certain small value, normalized by technical conditions (for example, 0.001%; ​​0.01%, etc.). Sometimes the elastic limit is designated according to the tolerance σ 0.001, σ 0.01, etc.

3. Yield strength Р t

σ t = P t /F o (318.2.5)

Limits the area of ​​the diagram in which the deformation increases without a significant increase in load (yield state). In this case, a partial rupture of internal bonds occurs throughout the entire volume of the sample, which leads to significant plastic deformations. The sample material is not completely destroyed, but its initial geometric dimensions undergo irreversible changes. On the polished surface of the samples, yield figures are observed - shear lines (discovered by Professor V.D. Chernov). For different metals, the angles of inclination of these lines are different, but are in the range of 40-50 o. In this case, part of the accumulated potential energy is irreversibly spent on the partial rupture of internal bonds. When testing for tension, it is customary to distinguish between the upper and lower yield limits - respectively, the highest and lowest stresses at which plastic (residual) deformation increases at an almost constant value of the effective load.

The stress diagrams indicate the lower yield strength. It is this limit for most materials that is taken as the standard resistance of the material.

Some materials do not have a pronounced yield plateau. For them, the conditional yield strength σ 0.2 is taken to be the stress at which the residual elongation of the sample reaches a value of ε ≈0.2%.

4. Tensile strength P max (temporary strength)

Normal stresses in the cross section of the sample when the ultimate strength is reached will be equal to:

σ in = P max /F o (318.2.6)

After overcoming the upper yield limit (not shown in the stress diagrams), the material again begins to resist loads. At maximum force P max, complete destruction of the internal bonds of the material begins. In this case, plastic deformations are concentrated in one place, forming a so-called neck in the sample.

The stress at maximum load is called the tensile strength or tensile strength of the material.

Tables 318.2 - 318.5 provide approximate strength values ​​for some materials:

Table 318.2 Approximate limits of compressive strength (temporary strength) of some building materials.

Note: For metals and alloys, the value of tensile strength should be determined in accordance with regulatory documents. The value of temporary resistances for some steel grades can be viewed.

Table 318.3. Approximate strength limits (tensile strengths) for some plastics

Table 318.4. Approximate tensile strengths for some fibers

Table 318.5. Approximate strength limits for some wood species

5. Material destruction P r

If you look at the stress diagram, it seems that the destruction of the material occurs as the load decreases. This impression is created because as a result of the formation of a “neck,” the cross-sectional area of ​​the sample in the area of ​​the “neck” changes significantly. If you construct a stress diagram for a sample made of low-carbon steel depending on the changing cross-sectional area, you will see that the stresses in the section under consideration increase to a certain limit:

Figure 318.3. Stress diagram: 2 - in relation to the initial cross-sectional area, 1 - in relation to the changing cross-sectional area in the neck area.

Nevertheless, it is more correct to consider the strength characteristics of the material in relation to the area of ​​the original section, since strength calculations rarely include changes in the original geometric shape.

One of the mechanical characteristics of metals is the relative change ψ of the cross-sectional area in the neck area, expressed as a percentage:

ψ = 100(F o - F)/F o (318.2.7)

where F o is the initial cross-sectional area of ​​the sample (cross-sectional area before deformation), F is the cross-sectional area in the “neck” area. The higher the value of ψ, the more pronounced the plastic properties of the material are. The lower the value of ψ, the greater the fragility of the material.

If you add up the torn parts of the sample and measure its elongation, it turns out that it is less than the elongation in the diagram (by the length of the segment NL), since after rupture the elastic deformations disappear and only plastic deformations remain. The amount of plastic deformation (elongation) is also an important characteristic of the mechanical properties of the material.

Beyond elasticity, up to fracture, total deformation consists of elastic and plastic components. If you bring the material to stresses exceeding the yield strength (in Fig. 318.1, some point between the yield strength and the tensile strength), and then unload it, then plastic deformations will remain in the sample, but when reloaded after some time, the elastic limit will become higher, since in this case, a change in the geometric shape of the sample as a result of plastic deformations becomes, as it were, the result of the action of internal connections, and the changed geometric shape becomes the initial one. This process of loading and unloading material can be repeated several times, and the strength properties of the material will increase:

Figure 318.4. Stress diagram during work hardening (inclined straight lines correspond to unloading and repeated loading)

This change in the strength properties of a material, obtained through repeated static loading, is called work hardening. However, when the strength of a metal increases by cold hardening, its plastic properties decrease and its fragility increases, so relatively small hardening is usually considered useful.

Work of deformation

The greater the internal forces of interaction between the particles of the material, the higher the strength of the material. Therefore, the value of elongation resistance per unit volume of a material can serve as a characteristic of its strength. In this case, the tensile strength is not an exhaustive characteristic of the strength properties of a given material, since it characterizes only the cross sections. When a rupture occurs, the interconnections are destroyed over the entire cross-sectional area, and during shears, which occur during any plastic deformation, only local interconnections are destroyed. To destroy these connections, a certain amount of work of internal interaction forces is expended, which is equal to the work of external forces expended on displacement:

A = РΔl/2 (318.4.1)

where 1/2 is the result of the static action of the load, increasing from 0 to P at the time of its application (average value (0 + P)/2)

During elastic deformation, the work of forces is determined by the area of ​​the triangle OAB (see Fig. 318.1). Total work expended on the deformation of the sample and its destruction:

A = ηР max Δl max (318.4.2)

where η is the coefficient of completeness of the diagram, equal to the ratio of the area of ​​the entire diagram, limited by the curve AM and straight lines OA, MN and ON, to the area of ​​a rectangle with sides 0P max (along the P axis) and Δl max (dotted line in Fig. 318.1). In this case, it is necessary to subtract the work determined by the area of ​​the triangle MNL (related to elastic deformations).

The work spent on plastic deformation and destruction of the sample is one of the important characteristics of the material that determines the degree of its fragility.

Compression strain

Compressive deformations are similar to tensile deformations: first, elastic deformations occur, to which plastic deformations are added beyond the elastic limit. The nature of deformation and fracture during compression is shown in Fig. 318.5:

Figure 318.5

a - for plastic materials; b - for fragile materials; c - for wood along the grain, d - for wood across the grain.

Compression tests are less convenient for determining the mechanical properties of plastic materials due to the difficulty of recording the moment of failure. Methods of mechanical testing of metals are regulated by GOST 25.503-97. When testing for compression, the shape of the sample and its dimensions may be different. Approximate values ​​of tensile strength for various materials are given in tables 318.2 - 318.5.

If the material is under load at a constant stress, then additional elastic deformation is gradually added to the almost instantaneous elastic deformation. When the load is completely removed, the elastic deformation decreases in proportion to the decreasing stresses, and the additional elastic deformation disappears more slowly.

The resulting additional elastic deformation under constant stress, which does not disappear immediately after unloading, is called elastic aftereffect.

The influence of temperature on changes in the mechanical properties of materials

The solid state is not the only state of aggregation of a substance. Solids exist only in a certain range of temperatures and pressures. An increase in temperature leads to a phase transition from solid to liquid, and the transition process itself is called melting. Melting points, like other physical characteristics of materials, depend on many factors and are also determined experimentally.

Table 318.6. Melting points of some substances

Note: The table shows the melting points at atmospheric pressure (except for helium).

The elastic and strength characteristics of materials given in tables 318.1-318.5 are determined, as a rule, at a temperature of +20 o C. GOST 25.503-97 allows testing of metal samples in the temperature range from +10 to +35 o C.

When the temperature changes, the potential energy of the body changes, which means the value of the internal interaction forces also changes. Therefore, the mechanical properties of materials depend not only on the absolute value of temperature, but also on the duration of its action. For most materials, when heated, the strength characteristics (σ p, σ t and σ v) decrease, while the plasticity of the material increases. As the temperature decreases, the strength characteristics increase, but at the same time the fragility increases. When heated, Young's modulus E decreases, and Poisson's ratio increases. When the temperature decreases, the reverse process occurs.

Figure 318.6. The influence of temperature on the mechanical characteristics of carbon steel.

When non-ferrous metals and alloys made from them are heated, their strength immediately drops and at a temperature close to 600° C, it is practically lost. The exception is aluminothermic chromium, the tensile strength of which increases with increasing temperature and at a temperature of 1100° C reaches a maximum σ in1100 = 2σ in20.

The ductility characteristics of copper, copper alloys and magnesium decrease with increasing temperature, and those of aluminum increase. When plastics and rubber are heated, their tensile strength decreases sharply, and when cooled, these materials become very brittle.

Effect of radioactive irradiation on changes in mechanical properties

Radiation exposure affects different materials differently. Irradiation of materials of inorganic origin in its effect on mechanical characteristics and plasticity characteristics is similar to a decrease in temperature: with an increase in the dose of radioactive irradiation, the tensile strength and especially the yield strength increase, and the plasticity characteristics decrease.

Irradiation of plastics also leads to an increase in fragility, and irradiation has different effects on the tensile strength of these materials: on some plastics it has almost no effect (polyethylene), in others it causes a significant decrease in tensile strength (katamen), and in others it increases the tensile strength (selectron ).

2. Elastic limit

3. Yield strength

4. Tensile strength or tensile strength

5. Voltage at break

Drawing. 2.3 – View of a cylindrical sample after fracture (a) and change in the sample zone near the fracture site (b)

In order for the diagram to reflect only the properties of the material (regardless of the size of the sample), it is rearranged in relative coordinates (stress-strain).

Arbitrary ordinates i-th the points of such a diagram (Fig. 2.4) are obtained by dividing the values ​​of the tensile force (Fig. 2.2) by the original cross-sectional area of ​​the sample (), and the abscissa by dividing the absolute elongation of the working part of the sample by its original length (). In particular, for characteristic points of the diagram, the ordinates are calculated using formulas (2.3)…(2.7).

The resulting diagram is called conventional voltage diagram (Fig. 2.4).

The convention of the diagram lies in the method of determining the stress not from the current cross-sectional area, which changes during testing, but from the original one - the stress diagram retains all the features of the original tensile diagram. The characteristic stresses in the diagram are called limiting stresses and reflect the strength properties of the material being tested. (formulas 2.3…2.7). Note that the yield strength of the metal taught in this case corresponds to the new physical state of the metal and is therefore called the physical yield strength

Drawing. 2.4 – Voltage diagram

From the voltage diagram (Fig. 2.4) it is clear that

i.e. tensile modulus E is numerically equal to the tangent of the angle of inclination of the initial straight section of the stress diagram to the abscissa axis. This is the geometric meaning of the tensile elastic modulus.

If we relate the forces acting on the sample at each moment of loading to the true value of the cross section at the corresponding moment in time, then we obtain a diagram of true stresses, often denoted by the letter S(Fig. 2.5, solid line). Since in the section of the diagram 0-1-2-3-4 the diameter of the sample decreases slightly (the neck has not yet formed), the true diagram, within this section, practically coincides with the conventional diagram (dashed curve), passing slightly higher.

Drawing. 2.5 – True voltage diagram

Constructing the remaining section of the true stress diagram (section 4-5 in Fig. 2.5) necessitates measuring the diameter of the sample during a tensile test, which is not always possible. There is an approximate way to construct this section of the diagram, based on determining the coordinates of point 5() of the true diagram (Fig. 2.5), corresponding to the moment of sample rupture. First, the true breaking stress is determined

where is the force on the sample at the moment of its rupture;

– cross-sectional area in the neck of the sample at the moment of rupture.

The second coordinate of the point - relative deformation - includes two components - true plastic - and elastic -. The value can be determined from the condition of equality of the volumes of material near the point of rupture of the sample before and after the test (Fig. 2.3). So before testing the volume of material of a sample of unit length will be equal to , and after rupture . Here is the elongation of a sample of unit length near the fracture site. Since the true deformation is here and , That . We find the elastic component using Hooke's law: . Then the abscissa of point 5 will be equal to . Drawing a smooth curve between points 4 and 5, we obtain a complete view of the true diagram.

For materials whose tensile diagram in the initial section does not have a clearly defined yield plateau (see Fig. 2.6), the yield strength is conventionally defined as the stress at which the residual deformation is the value established by GOST or technical specifications. According to GOST 1497–84, this value of residual deformation is 0.2% of the measured length of the sample, and proof strength is indicated by the symbol – .

When testing tensile samples, in addition to strength characteristics, plasticity characteristics are also determined, which include relative extension sample after rupture, defined as the ratio of the increment in the length of the sample after rupture to its original length:

And relative narrowing , calculated by the formula

% (2.10)

In these formulas - the initial calculated length and cross-sectional area of ​​the sample, - respectively, the length of the calculated part and the minimum cross-sectional area of ​​the sample after rupture.

Instead of relative deformation, in some cases the so-called logarithmic deformation is used. Since the length of the sample changes as the sample is stretched, the increment in length dl refer not to , but to the current value . If we integrate the increments of elongations when the length changes from to , we get the logarithmic or true deformation of the metal

Then – strain at break (i.e. . = k) will

.

It should also be taken into account that plastic deformation in the sample occurs unevenly along its length.

Depending on the nature of the metal, they are conventionally divided into very ductile (annealed copper, lead), ductile (low-carbon steels), brittle (gray cast iron), very brittle (white cast iron, ceramics).

Load Application Rate V deformation affects the appearance of the diagram and the characteristics of the material. σ T And σ V increases with increasing load speed. Deformations corresponding to the ultimate strength and failure point are reduced.

Conventional machines provide strain rate

10 -2 ...10 -5 1/sec.

As the temperature drops T isp for pearlitic steels increases σ T and decreases.

Austenitic steels, Al And Ti alloys react weaker to lowering T.

With increasing temperature, a change in deformation over time is observed at constant stresses, i.e. creep occurs, and more than > σ , those< .

There are usually three stages of creep. For mechanical engineering, stage II is of greatest interest, where έ = const (steady stage of creep).

To compare the creep resistance of various metals, a conditional characteristic has been introduced - the creep limit.

Creep limit σ pl is called the stress at which plastic deformation in a given period of time reaches the value established by the technical conditions.

Along with the concept of “creep”, the concept of “stress relaxation” is also known.

The process of stress relaxation occurs under constant deformations.

A sample under constant load at high T can fracture either with necking (ductile intercrystalline fracture) or without necking (brittle transcrystalline fracture). The first is typical for lower T and high σ .

Material strength at high T assessed by the long-term strength limit.

Long-term strength limit(σ dp) is the ratio of the load under which a tensile sample fails after a certain period of time to the original cross-sectional area.

When designing welded products operating at elevated T, are guided by the following values ​​when assigning [ σ ]:

a) when T 260 o C for tensile strength σ V ;

b) when T 420 o C for carbon steels T < 470 о С для стали 12Х1МФ, T< 550 о С для 1Х18Н10Т – на σ T ;

c) at higher T to the limit of long-term strength σ dp .

In addition to the listed test methods under static loads, bending, torsion, shear, compression, crushing, stability, and hardness tests are also performed.

Metals are characterized by high ductility, thermal and electrical conductivity. They have a characteristic metallic luster.

About 80 elements of the periodic table of D.I. have properties of metals. Mendeleev. For metals, as well as for metal alloys, especially structural ones, mechanical properties are of great importance, the main ones being strength, ductility, hardness and impact strength.

Under the influence of an external load, stress and deformation arise in a solid body. related to the original cross-sectional area of ​​the sample.

Deformation – this is a change in the shape and size of a solid body under the influence of external forces or as a result of physical processes that occur in the body during phase transformations, shrinkage, etc. Deformation may be elastic(disappears after the load is removed) and plastic(remains after the load is removed). With an ever-increasing load, elastic deformation, as a rule, turns into plastic, and then the sample collapses.

Depending on the method of applying the load, methods for testing the mechanical properties of metals, alloys and other materials are divided into static, dynamic and alternating.

Strength – the ability of metals to resist deformation or destruction under static, dynamic or alternating loads. The strength of metals under static loads is tested in tension, compression, bending and torsion. Tensile testing is mandatory. Strength under dynamic loads is assessed by specific impact strength, and under alternating loads - by fatigue strength.

To determine strength, elasticity and ductility, metals in the form of round or flat samples are tested for static tension. Tests are carried out on tensile testing machines. As a result of the tests, a tensile diagram is obtained (Fig. 3.1) . The abscissa axis of this diagram shows the strain values, and the ordinate axis shows the stress values ​​applied to the sample.

The graph shows that no matter how small the applied stress, it causes deformation, and the initial deformations are always elastic and their magnitude is directly dependent on the stress. On the curve shown in the diagram (Fig. 3.1), elastic deformation is characterized by the line OA and its continuation.

Rice. 3.1. Strain curve

Above the point A the proportionality between stress and strain is violated. Stress causes not only elastic, but also residual, plastic deformation. Its value is equal to the horizontal segment from the dashed line to the solid curve.

During elastic deformation under the influence of an external force, the distance between atoms in the crystal lattice changes. Removing the load eliminates the cause that caused the change in the interatomic distance, the atoms return to their original places and the deformation disappears.

Plastic deformation is a completely different, much more complex process. During plastic deformation, one part of the crystal moves relative to another. If the load is removed, the displaced part of the crystal will not return to its original location; the deformation will persist. These shifts are revealed by microstructural examination. In addition, plastic deformation is accompanied by crushing of mosaic blocks inside the grains, and at significant degrees of deformation, a noticeable change in the shape of the grains and their location in space is also observed, and voids (pores) appear between the grains (sometimes inside the grains).

Represented dependency OAV(see Fig. 3.1) between externally applied voltage ( σ ) and the relative deformation caused by it ( ε ) characterizes the mechanical properties of metals.

· straight line slope OA shows metal hardness, or a characteristic of how a load applied from the outside changes interatomic distances, which, to a first approximation, characterizes the forces of interatomic attraction;

· tangent of the angle of inclination of the straight line OA proportional to elastic modulus (E), which is numerically equal to the quotient of stress divided by relative elastic deformation:

voltage, which is called the limit of proportionality ( σ pc), corresponds to the moment of appearance of plastic deformation. The more accurate the deformation measurement method, the lower the point lies A;

· in technical measurements a characteristic called yield strength (σ 0.2). This is a stress that causes a residual deformation equal to 0.2% of the length or other size of the sample or product;

maximum voltage ( σ c) corresponds to the maximum stress achieved during tension and is called temporary resistance or tensile strength .

Another characteristic of the material is the amount of plastic deformation that precedes fracture and is defined as a relative change in length (or cross-section) - the so-called relative extension (δ ) or relative narrowing (ψ ), they characterize the plasticity of the metal. Area under the curve OAV proportional to the work that must be expended to destroy the metal. This indicator, determined in various ways (mainly by striking a cut sample), characterizes viscosity metal

When a sample is stretched to the point of failure, the relationships between the applied force and the elongation of the sample are recorded graphically (Fig. 3.2), resulting in so-called deformation diagrams.

Rice. 3.2. Diagram "force (tension) - elongation"

The deformation of the sample when the alloy is loaded is first macroelastic, and then gradually and in different grains under unequal loads transforms into plastic, occurring through shear through the dislocation mechanism. The accumulation of dislocations as a result of deformation leads to strengthening of the metal, but when their density is significant, especially in individual areas, centers of destruction arise, ultimately leading to the complete destruction of the sample as a whole.

Tensile strength is assessed by the following characteristics:

1) tensile strength;

2) the limit of proportionality;

3) yield strength;

4) elastic limit;

5) elastic modulus;

6) yield strength;

7) relative elongation;

8) relative uniform elongation;

9) relative narrowing after rupture.

Tensile strength (tensile strength or tensile strength) σ in, is the voltage corresponding to the greatest load R V preceding the destruction of the sample:

σ in = P in /F 0,

This characteristic is mandatory for metals.

Proportionality limit (σ pc) – this is the conditional voltage R pc, at which the deviation from the proportional dependence of the bridge between deformation and load begins. It is equal to:

σ pc = P pc /F 0.

Values σ pc is measured in kgf/mm 2 or in MPa .

Yield strength (σ t) is the voltage ( R T) in which the sample deforms (flows) without a noticeable increase in load. Calculated by the formula:

σ t = R T / F 0 .

Elastic limit (σ 0.05) is the stress at which the residual elongation reaches 0.05% of the length of the section of the working part of the sample, equal to the base of the strain gauge. Elastic limit σ 0.05 is calculated using the formula:

σ 0,05 = P 0,05 /F 0 .

Elastic modulus (E) – the ratio of the increment in stress to the corresponding increment in elongation within the limits of elastic deformation. It is equal to:

E = Pl 0 /l avg F 0 ,

Where ∆Р– load increment; l 0– initial estimated length of the sample; l wed– average increment of elongation; F 0 – initial cross-sectional area.

Yield strength (conditional) – stress at which the residual elongation reaches 0.2% of the length of the sample section on its working part, the elongation of which is taken into account when determining the specified characteristic.

Calculated by the formula:

σ 0,2 = P 0,2 /F 0 .

The conditional yield strength is determined only if there is no yield plateau on the tensile diagram.

Relative extension (after the breakup) – one of the characteristics of the plasticity of materials, equal to the ratio of the increment in the estimated length of the sample after destruction ( l to) to the initial effective length ( l 0) in percentages:

Relative uniform elongation (δ р)– the ratio of the increment in the length of sections in the working part of the sample after rupture to the length before testing, expressed as a percentage.

Relative narrowing after rupture (ψ ), as well as relative elongation, is a characteristic of the plasticity of the material. Defined as the difference ratio F 0 and minimum ( F to) cross-sectional area of ​​the sample after destruction to the initial cross-sectional area ( F 0), expressed as a percentage:

Elasticity – the property of metals to restore their previous shape after removal of external forces causing deformation. Elasticity is the opposite property of plasticity.

Very often, to determine strength, a simple, non-destructive, simplified method is used - measuring hardness.

Under hardness material is understood as resistance to penetration of a foreign body into it, i.e., in fact, hardness also characterizes resistance to deformation. There are many methods for determining hardness. The most common is Brinell method (Fig. 3.3, a), when the test body is subjected to force R a ball with a diameter of D. The Brinell hardness number (HH) is the load ( R), divided by the area of ​​the spherical surface of the print (diameter d).

Rice. 3.3. Hardness test:

a – according to Brinell; b – according to Rockwell; c – according to Vickers

When measuring hardness Vickers method (Fig. 3.3, b) the diamond pyramid is pressed in. By measuring the diagonal of the print ( d), judge the hardness (HV) of the material.

When measuring hardness Rockwell method (Fig. 3.3, c) the indenter is a diamond cone (sometimes a small steel ball). The hardness number is the reciprocal of the indentation depth ( h). There are three scales: A, B, C (Table 3.1).

Brinell and Rockwell B scale methods are used for soft materials, Rockwell C scale method for hard materials, and Rockwell A scale method and Vickers method for thin layers (sheets). The described methods for measuring hardness characterize the average hardness of the alloy. In order to determine the hardness of individual structural components of the alloy, it is necessary to sharply localize the deformation, press the diamond pyramid into a certain place, found on a thin section at a magnification of 100 - 400 times under a very small load (from 1 to 100 gf), followed by measuring the diagonal of the indentation under a microscope . The resulting characteristic ( N) is called microhardness , and characterizes the hardness of a certain structural component.

Table 3.1 Test conditions when measuring hardness using the Rockwell method

Test conditions		Designation t firmness
R= 150 kgf
When tested with diamond cone and load R= 60 kgf
When pressing the steel ball and loading R= 100 kgf

The NV value is measured in kgf/mm 2 (in this case, the units are often not indicated) or in SI - in MPa (1 kgf/mm 2 = 10 MPa).

Viscosity – the ability of metals to resist impact loads. Viscosity is the opposite property of brittleness. During operation, many parts experience not only static loads, but are also subject to shock (dynamic) loads. For example, such loads are experienced by the wheels of locomotives and cars at rail joints.

The main type of dynamic tests is impact loading of notched samples under bending conditions. Dynamic impact loading is carried out on pendulum impact drivers (Fig. 3.4), as well as with a falling load. In this case, the work expended on the deformation and destruction of the sample is determined.

Typically, in these tests, the specific work spent on deformation and destruction of the sample is determined. It is calculated using the formula:

KS =K/ S 0 ,

Where KS– specific work; TO– total work of deformation and destruction of the sample, J; S 0– cross-section of the sample at the incision site, m 2 or cm 2.

Rice. 3.4. Impact testing using a pendulum impact tester

The width of all types of specimens is measured before testing. The height of samples with a U- and V-shaped notch is measured before testing, and with a T-shaped notch after testing. Accordingly, the specific work of fracture deformation is denoted by KCU, KCV and KST.

Fragility metals at low temperatures are called cold brittleness . The value of impact strength is significantly lower than at room temperature.

Another characteristic of the mechanical properties of materials is fatigue strength. Some parts (shafts, connecting rods, springs, springs, rails, etc.) during operation experience loads that change in magnitude or simultaneously in magnitude and direction (sign). Under the influence of such alternating (vibration) loads, the metal seems to get tired, its strength decreases and the part collapses. This phenomenon is called tired metal, and the resulting fractures are fatigue. For such details you need to know endurance limit, those. the magnitude of the maximum stress that a metal can withstand without destruction for a given number of load changes (cycles) ( N).

Wear resistance – resistance of metals to wear due to friction processes. This is an important characteristic, for example, for contact materials and, in particular, for the contact wire and current-collecting elements of the current collector of electrified transport. Wear consists of the separation of individual particles from the rubbing surface and is determined by changes in the geometric dimensions or mass of the part.

Fatigue strength and wear resistance give the most complete picture of the durability of parts in structures, and toughness characterizes the reliability of these parts.