Introduction
In order to successfully study the material part of the equipment of the NBC protection troops, deep knowledge of general technical disciplines is necessary. Many machine parts are subjected to cyclic stresses during operation. Therefore, cadets should have an idea about the parameters and types of stress cycles, the phenomenon and endurance limit.
Therefore, the material of this lecture is of great importance. The purpose of this lecture is to give students the basic terms and definitions related to cyclic stresses, to study the issue of calculating structural elements for strength under a given type of loading.
The concept of cyclic stresses. Parameters and types of stress cycles
Dynamic loads, despite the absence of significant inertial forces, include periodic multiple repeated (cyclic) loads acting on structural elements. This kind of loading is typical for most engineering structures, such as axles, shafts, rods, springs, connecting rods, etc.
The strength of materials under repeated-variable loading largely depends on the nature of the change in stresses over time.
- variable load with a time-established nature of change, the values of which are repeated after a certain period (period) of time.Stress cycle- the totality of all values of variable stresses during one period of load change.
Typically, a stress cycle is characterized by two main cycle parameters: and - the maximum and minimum stresses of the cycle.
Average cycle stress 
.
Amplitude cycle voltage 
.
Coefficient of asymmetry of the stress cycle.
Depending on the magnitude of the listed characteristics, stress cycles can be divided into the following main types:
Symmetric cycle- the maximum and minimum voltages are equal in absolute value and opposite in sign, R = -1.
Asymmetric cycle- the maximum and minimum voltages are not equal in absolute value, while the asymmetric cycle can be sign-alternating or sign-constant.
alternating cycle– the maximum and minimum voltages are not equal in absolute value and opposite in sign , , .
Constant-sign cycle– the maximum and minimum voltages are not equal in absolute value and have the same sign , , .
Zero (pulsating) cycle– the maximum or minimum voltages are equal to zero or , or .
The phenomenon of fatigue. fatigue curve. endurance limit
As practice shows, loads that cyclically change in time in magnitude or in magnitude and sign can lead to structural failure at stresses significantly lower than the yield strength (or tensile strength). Such destruction is called "fatigue". The material seems to "get tired" under the action of repeated periodic loads.
fatigue failure- destruction of the material under the action of repetitively alternating stresses.
Material fatigue- gradual accumulation of damage in the material under the action of variable stresses, leading to the formation of cracks in the material and destruction.
Endurance is the ability of a material to resist fatigue failure.
The physical causes of fatigue failure of materials are quite complex and not yet fully understood. One of the main causes of fatigue failure is considered to be the formation and development of cracks.
The mechanism of fatigue failure is largely related to the heterogeneity of the real structure of materials (difference in size, shape, orientation of neighboring metal grains; the presence of various inclusions - slags, impurities; crystal lattice defects, material surface defects - scratches, corrosion, etc.). In connection with the indicated inhomogeneity at variable stresses at the boundaries of individual inclusions and near microscopic voids and various defects, a stress concentration arises, which leads to microplastic shear deformations of some metal grains, while slip bands may appear on the surface of the grains, and shear accumulation, which on some materials manifests itself in the form of microscopic tubercles and depressions - extrusions and intrusions. Then there is the development of shifts into microcracks, their growth and merging; at the last stage, one or several macrocracks appear, which develop (grow) quite intensively. Crack edges under action variable load rub against each other, and therefore the crack growth zone has a smooth (polished) surface. As the crack grows, the cross section of the part weakens more and more, and finally, a sudden brittle fracture of the part occurs, while the brittle fracture zone has a coarse-grained crystalline structure, as in brittle fracture.
The fatigue curve (Weller curve) is built on the basis of the results of fatigue tests with a symmetrical cycle. It shows that with an increase in the number of cycles, the maximum stress at which the destruction of the material occurs decreases significantly. At the same time, for many materials, for example, carbon steel, it is possible to set such a maximum cycle stress at which the sample does not collapse after any number of cycles (horizontal section of the diagram), called the endurance limit ().
Limit of endurance (fatigue) is the maximum (limiting) stress of the cycle, at which there is no fatigue failure of the sample after an arbitrarily large number of cycles.
Since tests cannot be carried out indefinitely, the number of cycles is limited by a certain limit, which is called the base number of cycles. In this case, if the sample withstands the base number of cycles (for ferrous metals - N= 10 7), then it is considered that the voltage in it is not higher than the endurance limit.
Fatigue curves for non-ferrous metals do not have horizontal sections, so for them for the base number of cycles it increases to N= 10 8 and the limit of limited endurance is set.
In real structures, the vast majority of parts operate under asymmetric loading.
Diagram ultimate stresses(Smith diagram) is built on at least three loading modes (on three points), for each of which the endurance limit is determined.
The first mode (point 1) is the usual symmetrical loading cycle ( , , , ).
The second mode (point 2) is an asymmetric loading cycle, usually zero ( , , , ).
The third mode (point 3) is simple static stretching ( , ).
The obtained points are connected by a smooth line, the ordinates of the points of which correspond to the endurance limits of the material at different values cycle asymmetry coefficient.
A beam passing at an angle through the origin of the limit stress diagram characterizes cycles with the same asymmetry coefficient R :
.
Diagram limiting amplitudes(Haig diagram) is plotted in coordinates: average stress of the cycle – amplitude of the cycle (Figure 7). At the same time, to build it, it is necessary to carry out fatigue tests for at least three modes: 1 - symmetrical cycle; 2 – zero cycle; 3 - static stretching.
Connecting the obtained points with a smooth curve, a graph is obtained that characterizes the relationship between the values of the limiting amplitudes and the values of the limiting average stresses in the cycle.
In addition to material properties, the following factors influence fatigue strength: 1) the presence of stress concentrators; 2) scale factor, that is, the influence of the absolute dimensions of the part (the larger the size of the part, the lower the fatigue strength); 3) the quality of surface treatment (with a decrease in the surface roughness of the part, fatigue strength increases); 4) operational factors (temperature, corrosion, frequency of loading, radiation exposure, etc.); 5) the presence of a surface layer hardened by various technological methods.
stress fatigue strength curve
endurance limit denoted by (or ), where the index R corresponds to the cycle asymmetry coefficient. So, for example, for a symmetrical cycle it is denoted , for a zero cycle (at ), for a constant cycle .
Endurance limit for a symmetrical cycle is the smallest compared to other types of cycles, that is, .
For example, 
; 
.
endurance limit
To calculate parts that are not intended for long-term operation, it becomes necessary to determine the highest stress value that the material can withstand for a given number of cycles (N), the value of which is less than the base value (). In this case, according to the fatigue curve and a given number of cycles (N), the corresponding stress (), called limit of limited endurance.
Endurance limit factors for a symmetrical cycle
When assessing the strength of a part operating under static loading, the mechanical characteristics of the material of the part are completely identified with the mechanical characteristics of the sample material obtained as a result of the experiment. This does not take into account the difference in either the shape or size of the part and sample, or some other differences.
When designing a part for fatigue, these factors must be taken into account. The most significant factors that affect the endurance limit in a symmetrical cycle are the stress concentration, the absolute dimensions of the cross section of the part, and the roughness of its surface. This is easily explained by the fact that all of the above factors contribute to the emergence and propagation of microcracks.
Influence of stress concentration
Near undercuts, at the edges of holes, at places where the shape of the rod changes, at cuts, etc. there is a sharp increase in stresses compared to the nominal stresses calculated using the usual formulas for the resistance of materials. Such a phenomenon is called stress concentration, and the reason that causes a significant increase in stresses is stress concentrator.
The zone of distribution of increased stresses is purely local in nature, therefore these stresses are often called local.
At stresses that are variable in time, the presence of a stress concentrator on the sample leads to a decrease in the endurance limit. This is explained by the fact that a multiple change in stresses in the stress concentration zone leads to the formation and further development of a crack, followed by fatigue failure of the specimen.
In order to evaluate the effect of stress concentration on reducing the fatigue resistance of a sample, taking into account the sensitivity of the material to stress concentration, the concept of effective concentration coefficient is introduced, which is the ratio of the endurance limit of a standard sample without stress concentration to the fatigue limit of a sample with stress concentration: 
(or 
).
Influence of the absolute dimensions of the cross section
With an increase in the size of the cross sections of the samples, reduction in endurance limit. This influence is taken into account by the coefficient of influence of the absolute dimensions of the cross section (previously this coefficient was called the scale factor). The mentioned coefficient is equal to the ratio of the endurance limit of smooth samples with a diameter d to the endurance limit of a smooth standard sample with a diameter equal to 7.5 mm: 
(or 
).
Surface roughness
Machining the surface of the part has a significant impact on the endurance limit. This is due to the fact that a rougher surface treatment of the part creates additional places for stress concentrators and, therefore, leads to additional conditions for the appearance of microcracks.
The main parameter characterizing the fatigue strength of materials, i.e. strength under repeated alternating loads, is endurance limit of R is the maximum absolute value of the cycle stress at which the fatigue failure of the material does not yet occur to the base number N at loading cycles. For the basic, i.e. the largest number of cycles specified during testing is taken for ferrous metals 10 7 loading cycles, and for non-ferrous - 10 8 . The index in the designation of the endurance limit corresponds to the coefficient of asymmetry of the stress cycle during testing. So, for a symmetrical cycle, the endurance limit is denoted by y-1, and for a zero cycle - y 0. The endurance limit of a material is determined by testing samples for fatigue on testing machines. The most common is the testing of specimens under a symmetrical stress cycle. The layout of the installation for testing specimens for bending is shown in fig. 5. Sample 1 together with clamp 2 rotates at a constant angular velocity. At the end of the sample, there is a bearing 3 loaded with a force F permanent direction. The sample is subjected to bending deformation with a symmetrical cycle. Maximum stresses occur on the sample surface in the most dangerous section I - I and are defined as y = M and /W, where M and = F?? - bending moment in the section; W \u003d 0.1d 3 - moment of resistance relative to the neutral axis of the cross section of the sample, a circle with a diameter d. In the presented position at the point BUT tensile stresses act, as the sample is bent with a convexity upwards. After rotating the sample by 180° at the point BUT compressive stresses of the same magnitude will act, i.e. -y. When passing through the neutral axis, the voltage at the point BUT will be zero.
By testing to fatigue failure of identical samples at different values of cycle stresses, a graph is constructed that characterizes the relationship between maximum stresses y and the number of cycles to failure (cycle life N). This dependence (Fig. 6) is called fatigue curve or weller curve, in honor of the German scientist who first built it. To build a fatigue curve in coordinates at max - N at least 10 identical samples are required, for which strict requirements dimensional accuracy, surface roughness. The first of the samples is loaded with force F so that the maximum stress of the cycle at 1 was somewhat less than the ultimate strength of the material (at 1< у u) и испытывают до разрушения, отмечая (рис. 6) точку BUT with coordinates y 1 and the number of cycles to destruction N 1 .
The second sample is tested by creating a voltage in it at 2 less than in the first (at 2< у 1) образце. Число циклов до разрушения этого образца будет N 2 (N 2 > N 1). Mark a point on the graph AT with coordinates at 2 , N 2 . By gradually reducing the maximum cycle stress in the tested samples, the tests are carried out until the destruction of the samples, until one of them collapses to the base number N at loading cycles. By connecting the dots in series with a smooth line BUT, AT, FROM, …, constructed during testing of samples, we obtain a fatigue curve. Voltage corresponding to the base number N at cycles, and there is an endurance limit at- 1 bending material. On other testing machines, similarly to the bending test, the endurance limits of the material are determined during torsion (f - 1), during tension - compression (y - 1r). For many materials, the ratios between the limits of endurance in bending, torsion and tension-compression have been experimentally established. For example, for steels f-1 = 0.55y-1; y-1p = 0.7y-1. The endurance limit under a symmetrical loading cycle for all metals, except for very ductile ones (copper, commercial iron), is less than the elastic limit, with increasing loading frequency it slightly increases.
Dozens of equations are proposed in the literature that describe the fatigue curves of various materials and specimens. In engineering calculations, the power equation of the fatigue curve is most often used
y m N = const, (10)
where N- the number of cycles before failure at the maximum stress of the cycle; m- exponent depending on the material, sample parameters, for metals m = 5 ... 10.
Often, the service life of products, especially special one-time use, is limited, the number of loading cycles N during operation is less than the base (N< N у). Уравнение (10)позволяет при расчетах таких изделий на усталостную прочность определять предельно максимальные напряжения в циклах или ограниченный предел выносливости at- 1N corresponding to the given number of cycles N loading
N \u003d N y (y- 1 / y- 1N) m , (12)
where the quantities at- 1 , N at , m taken from reference data on materials. The use of equations (11) and (12) is possible only if the physics and mechanism of fatigue damage remain unchanged while maintaining mechanism high cycle fatigue. High-cycle fatigue is guaranteed to take place if the number of cycles before failure is at least 10 4 , i.e. N? 10 4 .
Determination of the fatigue strength characteristics of materials by fatigue testing is a laborious and expensive process due to the length and significant scatter of test results. Empirical dependences of an approximate estimate of the values of the endurance limit on the magnitude of the mechanical properties of the material under static loading are sought. So, the value of the endurance limit in bending with a symmetrical loading cycle for carbon steel is y-1 = (0.4 ... 0.45) y ut ; for non-ferrous metals y- 1 = = (0.24 ... 0.5) y ut , where at ut is the tensile strength of the material.
As experiments show, the value of the endurance limit of the material largely depends on the ratio between the extreme values R max and p min changing voltage. If these values are equal in magnitude R a and are opposite in sign (Fig. 14.1), then we have symmetrical cycle, at which the endurance limit is the lowest.
Rice. 14.1
If we add to the symmetrically oscillating within + R a and - R a voltage is also a constant voltage R m (Fig. 14.2), then we get the case asymmetrical cycle; in this case, the endurance limit is higher than for the symmetrical cycle.
Voltage extremes for unbalanced cycle R max and p min will be (Fig. 14.2):
R max = p m + p a and p min = p m - p a ;
in its turn
Voltage R t is called the average stress of the cycle, and R a - the amplitude of cycle voltage fluctuations. The relation is called cycle characteristics. With a symmetrical cycle R t = 0, p min = -p max and r=-1; at constant static voltage R a = 0, p min =p max and r= +1; if p min =0, then and r = 0. Here are some examples of asymmetric cycles:
Double the magnitude of the amplitude of voltage fluctuations R a
is called the "span" of the cycle.
The value of the endurance limit for any cycle of alternating stresses will be denoted by R, or with an icon at the bottom indicating the corresponding cycle characteristic. So, p -1 - endurance limit for a symmetrical cycle with characteristic r=-1, p 0,2 - endurance limit for an asymmetric cycle with a characteristic r= +0.2, etc.
Of greatest interest is the determination of the value of the endurance limit for symmetrical ( R m= 0) cycle as the smallest. This value turns out to be different for the case of bending deformation, axial deformation (tension and compression) and torsion.
To determine the endurance limit in bending, machines are used in which a sample of a circular cross section is loaded through ball bearings, either as a cantilever - by force at the end, or as a hinged beam - by symmetrically located equal forces; the sample rotates at a speed of about 2000-3000 rpm. With each revolution, the sample material in the most stressed places experiences a symmetrical cycle of stress changes from the highest compression to the same highest tension, and vice versa. The number of cycles tested by the sample is determined by the number of its revolutions N, marked with a special counter.
The samples are given a shape with very smooth outlines, excluding the possibility of local stresses. Experience to determine the endurance limit is as follows. A batch of samples of the tested material is prepared in the amount of 6-10 pieces; samples are numbered sequentially: 1, 2, 3…
The first sample is placed in the machine and loaded in such a way as to obtain a certain value of the highest normal stress "; this value is usually taken equal to 0.5-0.6 of the material's tensile strength; then the machine starts up, and the sample rotates, testing variable voltages from +" to -" until a break occurs. At this moment, a special device turns off the motor, the machine stops, and the rev counter shows the number of cycles N 1 required to break the sample under stress ".
The second sample is tested in the same order at a voltage ", smaller", the third - at voltage ""<", и т.д. Соответственно возрастает число циклов, необходимое для излома. Уменьшая для каждого нового образца рабочее напряжение, мы, наконец, для какого-то из них не получаем излома, даже при очень большом числе оборотов образца. Соответствующее напряжение будет очень близко к пределу выносливости.
Experiments have shown that if a steel sample has not collapsed after 1010 6 cycles, then it can withstand an almost unlimited number of cycles (10010 6 - 20010 6). Therefore, when determining the endurance limit for a particular steel grade, the experiment is stopped if the sample has experienced
1010 6 cycles and didn't break. In some cases, during testing, they are limited to a smaller limit number of cycles, however, not less than 510 6 .
There is no such dependence for non-ferrous metals, and in order to find out whether the sample can actually withstand a very large number of sign changes at a given voltage, one has to give up to 20010 6 and even 50010 6 cycles. In this case, we can talk about the conditional endurance limit corresponding to the absence of a break at a certain number of stress sign changes - at 1010 6 , 3010 6 , etc.
To find the numerical value of the endurance limit, the obtained results are processed graphically. Figure 14.3 and Figure 14.4 show two methods for such processing. On the first of them, along the ordinate axis, the quantities ", ",. .., and along the abscissa N 1 , N 2 etc. The ordinate of the horizontal tangent to the resulting curve (asymptotes) will be equal to the endurance limit. In the second drawing, the x-axis plots equal values. In this case, the endurance limit is defined as a segment cut off on the ordinate axis by the continuation of the resulting curve, since the origin of coordinates corresponds to N=. At present, the second method is more common.
Similarly, the endurance limit for axial forces (tension and compression) and for torsion is determined; special testing machines (pulsators, etc.) are also used for this purpose.
At present, a huge number of experimental results have been obtained to determine the endurance limit of various materials. Most of the research carried out relates to steel, as the most commonly used material in mechanical engineering. The results of these studies showed that the endurance limit steel of all grades is connected by a more or less definite relationship only with the magnitude of the ultimate tensile strength c. For rolled and forged material, the endurance limit for a symmetrical cycle in the case of bending is from 0.40 to 0.60 V; for casting, this ratio is in the range from 0.40 to 0.46.
In this way, safety margin with sufficient accuracy for the purposes of practice can be accepted for all grades of steel
If a steel sample is subjected axial forces in a symmetrical cycle (alternate tension and compression), then the corresponding endurance limit, as experiments show, will be lower than in bending; the ratio between these endurance limits can be taken equal, as experiments show, to 0.7, i.e. .
This decrease is explained by the fact that during tension and compression, the entire section is subjected to the same stresses; in bending, the greatest stresses occur only in the outermost fibers; the rest of the material works weaker and thus somewhat more difficult to form fatigue cracks; in addition, in practice there is always some eccentricity of the axial load.
Finally, in torsion for a symmetrical cycle, the endurance limit for shear stresses is on average 0.55 of the bending endurance limit. Thus, for steel with a symmetrical cycle
These data can be used as the basis for calculation formulas for strength testing.
For non-ferrous metals, we have a less stable relationship between endurance limit and tensile strength; experiences give
= (0.24 0.50) c.
When using the above relations (14.1), it must be borne in mind that the endurance limit for a given material is a characteristic that depends on a very large number of factors; data (14.1) refer to experiments with samples of relatively small diameter(7-10 mm) with a polished surface and the absence of sharp changes in the shape of the cross section.
The ability of a material to perceive the repeated action of alternating stresses is called endurance, and the verification of the strength of structural elements under the action of such stresses is called endurance calculation (or fatigue strength calculation).
To obtain the mechanical characteristics of the material required for strength calculations at alternating stresses, special endurance (fatigue) tests are carried out. For these tests, a series of completely identical samples is made (at least 10 pieces).
The most common tests are for pure bending under a symmetrical stress cycle; they are carried out in the following order.
In the first sample, using a special machine, stress cycles are created, characterized by stress values taken large enough (slightly less than the ultimate strength of the material) so that the destruction of the sample occurs after a relatively small number of cycles. on the accepted scale) the number of cycles that caused the destruction of the sample, and the ordinate - the stress value (Fig. 5.15).
Then another sample is tested to failure under stress, the test result of this sample is depicted on the graph by a point. Testing the rest of the samples from the same series, points IV, V, etc. are similarly obtained. By connecting the points of a smooth curve obtained from the experiments, the so-called fatigue curve is obtained, or the Wöhler curve (Fig. 5.15), corresponding to symmetrical cycles
Similarly, fatigue curves can be obtained corresponding to cycles with other values of the asymmetry coefficient
The destruction of the material under a single loading occurs at the moment when the stresses arising in it are equal to the ultimate strength. Therefore, the fatigue curves at have ordinates atax equal to
The endurance curve (Fig. 5.15) shows that with an increase in the number of cycles, the maximum stress at which the material is destroyed decreases. The fatigue curve for low or medium carbon, as well as for some grades of alloy steel, has a horizontal asymptote. Therefore, at a given value of the asymmetry coefficient R and a maximum stress less than a certain value, the material does not fail, no matter how large the number of cycles is.
The highest (limiting) maximum cycle stress at which there is no fatigue failure of a sample of a given material after an arbitrarily large number of cycles is called the endurance limit. Thus, the endurance limit is equal to the ordinate of the asymptote of the fatigue curve. It is designated hell; with a symmetrical cycle, the asymmetry coefficient and the endurance limit during this cycle are denoted (see Fig. 5.15).
It is quite obvious that when testing a sample it is impossible to repeat the same cycle of stresses indefinitely many times, but this is not necessary. The ordinates atax of the fatigue curve for some materials (low and medium carbon steel, etc.) after a certain number of cycles (equal to several millions) hardly change; therefore, the same maximum stresses correspond to the number of cycles, even several times greater, on the fatigue curve. In this regard, the number of cycles (when testing a material for endurance) is limited by a certain limit, which is called the base number of cycles. If the sample withstands the base number of cycles, then it is considered that the stress in it is not higher than the endurance limit. For steel and cast iron, the base number of cycles is assumed to be 107.
The endurance limit for steel in a symmetrical cycle is several times less than the tensile strength (in particular, for carbon steel 00.430).
Fatigue curves for non-ferrous metals and alloys and some alloy steels do not have a horizontal asymptote, and, therefore, such materials can fail with a sufficiently large number of cycles, even at relatively low stresses.
Therefore, the concept of endurance limit for these materials is conditional. More precisely, for these materials, one can only use the concept of limited endurance limit, which is the maximum value of the maximum (in absolute value) cycle stress at which the sample is not yet destroyed at a certain (basic) number of cycles. The base number of cycles in the cases under consideration is taken very large - up to .
In cases where the service life of a structural element in which alternating stresses occur is limited, the maximum stresses may exceed the endurance limit; they, however, should not exceed the limit of limited endurance corresponding to the number of cycles during the operation of the calculated element.
It should be noted that the endurance limit for central tension-compression of the sample is approximately 0.7-0.9 of the endurance limit for a symmetrical bending cycle. This is explained by the fact that during bending, the internal points of the cross section are less stressed than the external ones, and during central tension-compression, the stress state is uniform. Therefore, during bending, the development of fatigue cracks occurs less intensively.
The fatigue limit for a symmetrical torsion cycle for steel is on average 0.58 (58% of the fatigue limit for a symmetrical bending cycle).
