To determine the endurance limit under the action of stresses with asymmetric cycles, diagrams of various types are constructed. The most common of these are:
ultimate stress diagram, in coordinates dmax - dm (Smith diagram);
diagram of limiting amplitudes, in coordinates yes - gt (Hay diagram).
Consider these diagrams of ultimate stresses. In the Smith chart, the limit stress of the cycle, corresponding to the endurance limit, is plotted vertically, the average stress is plotted along the horizontal axis (Fig. 12.6).
First, point C is plotted on the dmax axis, the ordinate of which is the endurance limit for a symmetrical cycle d-1 (with a symmetrical cycle, the average stress is zero). Then, experimentally, the endurance limit is determined for some asymmetric load, for example, for zero, in which the maximum stress is always twice the average. Let's plot the point P on the diagram, the ordinate of which is the endurance limit for the zero cycle q0. For many materials, the values of d-1 and d0 are defined and given in reference books.
Similarly, the endurance limit for asymmetric cycles with other parameters is determined experimentally.
The results are plotted on a diagram as points A, B, etc., the ordinates of which are the endurance limits for the respective stress cycles. Point D, which lies simultaneously on the bisector OD, characterizes the ultimate stress (ultimate strength) for a constant load, for which dmax = dt.
Since for plastic materials the yield strength o * is also a dangerous stress, a horizontal line KL is plotted on the diagram, the ordinate of which is equal to dt. (For plastic materials, the tension diagram of which does not have a yield plateau, the role of dt is played by the conditional yield strength d0.2.) Therefore, the ultimate stress diagram will finally have a VPD CAPKL.
Usually this diagram is simplified by replacing it with two straight lines CM and ML, and the straight line CM is drawn through point C (corresponding to a symmetric cycle) and point P (corresponding to a zero cycle).
The indicated method of schematization of the limit stress diagram was proposed by S. V. Serensen and R. S. Kinasoshvili.
In this case, within the direct SM, the limit stress of the cycle (limit "endurance") will be expressed by the equation
The coefficient characterizes the sensitivity of the material to the asymmetry of the cycle.
When calculating endurance, they also often use the diagram of limiting amplitudes, which is plotted in coordinates -- (Hay's diagram). To do this, the amplitude voltage is plotted along the vertical axis, and the average voltage along the horizontal axis (Fig. 12.7).
Point A of the diagram corresponds to the endurance limit for a symmetrical cycle, since with such a cycle dm = 0.
Point B corresponds to the tensile strength at constant stress, since in this case yes \u003d 0.
Point C corresponds to the endurance limit during a pulsating cycle, since with such a cycle yes \u003d dt.
Other points of the diagram correspond to endurance limits for cycles with different ratios of yes and dm.
The sum of the coordinates of any point of the limit curve ASV gives the value of the endurance limit at a given average cycle stress
For ductile materials, the ultimate stress should not exceed the yield strength
Therefore, on the diagram of limiting stresses, we plot the straight line DE, constructed according to the equation
The final stress limit diagram is AKD.
In practice, they usually use an approximate diagram yes - dt, built on three points A, C and D and consisting of two straight sections AL and LD (Sorensen-Kinaso-shvili method). Point L is obtained as a result of the intersection of two lines: line DE and line AC. Calculations according to the Smith and Hay diagrams with the same methods of approximation lead to the same results.
To construct a diagram of limiting amplitudes, it is necessary to have endurance limits at various values parameter “ ” (asymmetry coefficient). The introduction significantly complicates the experiment, since now it is already necessary to have several dozen samples, each ten of which is tested at . By setting a constant value , we find, by successive tests of samples, such a maximum amplitude value at which the material is still able to withstand an unlimited number of cycles. As a result of testing one dozen samples, we get one point on the diagram of limiting amplitudes. Having tested the next group of samples, we get one more point, and so on. (fig.11.7).
The meaning of the diagram of limiting amplitudes is obvious. Let the cycle be characterized by stresses and , which we will consider as the coordinates of the operating point. By plotting the operating point on the diagram, we can judge the strength of the sample. If the operating point is located below the limit curve, then the sample will withstand an infinite number of cycles (not less than the base). If R.T. is above the curve, then the sample will fail at a certain number of cycles less than the base one.
The construction of a diagram of limiting amplitudes is very laborious, therefore it is often schematized by straight line segments. The dot reflects the corresponding test of samples under a symmetrical cycle. The point corresponds to the static test of the samples. For brittle materials, it is determined by the tensile strength. For ductile materials, the limitation can be either yield strength or tensile strength.
To build the left side of the diagram, you need at least one more point, for example, for a pulsating cycle, or you need to know the angle of inclination of the straight line. Let us introduce the concept of slope = . Experiments have shown that the value of the angular coefficient for carbon steels lies within 0.1 ÷ 0.2 and for alloyed 0.2 ÷ 0.3.
Thus, the equation of the left straight line has the form 
. The right side of the diagram is approximated by a straight line passing through the point and making an angle of 45 with the axes and
Therefore, when schematizing, the diagram of limiting amplitudes is replaced by two straight lines and .
The constructed diagram does not yet allow calculating the strength of parts, because fatigue strength depends on many other factors.
Factors Affecting Fatigue Strength
Stress concentration
Concentration is the phenomenon of an abrupt increase in stresses near sharp changes in the shape of a part, holes, grooves (Fig. 11.8)
The measure of concentration is the theoretical stress concentration factor equal to:
Tensile, bending, torsion,
The so-called nominal voltage, determined by the formulas of the resistance of materials, is the highest local stress. Data on the theoretical stress concentration factor are given in manuals on mechanical engineering. The stress concentration has a different effect on the strength of the part, depending on the properties of the material and the loading conditions. Therefore, instead of the theoretical stress concentration factor, the effective stress concentration factor and is introduced.
For a symmetrical cycle, the effective stress concentration factor is determined by the ratio
where are the endurance limits of a smooth sample,
Endurance limits calculated from nominal stresses for samples having stress concentration, but the same cross-sectional dimensions as for a smooth sample. determined from tables.
In those cases where there are no experimental data, by direct definition, they resort to approximate estimates. For example, according to the formula
Coefficient of material sensitivity to stress concentration. It depends mainly on the material. For structural steels.
scale effect
If several samples of different diameters are made from the same material, then after a fatigue test it can be found that the endurance limit decreases with increasing diameter. The decrease in the endurance limit with an increase in the size of the part is called the scale effect.
The measure of this reduction is the scale factor
Endurance limit of a specimen with a diameter similar to that of the workpiece
Sample Endurance Limit d= 7.5mm.
On fig. 11.9 gives an approximate dependence of the scale factor on the shaft diameter for the case of bending and torsion.
Curve 1 was obtained for carbon steel, curve 2 for alloy steel.
During the operation of machines and engineering structures, stresses arise in their elements that change over time in a variety of cycles. To calculate elements for strength, it is necessary to have data on the values of endurance limits during cycles with different asymmetry coefficients. Therefore, along with tests with symmetrical cycles, tests are also carried out with asymmetric cycles.
It should be borne in mind that endurance tests with asymmetric cycles are carried out on special machines, the designs of which are much more complicated than the designs of machines for testing specimens with a symmetrical bending cycle.
The results of tests for endurance in cycles with different coefficients of asymmetry are usually presented in the form of diagrams (graphs) depicting the relationship between any two parameters of the limit cycles.
These diagrams can be constructed, for example, in coordinates from, they are called limit amplitude diagrams, they show the relationship between average stresses and amplitudes of limit cycles-cycles for which the maximum stresses are equal to the endurance limits: Here and below the maximum, minimum, average and amplitude limit stresses cycle will be denoted
A diagram of the dependence between the parameters of the limit cycle can also be constructed in coordinates. Such a diagram is called the limit stress diagram.
When calculating steel structures in industrial and civil engineering, diagrams are used that give the relationship between the cycle asymmetry coefficient R and the endurance limit otax
Let us consider in detail the diagram of limiting amplitudes (it is sometimes called the diagram), which is further used to obtain the dependencies used in strength calculations at variable voltages.
To obtain one point of the diagram under consideration, it is necessary to test a series of identical samples (at least 10 pieces) and construct a Wöhler curve, which will determine the value of the endurance limit for a cycle with a given asymmetry coefficient (this also applies to all other types of diagrams for limit cycles).
Assume that tests have been carried out with a symmetrical bending cycle; as a result, the value of the endurance limit was obtained. The coordinates of the point depicting this limit cycle are: [see. formulas (1.15) - (3.15)], i.e., the point is on the y-axis (point A in Fig. 6.15). For an arbitrary asymmetric cycle, according to the endurance limit determined from experiments, it is not difficult to find from. By formula (3.15),
but [see formula (5.15)], therefore,
In particular, for a zero cycle with an endurance limit equal to
This cycle corresponds to point C in the diagram shown in fig. 6.15.
Having determined the experimental value for five or six different cycles, by formulas (7.15) and (8.15) one obtains the coordinates of and individual points belonging to the limit curve. In addition, as a result of testing at a constant load, the tensile strength of the material is determined, which, for the sake of generality of reasoning, can be considered as the endurance limit for the cycle with . Point B corresponds to this cycle in the diagram. By connecting the points whose coordinates are found from experimental data with a smooth curve, a diagram of limiting amplitudes is obtained (Fig. 6.15).
The arguments about the construction of the diagram, carried out for cycles of normal stresses, are applicable to cycles of shear stresses (during torsion), but the designations are changed instead of from, etc.).
The diagram shown in fig. 6.15 is built for cycles with positive (tensile) average stresses from 0. Of course, it is fundamentally possible to construct a similar diagram in the region of negative (compressive) average stresses, but practically at present there are very few experimental data on fatigue strength at For low- and medium-carbon steels, it can be approximately assumed that in the region of negative average stresses, the limit curve is parallel to the abscissa axis.
Consider now the question of using the constructed diagram. Let the point N with coordinates correspond to the working cycle of stresses (i.e., when working at the considered point of the part, stresses arise, the cycle of change of which is specified by any two parameters, which makes it possible to find all the parameters of the cycle and, in particular, ).
Let's draw a ray from the origin through the point N. The tangent of the angle of inclination of this ray to the abscissa axis is equal to the characteristic of the cycle:
It is obvious that any other point lying in the same ray corresponds to a cycle similar to the given one (a cycle having the same values ). So, any ray drawn through the origin is the locus of points corresponding to such cycles. All cycles depicted by the points of the beam that lie not above the limit curve (i.e., the points of the segment (Ж) are safe with respect to fatigue failure. In this case, the cycle depicted by the point of the KU is its maximum stress for a given asymmetry coefficient, defined as the sum of the abscissa and the ordinates of the point K (otax), is equal to the endurance limit:
Similarly, for a given cycle, the maximum stress is equal to the sum of the abscissa and the ordinate of the point
Assuming that the working cycle of stresses in the calculated part and the limiting cycle are similar, we determine the safety factor as the ratio of the endurance limit to the maximum stress of a given cycle:
As follows from the foregoing, the safety factor in the presence of a diagram of limiting amplitudes constructed from experimental data can be determined by a graphical-analytical method. However, this method is suitable only on condition that the calculated part and the samples, as a result of which the diagram was tested, are identical in shape, size and quality of processing (this is described in detail in § 4.15, 5.15).
For parts made of plastic materials, not only fatigue failure is dangerous, but also the occurrence of noticeable residual deformations, i.e., the onset of yield. Therefore, from the area bounded by the line AB (Fig. 7.15), all points of which correspond to cycles that are safe with respect to fatigue failure, it is necessary to select a zone corresponding to cycles with maximum stresses that are less than the yield strength. To do this, from the point L, the abscissa of which is equal to the yield strength, a straight line is drawn, inclined to the abscissa axis at an angle of 45 °. This direct reading on the y-axis is the segment OM, equal (in the scale of the diagram) to the yield strength. Therefore, the equation of the straight line LM (the equation in segments) will look like
i.e. for any cycle represented by the points of the LM line, the maximum stress is equal to the yield strength. Points lying above the LM line correspond to cycles with maximum stresses greater than the yield strength. Thus, cycles that are safe both in terms of fatigue failure and in terms of yielding are represented by points
To determine the endurance limit under the action of stresses with asymmetric cycles, diagrams of various types are constructed. The most common of these are:
1) diagram of the limit stresses of the cycle in the coordinates max - m
2) a diagram of the limiting amplitudes of the cycle in the coordinates a - m .
Consider a diagram of the second type.
To plot the diagram of the limiting amplitudes of the cycle, the amplitude of the stress cycle a is plotted along the vertical axis, and the average stress of the cycle m is plotted along the horizontal axis (Fig. 8.3).
Dot BUT diagram corresponds to the endurance limit for a symmetrical cycle, since with such a cycle m = 0.
Dot AT corresponds to the tensile strength at constant stress, since in this case a \u003d 0.
Point C corresponds to the endurance limit during a pulsating cycle, since with such a cycle a = m .
Other points of the diagram correspond to endurance limits for cycles with different ratios a and m .
The sum of the coordinates of any point of the limit curve DIA gives the endurance limit at a given average cycle stress
.
For ductile materials, the ultimate stress should not exceed the yield strength i.e. Therefore, we plot the straight line DE on the limit stress diagram , constructed according to the equation
The final stress limit diagram is AKD .
The workloads must be inside the diagram. The endurance limit is less than the tensile strength, for example, for steel σ -1 \u003d 0.43 σ in.
In practice, an approximate diagram a - m is usually used, built on three points A, L and D, consisting of two straight sections AL and LD. Point L is obtained as a result of the intersection of two lines DE and AC . The approximate diagram increases the margin of fatigue strength and cuts off the region with a scatter of experimental points.
Factors affecting the endurance limit
Experiments show that the following factors significantly affect the endurance limit: stress concentration, cross-sectional dimensions of parts, surface condition, nature of technological processing, etc.
Influence of stress concentration.
To 
concentration (local increase) of stresses occurs due to cuts, sharp changes in size, holes, etc. In fig. 8.4 shows stress diagrams without a concentrator and with a concentrator. The influence of the concentrator on strength takes into account the theoretical stress concentration factor.
where 
- voltage without concentrator.
The values of K t are given in reference books.
Stress concentrators significantly reduce the fatigue limit compared to the fatigue limit for smooth cylindrical specimens. At the same time, concentrators affect the fatigue limit differently depending on the material and the loading cycle. Therefore, the concept of the effective concentration coefficient is introduced. The effective stress concentration factor is determined experimentally. To do this, take two series of identical samples (10 samples each), but the first without a stress concentrator, and the second with a concentrator, and determine the endurance limits for a symmetrical cycle for samples without a stress concentrator σ -1 and for samples with a stress concentrator σ -1 ".
Attitude
determines the effective stress concentration factor.
Values K - are given in reference books
Sometimes the following expression is used to determine the effective stress concentration factor
where g is the coefficient of material sensitivity to stress concentration: for structural steels - g = 0.6 0.8; for cast iron - g = 0.
Influence of the state of the surface.
Experiments show that rough surface treatment of a part reduces endurance limit . The influence of surface quality is associated with a change in microgeometry (roughness) and the state of the metal in the surface layer, which, in turn, depends on the method of machining.
To assess the effect of surface quality on the endurance limit, the coefficient p is introduced, called the surface quality factor and equal to the ratio of the endurance limit of a sample with a given surface roughness σ -1 n to the endurance limit of a sample with a standard surface σ -1
H 
and fig. 8.5 shows a graph of values p
depending on the tensile strength σ in
steel and surface treatment. In this case, the curves correspond to the following types of surface treatment: 1 - polishing, 2
-
grinding, 3
-
fine turning, 4 - rough turning, 5 - the presence of scale.
Various methods of surface hardening (hardening, carburizing, nitriding, surface hardening with high frequency currents, etc.) greatly increase the fatigue limit values. This is taken into account by introducing the coefficient of influence of surface hardening . By surface hardening of parts, it is possible to increase the fatigue resistance of machine parts by 2-3 times.
Influence of part dimensions (scale factor).
Experiments show that the larger the absolute dimensions the cross section of the part, the lower the endurance limit , because with the increase size increases the likelihood of defects in the hazardous area . The ratio of the endurance limit of the part with a diameter d σ -1 d to the endurance limit of a laboratory sample with a diameter d 0 = 7 - 10 σ -1 mm is called the scale factor
experimental data to determine m still not enough.
It has been experimentally established that the endurance limit with an asymmetric cycle is greater than with a symmetric one, and depends on the degree of cycle asymmetry:
With a graphical representation of the dependence of the endurance limit on the asymmetry coefficient, it is necessary for each R determine your endurance limit. It is difficult to do this, since in the range from a symmetric cycle to a simple stretching, an infinite number of the most diverse cycles fit. An experimental determination for each type of cycle is almost impossible due to the large number of samples and the long time of their testing.
Due to specified reasons for a limited number of experiments for three to four values R build a diagram of limit cycles.
| Rice. 445 |
A limit cycle is one in which the maximum stress is equal to the endurance limit, i.e. . On the ordinate axis of the diagram, we plot the value of the amplitude, and on the abscissa axis, the average stress of the limit cycle. Each pair of voltages and , defining the limit cycle, is represented by a certain point on the diagram (Fig. 445). As experience has shown, these points are generally located on the curve AB, which, on the ordinate axis, cuts off a segment equal to the endurance limit of a symmetrical cycle (with this cycle = 0), and on the abscissa axis, a segment equal to the ultimate strength. In this case, constant voltages apply:
Thus, the diagram of limit cycles characterizes the relationship between the values of average stresses and the values of the limit cycle amplitudes.
Any point M, located inside this diagram corresponds to a certain cycle defined by the quantities (CM) and (ME).
To determine , a cycle from a point M spend segments MN and MD to the intersection with the x-axis at an angle of 45° to it. Then (Fig. 445):
Cycles whose skewness coefficients are the same (similar cycles) will be characterized by points located on a straight line 01, the angle of inclination of which is determined by the formula
| Rice. 446 |
Dot 1 corresponds limit cycle of all the mentioned cycles. Using the diagram, you can determine the limiting stresses for any cycle, for example, for a pulsating (zero) one, for which, a (Fig. 446). To do this, from the origin (Fig. 445) draw a straight line at an angle α 1 = 45°() until it intersects with the curve at a point 2. Coordinates of this point: ordinate H2 is equal to the limiting amplitude voltage, and the abscissa K2– limit average stress of this cycle. The limiting maximum voltage of the pulsating cycle is equal to the sum of the coordinates of the point 2:
In this way, the issue of limit stresses any cycle.
If the part of the machine that experiences alternating stresses is made of plastic material, then not only fatigue failure will be dangerous, but also the occurrence of plastic deformations. The maximum cycle stresses in this case are determined by the equality
where - betrayed fluidity.
Points that satisfy this condition are located on a straight line. DC, inclined at an angle of 45 ° to the x-axis (Fig. 447, a), since the sum of the coordinates of any point on this line is equal to .
If straight 01 (Fig. 447, a), corresponding this species cycle, with increasing loads on the machine part, crosses the curve AU, then fatigue failure of the part will occur. If a straight line 01 crosses the line CD, then the part will fail as a result of the appearance of plastic deformations.
Often in practice, schematized diagrams of limiting amplitudes are used. curve ACD(Fig. 447, a) for plastic materials approximately replace the straight line AD. This straight line cuts off segments and on the coordinate axes. The equation looks like
| Rice. 447 |
For Brittle Materials Chart restrict straight A B with the equation
The most widely used diagrams of limiting amplitudes, built on the basis of the results of three series of tests of samples: with a symmetrical cycle ( point A) with a zero cycle (point C) and a static break (point D)(Fig. 447, b). Connecting the dots BUT and FROM straight and swiping out D straight line at an angle of 45°, we obtain an approximate diagram of the limiting amplitudes. Knowing the coordinates of the point BUT and FROM, you can write the equation of a straight line AB. Take an arbitrary point on a straight line To with coordinates and . From the similarity of triangles ASA 1 and KSK 1 we get
from where we find the equation of a straight line AB in form
End of work -
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