3.1. Statistical Assessment of the Low-Cycle Fatigue Curves Parameters
Heat-resistant steel 15Cr2MoVA has been anisotropic during cyclic loading, that is, it accumulates plastic strain in the direction of tension. Structural steel C45 has been a cyclically stable anisotropic material. The D16T1 aluminum alloy has been cyclically hardened during cyclic loading with stress-limited loading and has not unilaterally accumulated plastic deformation.
As follows of the equations from the previous paragraph, the values of parameters
are necessary in the analytical description of the cyclic deformation curves. According to the methodology provided in [34
], the histograms for the cyclic deformation parameters mentioned above were defined (Figure 4
, Figure A1
, Figure A2
and Figure A3
). The histograms of parameter
for 15Cr2MoVA steel indicate a significant scatter of the parameter, and with an increase in the volume of the statistical series (the level of loading
—20 samples and
—40 samples), their form approaches the normal distribution. For all loading levels, the histograms of parameter
of steel 15Cr2MoVA are shifted to the left, that is, they have a positive asymmetry.
The almost same nature of the histograms of parameter is observed for steel C45. For loading levels , there is positive asymmetry, and for loading level , it is negative.
From the analysis of the parameter distribution histograms for steel C45, it follows that they differ little from the parameter histograms. For this parameter, the sample size also influences the shape of the histogram, i.e., with increasing samples, the fit to the normal distribution improves. For loading levels , slight negative asymmetry is observed, and for , the asymmetry is positive.
The parameter histograms for steel C45, compared to the histograms for steel 15Cr2MoVA, do not correspond exactly to the normal distribution. Furthermore, a negative asymmetry is observed at loading levels , and at loading level , the asymmetry is positive.
The parameter histograms are similar to those of parameter . For steel 15Cr2MoVA, with , the asymmetry is slightly negative, for —negative, while with —positive. For steel C45, the parameter histograms are identical to those of parameter . At loading levels , negative asymmetry is observed, and at level , the asymmetry is positive.
It should be noted that the histograms for the parameter distribution depend on the loading level and sample size, that is, with an increase in the samples and loading level, the shape of the histograms approaches the normal distribution. With an increase in the loading level for steel 15Cr2MoVA, the quasi-static fracture zone is approached, as more steady change in cyclic deformation curves is established compared to the transient fracture zone, leading to a smaller scatter of parameters reflecting this process. For 15Cr2MoVA steel, the parameter histograms have positive asymmetry at all loading levels.
is a function of the parameter
, consequently the parameter
histograms do not vary much from the parameter
histograms (Figure A3
From the consideration of the histograms for the cyclic proportionality limit
, it follows that, with the increasing sample size, the shape of the histograms approaches the normal distribution. For the 15Cr2MoVA and C45 steels with a sample size of 80 points (Figure 4
), the histograms in their shape agree with the normal distribution. The asymmetry of the distribution of the parameter for steel 15Cr2MoVA is negative, while for steel C45, it is positive. For the aluminum alloy D16T1, the sample size of
is 20 samples (Figure 4
), so the histogram does not have a clearly defined distribution law.
From the analysis of the distribution of cyclic deformation parameters , it can be concluded that the shape of histograms mainly depends on the sample size, i.e., the number of tests, and, with an increase i the sample size of the histogram for most of the parameters, the shapes of the histograms approach the normal distribution.
For a more accurate definition of the distribution law, a computer-aided study has been conducted to confirm the agreement between the hypotheses of the empirical distribution and the theoretical law of normal distribution according to the Smirnov compatibility criterion
The criterion for the results of a sample to satisfy the law of the normal or log-normal distribution is expressed by the inequality:
The results of agreement with the normal distribution according to the Smirnov compatibility criterion
are provided in Table 3
In the analysis of cyclic deformation characteristics, the level of significance has been accepted. The results of the reports confirm the inequality of , indicating that the experimental data correspond to the theoretical law of normal distribution. Comparison of the calculation results of the criterion and the shape of the histograms of cyclic deformation parameters for all materials has suggested that the values of function correspond to the histograms similarly to the normal distribution law. This is apparently due to the greater sensitivity of the cyclic deformation curves, compared to monotonous tensile curves, chemical composition, and conditions of heat treatment of the material, as well as to the conditions of a more complex cyclic experiment.
The calculation was performed using statistical characteristics (arithmetic mean
, standard deviation s
, dispersion D
, skewness Sk
, and coefficient of variation V
) for the three distribution laws: normal, log-normal, and Weibull law. The results of the calculations are given in Table A1
. From the analysis of statistical characteristics, it can be established that it is more appropriate to describe the cyclic deformation parameters
as normal, and parameters
as log-normal distribution laws. When using these laws to describe cyclic parameters, the coefficient of variation takes the minimum values. The statistical characteristics of the Weibull law are little influenced by the characteristics of the normal distribution. Interestingly, the coefficient of variation of the normal distribution of deformation parameters
clearly depends on the level of loading and decreases with the loading level increasing. The same trend is observed when histograms of the same parameters are considered. This is apparently due to the lower stability of the processes of cyclic hardening, and, in particular, the softening of the stages of fatigue and quasi-static damage.
Based on the chosen distribution law, the possible limits of the confidence intervals have been calculated at a designated confidence level
and 0.99, as well as for cyclic loading curves parameters. It is noteworthy that the limits of the confidence intervals of the parameters of cyclic deformation, as compared to the mechanical characteristics, also occupy a rather narrow band. The results of the calculations are presented in Table A2
provides a comparison of the variation coefficients of the three distribution laws. From the analysis it also follows that, to describe the parameters of cyclic deformation
it is more appropriate to use the normal distribution law, for
using log-normal is more appropriate.
The distributions of the cyclic deformation parameters investigated are shown in the probabilistic grid in Figure 6
The deformation parameters of the low-cycle fatigue diagrams shown in the probability grid of the normal distribution confirm that these parameters are consistent with this law.
To estimate the variation of the parameters of cyclic deformation
, ratio K
of the maximum values of statistical series of the cyclic parameter with its minimum value has been determined. The results of the calculation of relationship K
for the materials at the coordinates s
are shown in Figure 7
. The resulting curves for parameters
a,b) indicate the following mathematical relationships between:
Subjected to the condition: and .
The resulting curves for parameters
c,d) designate the following mathematical relationships between:
Subjected to the condition: and .
Equations (23)–(26) allow preliminary to estimate values of the cyclic deformation parameters.
3.2. Probability Evaluation of Low-Cycle Fatigue Curves
For statistically investigated characteristics of the low cycle fatigue diagrams, probabilistic values have been established, allowing theoretical low-cycle fatigue curves to be estimated and assessed probabilistically. The specimen under stress-controlled loading fails due to quasi-static damage dk
resulting from accumulated unilateral plastic deformation and fatigue damage dN
resulting from cyclic plastic deformation, characterized by the width of the hysteresis loop
where fatigue damage:
—fatigue damage accumulated over k
—fatigue damage accumulated before failure. Quasi-static damage:
To make the calculations more straightforward, the following was adopted:
Under strain-controlled loading strain is constrained, thus, no unilateral strain is accumulated and there is no quasi-static damage.
The fatigue curve at coordinates
is a straight line. Using the equation of the line [36
] one obtains the following.
Using Equations (28)–(32), the damage condition can be written:
In the formulation of the theoretical curves for the low-cycle fatigue, only the fatigue damage is considered, and stress-controlled loading is treated as a nonstationary strain-controlled loading, with the cumulative damage of one half-cycle expressed by the equation:
In this case, the condition for the crack initiation is:
accumulated fatigue damage prior to failure with hysteresis loop width
accumulated fatigue damage up to the crack initiation
etc. The fatigue curve under strain-controlled loading
By using the coordinates
, one obtains
By writing Equation (37) to Equation (36) gives:
and inserted into Equation (35) one obtains:
Under stress-controlled loading for cyclically stable material:
Therefore, the summation of the comparable cyclic deformations may be substituted by the summation of the comparable lifetimes:
In this case
, using Equations (36) and (41), one obtains:
Using Equation (36), it is achievable to calculate the fatigue damage of cyclically stable, cyclically softening, and cyclically hardening materials. A piecewise approximation of the generalized strain diagram produces Equation (36) as follows:
For the determination of the probabilistic curves, the values of the probabilistic coefficients
were used (Table 4
). The comparison of the calculation and experimental data of these graphs is shown in Figure 8
Using Equation (37) for calculation methodology, it was obtained that the low-cycle curves reach a suitable order, that is, the curves for steel 15Cr2MoVa and steel C45 of 1% show the lowest lifetime, while 99% the highest. Analytically determined curves for steel 15Cr2MoVa produce a relatively narrow range, whose magnitude is dependent on the loading level
. From the generated graphs, it was determined that for a curve of 1% curve at a loading level
, the durability
, and for a curve of 99%, the durability
; at a loading level
for a 1% curve, durability
and for a curve of 99%, the durability
; and at a loading level
for a 1% curve, durability
and for a curve of 99%, the durability
. The experimentally determined durability curves for steel 15Cr2MoVa are shown in Figure 8
a. It can be observed that the distribution and scatter of the curves are significantly higher compared to the analytically calculated ones. The analytical durability curves have a distribution between 70% and 90% of the probabilistic experimental curves at a loading level of
and between 15% and 50% of the probabilistic curves at a loading level of
The comparison of the experimental and theoretical curves of steel C45 under controlled strain loading is shown in Figure 8
b. Probability curves were calculated analytically using Equation (37) and the parameters in Table 4
. The probabilistic values of the constants C2
were derived from the probability curves
for strain-controlled loading and are presented in Table 4
. Hysteresis loop width
was determined using Equation (30). It can be seen that the probability curves reach a suitable order, i.e., the 1% curve is the lowest and the 99% curve the highest. It can be seen that there is a large scatter (Figure 8
b), which indicates a high dependence of durability N
on the loading level
The range of curves becomes narrower with increasing durability. When comparing the experimental results for steel 15Cr2MoVa and C45 with the analytical results, the opposite effect can be observed: the experimental curves have a much tighter range and scatter than the analytically estimated ones. At loading level
the experimentally estimated durability of 1% to 99% is distributed between the analytical curves of 15% and 90%.