INTRODUCTION

Due to the fact that the process of fatigue crack development under irregular operational loads depends on a number of factors that are far from being studied [1–3], the issue of predicting the achievement of the strength limit state of load-bearing structures remains largely open. Studies of the structural survivability characteristics show that, under irregular loading, fatigue damage is not determined by a linear summation of the damage from individual cycles. Therefore, in problems of an applied nature, experimental results obtained under conditions close to operational ones are necessary [4]. The nature of the accumulation of fatigue damage is such that the relative damage value, being close to unity according to the linear summation hypothesis, has limits from 0.1 to 10.0 according to the experimental results [5]. It has been found that, while maintaining the statistical parameters of the load spectrum, a change in the sequence of cycles significantly affects the crack development pattern [6]. However, in many laboratories following the normative recommendations (ASTM E606, E466, and E647 [7]) tests are carried out under regular loading conditions.

The most complete understanding of the regularities of the development of fatigue cracks under irregular loading processes can be found in the works by R. Sunder [1–3, 8–10]. Sunder’s model highlights several factors involved in the development of fatigue cracks under operational loading and determines their interaction. These are crack closure, crack tip blunting under irregular cycles, residual stresses arising near the tip, and the influence of atmospheric factors on the fracture mechanism. Sunder considers the local stress at the crack tip one of the main factors determining the development of fatigue cracks. Local stress is in complex dependence on the diffusion kinetics of the material. It occurs on the scale of several atomic layers at the top. The kinetics of such mutual penetration, which causes “sticking” of the crack tip edges, is accompanied by the impact of atmospheric factors on the newly formed surfaces, in the course of crack development. In general, the local stress component at the crack tip is due to the mechanics of cyclic stress hysteresis, deformation at the crack tip, and interaction with the cyclic influence of the environment [11]. The following conclusions are drawn regarding the region of the stress intensity factor (SIF) values close to the threshold value. ∆Kth is strongly affected by atmospheric fatigue. Surface diffusion slows down. But this influence decreases with an increase in the slip component in the fracture mechanism. The result of exposure to atmospheric fatigue completely disappears in deep vacuum tests [1, 3]. Thus, the stresses in the crack tip region determine the sensitivity to the hysteresis load sequence, even at a low level of SIF, where near-threshold conditions prevail. Thus, the stresses at the crack tip are sensitive to the cyclic sequence. Hence, it follows that the threshold level of SIF ∆Kth of the pseudo-random load sequence can change from cycle to cycle. This means that the technique for estimating the operating unit threshold level as a whole, proposed in [12] for applied problems, carries more information than the threshold value traditionally determined from the standard fatigue fracture kinetic diagram (FFKD) of a material under harmonic loading.

In taking into account such complex processes at the microscale level that determine the development of fatigue cracks, it should be recognized that the experimental testing of materials, while reproducing significant fragments of operational loading, is becoming increasingly important.

Experimental testing is facilitated by the creation of banks of models of operational loading modes inherent in various branches of mechanical engineering and various operating conditions, up to individual load-bearing structural elements [13]. At the same time, experimental techniques are being developed that are capable of reproducing loading with a high degree of accuracy. Such a characteristic as the fatigue fracture kinetic diagram (FFKD) of a material, obtained according to standard normative recommendations under harmonic loading modes [7], can be taken as the basis that determines the homogeneous fatigue pattern that the material under study has. Deviations caused by irregular loading and the influence of atmospheric or other factors can be conveniently correlated with the kinetic diagram of the fatigue fracture, and we can consider the possibility of correcting the parameters of the initial FFKD of the material. There are known works on modeling the fatigue of materials under irregular loading [14, 15]. The standard fatigue curves of material were corrected in them (fatigue was studied without taking into account crack development). The endurance limit and the exponent of the degree of the fatigue curve were subject to correction. If an experiment with an irregular loading process gave results incompatible with the original fatigue curve, then a separate, so-called secondary fatigue curve was constructed, taking into account the experimentally determined regularities.

According to the same algorithm, only with respect to the kinetic fracture diagram, the work presented in this article was conducted. The experimental results of [12] were analyzed in the calculations. It proposes a method for estimating the threshold level of operational loading. A feature of this method is that the reduction in the level of operational loading, in order to approach the threshold conditions, is carried out as the fatigue crack develops, i.e., with an increase in the sample compliance. The test is carried out by reproducing a constant block of opening crack edges in the sample (according to the hard loading method). This method was applied to preserve the similarity of the loading process during operation, but at the same time to ensure a gradual decrease in its level. The loading block recorded under operational conditions at a rolling stock speed of 81.5–90.4 km/h on straight and curved track sections was reproduced. The initial value of the fatigue crack velocity in the experiment was V = 22.8 × 10–7 m/block. During the tests, to determine the changes in the ratios of cycle extrema, the processed signals were recorded both from the load channel and from the extensometer channel. This was correlated with the change in crack length and the number of spent blocks. A standard sample of C(T) type (eccentric tension) (GOST 25.506-85) with dimensions of 125 × 120 × 10 mm was used for testing. The sample was cut from a cast bolster of a freight car [16]. The sample material is low-alloy steel (type 20L). The chemical composition of the steel is the followin: C, 0.2104; Mn, 1.085; Si, 0.292; P < 0.003; S, 0.020; Cr, 0.108; Ni, 0.13; Cu, 0.215; V < 0.0030; Fe, 97.91). With the number of spent blocks and crack growth, the patterns of load reduction ∆P and the change, at the same time, of the crack development rate V were recorded. The maximum value in the first spent block Pmax was chosen as the starting point for the load reduction value. Next, the change in this value with crack growth was monitored. When analyzing the results, it was found that with the crack growth, the ratio of the maximum and minimum values of the loading cycles, relative to the initial process, changes with the similarity coefficient KP for all cycles of the block. The asymmetry coefficient R for each cycle remains constant with a change in the crack length, while the average load value decreases.

Thus, the similarity coefficient is the value

$$K{{P}_{i}} = \left( {{{P}_{{{\text{max}}}}}-\sum {\Delta {{P}_{i}}} } \right){\text{/}}{{P}_{{{\text{max}}}}},$$

where KРi corresponds to the crack length Li; Pmax is the maximum load value in the first (initial) block, when the crack has not yet advanced; ΣΔPi is the total decrease in the Рmax value. Thus, the product of coefficient KРi by all extrema of initial block load cycles determines the load block at the crack length Li in the sample С(Т). Figure 1 shows the dependence of the similarity coefficient KР(L) on the total increase in the crack length L = 42 + ΣΔL, i.e., at its initial value L = 42 mm from the load application line, and the change in the crack growth rate on the number of loading blocks.

Fig. 1.
figure 1

(a) Similarity coefficient, depending on the crack length L: 1, approximating dependence, dots, experimental values; (b) change in the crack growth rate over the interval of its length L = 42–46.69 mm, depending on the number of loading blocks.

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The quadratic approximation equation (Fig. 1a, curve 1) has the form

$$KP\left( L \right) = -0.00393{{\left( L \right)}^{2}} + 0.2996L-4.664.$$
(1)

This relationship was used to determine the operating load as the fracture length changed.

The experiment with the operational loading process was carried out with the crack growing from the length L0 = 42 mm to L = 46.69 mm, while 2857 of the loading blocks were worked out. Each block consists of 325 cycles. The results of [16] were used to compare the patterns of crack development, which is determined by the standard fatigue fracture kinetic diagram. The diagram for the same steel grade (type 20L) is taken as the equation

$$V(\Delta K) = \frac{{{{V}_{0}}{{{\left[ {\Delta K - {{K}_{{{\text{th}}}}}{{{(1 - R)}}^{\alpha }}} \right]}}^{n}}}}{{{{{\left[ {{{K}_{{{\text{fc}}}}}(1 - R) - \Delta K} \right]}}^{n}}}},\quad {\text{m/cycle}}{\kern 1pt} -{\kern 1pt} {\text{MPa}}\sqrt {\text{m}} ,$$
(2)

where the parameters α(R) = 0.846 – 0.37R; V0(R) = 10–(1.565 + 4.04R); n(R) = 2.735 – 2.17R; Kfc = 105 MPa \(\sqrt {\text{m}} \); and Kth = 12.23 MPa \(\sqrt {\text{m}} \) (at R = 0).

CALCULATION ALGORITHM

To calculate the duration of crack development using Eq. (2), the well-known algorithm of linear damage summation for each cycle from the operational loading block was applied. To do this, the operational block of loads is subjected to schematization to highlight the full cycles [GOST 25.101-83]. As a result, the sequences of cycle amplitude values and asymmetry coefficients corresponding to each cycle are determined. They are shown in Fig. 2a and Fig. 2b, respectively.

Fig. 2.
figure 2

(a) Block of the loading process, after the selection of complete cycles; (b) asymmetry coefficients R of each cycle.

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Further, referring to the dependence that determines the stress intensity factor for a given crack length in a compact sample [7], the SIF amplitude was found and, according to the fatigue fracture kinetic diagram, expressed by Eq. (2), the crack growth value was found for each cycle.

Before calculating the crack increase from the subsequent block, the loading process was determined corresponding to the modified crack length. This was achieved by multiplying all extrema by the similarity coefficient (1). The general crack growth pattern was found by summation of cycle-by-cycle growth. The used model of the loading process according to the values of the asymmetry coefficients has a high level, which corresponds to a high traffic loading. The average value, after the selection of complete cycles, was R = 0.94. The values of R cycles in the schematized block are shown in Fig. 2b. Experimentally, the fatigue fracture kinetic diagram equation used in the calculation was found at the highest value of R = 0.8. For higher values of R, the velocity V was calculated from the dependences included in the definition of the coefficients of Eq. (2). Thus, extrapolation to higher values of R has been carried out to a certain extent. Figure 3 shows the kinetic diagrams of fatigue failure: experimental (points), at R = 0.8, and its approximation by Eq. (2) and, for comparison, the calculated dependence for R = 0.94 (the average value of the operational loading process) (Fig. 3b).

Fig. 3.
figure 3

Fatigue fracture kinetic diagram: (a) standard FFKD of steel 20GFL: 1, approximation according to Eq. (2), dots are the experiment; (b) calculated fragments of FFKD: 1, at R = 0.8; 2, for R = 0.94, Eq. (2).

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Figure 4a shows the calculated dependence of the crack length on the number of loading blocks. It is determined using the fatigue fracture kinetic diagram expressed by Eq. (2). The crack growth value for the total number of spent blocks 2857 (325 cycles in each block) was 0.032 mm. In the experiment, for the same number of blocks, the increase in the crack length was 4.69 mm. Thus, according to the calculated dependence, with the use of the fatigue fracture kinetic diagram, the crack should develop much more slowly. The first amendment that can be made is to consider all cycles in the calculation as valid. Take the threshold level in the fatigue fracture kinetic diagram of Eq. (2) equal to zero ∆Kth = 0. The calculation result with such a correction is shown in Fig. 4b, curve 2. Comparing the calculated and experimental dependences (Fig. 4b, curves 1, 2), it can be seen that such a correction is not sufficient.

Fig. 4.
figure 4

Dependences of fatigue crack growth on the number of operational loading blocks: (a) calculated, using the FFKD; (b): 1, experimental; 2, when using the FFKD with a threshold value ∆Kth = 0; 3, with the corrected FFKD: ∆Kth = 0 and V0 = 10–(1.088 + 4.04R) in Eq. (2).

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Taking into account that the operational loading process consists of cycles with a high asymmetry coefficient R (Fig. 2b) and the upper limit of experimental testing of Eq. (2) was carried out only at R = 0.8, it is possible to introduce a correction into the dependence V0(R): V0(R) = 10–(1.565 + 4.04R) and obtain the dependence V0(R) = 10–(1.088 + 4.04R). With these two joint corrections, the calculated dependence of the crack length on the number of operational loading blocks (Fig. 4b, curve 3) is obtained, which is close to the experimental one. Figure 5 shows the dependences of parameter V0 in Eq. (2) on the asymmetry coefficient R.

Fig. 5.
figure 5

Dependences of parameter V0 in Eq. (2) on the asymmetry coefficient R: 1, initial dependence of parameter V0, under harmonic loading; 2, corrected dependence.

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CONCLUSIONS

(1) A method for testing materials was proposed and tested when simulating an operational process with a decrease in the average level and maintaining its similarity. This method makes it possible to obtain both an estimate of the threshold level of process [14] and to reveal the features of damage accumulation associated with the irregularity of cyclic loads.

(2) The experimental regularity of fatigue crack development when simulating a load on a curved section of a railway track differs significantly from the result of a calculation using a standard fatigue fracture kinetic diagram of the material. The difference shows faster crack growth under the operational loading process.

(3) To bring these regularities closer, a correction is proposed for the standard fatigue fracture kinetic diagram of material with a decrease in the stress intensity factor threshold level ∆Kth to zero and the introduction of a stronger dependence of parameter V0 in Eq. (2) of the fatigue fracture kinetic diagram on the asymmetry coefficient R (Fig. 5). Thus, in the calculation of the development of fatigue cracks, in the operational process of this type, it is proposed to use the dependence V0(R) = 10–(1.088 + 4.04R), instead of V0(R) = 10–(1.565 + 4.04R) of Eq. (2) of the standard fatigue fracture kinetic diagram. This gives a pattern of crack development close to the experimental one.