Keywords

1 Introduction

While in operation, vehicles experience variable loads from various types of road surfaces, maneuvering during movement, changes in temperature conditions, impact action in the obstacle crossing process. It leads to fatigue damage accumulation in the load-bearing design elements, leading to the fatigue cracks growth. Microcracks are formed in stress raiser of a part, and it is often difficult to detect them without special equipment. Gradually, this process takes on the character of an avalanche effect, and the part may be destroyed.

To prevent such a scenario, it is necessary, knowing where such fatigue cracks can occur, to check the maintenance inspection to detect them. To establish the examination of such rates, it is necessary to know the crack growth kinetics so that the emerging crack can be detect before its critical growth.

In the suggested work, on the low-alloy steel specimens used in the automotive components production, and the fatigue crack kinetics growth analysis under regular and irregular loading is carried out, considering the effect on of different asymmetry ratio R and maximum loading Pmax. Irregular loading is modeled based on standard random spectra using in the engineering design of automotive load-bearing structural elements.

2 Material, Research Technique

The compact specimens 60 × 62.5 × 5 mm in size with an edge crack are investigated, cut from low-alloy steel, close to the analog Russian steel—09G2. This alloy is used in the automotive industry, concretely for the manufacture of front suspension for passenger cars [1,2,3,4,5].

The thickness of this material specimen is 5 mm that give it possible to assert that the fatigue crack propagation under a plane stress condition. The drawing is shown in Fig. 1.

Fig. 1
figure 1

Geometry of test specimen

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Chemical composition of the alloy % (mass): Fe 98.7; C 0.078; Cr 0.1; Cu 0.016; Mn 0.96; Nb 0.04; Si 0.1; Ti 0.016; Al 0.036. Mechanical properties of alloy: ultimate strength σu = 496 MPa, conditional yield strength σy = 345 MPa, modulus of elongation E = 210 GPa.

Experimental studies were carried out using specialized software MTL-32, controls the test under load control mode. VAFCP (Variable Amplitude Fatigue Crack Propagation) software, which allows testing under various loading conditions in automatic mode, including loading level control to provide a set value of K, as well as automatic recording of the crack length and other test results. The compliance method was used to measure crack length, using the BISS Bi-06–201 Crack Opening Displacement (COD) gauge, which measures distance between edges of the specimen. The tests were carried out in air under normal ambient conditions, and the frequency of the main type of loading was 10 Hz. The test procedure conforms to the specification of ASTM E 647–08. The crack growth kinetics were recorded depending on the number of cycles during the test.

The crack resistance testing program consists of testing at constant amplitude (CAL) and variable amplitude loading (VAL) with different asymmetry ratio R from 0 to 0.75 and a maximum load of 3.5–7 kN. For variable amplitude tests, spectra of a quasi-random nature are formed on the basis of standard loading spectra typical for variable loading of various technical objects and contractual elements [6,7,8,9,10]. Figure 2 shows examples of normalized loading spectra.

Fig. 2
figure 2

Examples of selection from normalized loading spectrums: a—SAESUS; b—SAETRANS; c—SAEBRACKET

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SAESUS, SAEBRACKET, SAETRANS spectra—shortened load front suspension spectrums, brake gear and transmission of a passenger car. Model spectra A and C are formed on the basis of an autocorrelation approach for the Rayleigh distribution, which is most typical for failures of technical objects and constructions fatigue-prone and wear [11,12,13]. The test schedule is shown in Table 1. Before testing, the fatigue crack was performed by precraking to a value of a0 = 15 mm under constant amplitude loading with a nominal load at R = 0, P = 0.6·Pmax from the subsequent test. The critical size, of the crack length corresponding to the specimen failure at the provided tests, is acr = 31–34 mm.

Table 1 Test schedule
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Fig.3
figure 3

Curves of crack growth (a), experimental (b) and effective SIF (c)

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3 Research Results

Figure 3 shows the test results (Fig. 3a). Based on the dependences of the crack growth kinetics a-lgN, obtained using a crack opening displacement gauge, the testing machine software automatically estimates the growth rate da/dN and the amplitude of the stress intensity factor (SIF) ∆K at the crack mouth. These results make it possible to plot the fatigue diagrams. Item numbers in Fig. 3 correspond to those numbers in Table 1.

Test results at different types of loading and different values of the asymmetry ratio of the loading block R = 0–0.75 show that all fatigue diagram curves have tendency to parallelism in logarithmical coordinates (lg(da/dN)-lg(ΔK)). Comparison of tests under constant amplitude loading is carried out with the same Pmax but different R, and show that an asymmetry ratio increasing leads to an increase in crack growth life, because of a decrease in the range of the ΔK. In Fig. 3a, it can be seen that at a constant Pmax = 5 kN and R ≤ 0.5 for a given material, the rate hardly exceeds 300 thousand cycles, and at R ≥ 0.7 the durability essentially increases to million cycles.

The asymmetry ratio is also influencing on the fatigue diagram position. In Fig. 3b, it is clearly seen that at increase asymmetry ratio, the curves from 1 to 6 are located lower and more to the left. At a constant value Pmax and regular loading, the range of coefficient ΔK decreases and the duration of crack growth increases. Growth curves and KDFC 5 and 6 in Fig. 3 correspond to tests at asymmetry of 0.7 and 0.75. It is assumed that at such asymmetries R, the crack closure effect is not observed [14,15,16], which effects on the crack growth kinetics.

The difference between variable and constant amplitude loading is estimated with measuring of irregularity (completeness). The loading completeness ratio V for the presented tests is shown in Table 1, for constant amplitude loading the completeness ratio is equal to one.

$$ V = \left[ {\frac{1}{{\nu_{b} }}\left[ {\sum\limits_{i = 1}^{{\nu_{b} }} {\nu_{ai} \left( {\frac{{\Delta P_{ai} }}{{P_{\max } }}} \right)^{n} } } \right]} \right]^{\frac{1}{n}} $$
(1)

where vb, va—number of cycles of the sample and block of variable loading with load \(\Delta P_{ai} ;\Delta P_{ai} /P_{\max }\)—is the normalized i loading amplitude. This approach corresponds to the replacement of variable amplitude loading with equivalent constant ones, corresponding to irregular random loading in terms of the damaged ability. The calculated V ratio is shown in Table 1. It is clearly seen from the position of the initial part of fatigue diagram for different spectra, where, at a decrease V ratio in the SAESUS spectrum, fatigue diagram settle regardless of R and Pmax that crack growth life reduces, but retains parallelism in these coordinates, as and fatigue diagrams at CAL. On the fatigue diagram, position of VAL is influenced by the value of the maximum load Pmax. At the increase the load from 3.5 to 7 kN and maintaining a low asymmetry R = 0–0.1, the fatigue diagram curves shift toward the region of increased crack growth rates. This is clearly seen in the SAESUS spectrum (13–15 curves) and the SAETRANS spectrum (7–9 curves).

Analytical fatigue diagram curves in the middle zone are described by the Paris equation [5]:

$$ \frac{da}{{aN}} = C \cdot \Delta K^{n} $$
(2)

where da/dN is the crack growth rate during the loading cycle; ΔК is the range of SIF; C and n are empirically determined coefficients that are the cyclic crack resistance characteristics of the material. In the present work, for the steel under study, the values C = 5·10–14 and n = 3.5 are obtained to estimate the speed da/dN (mm / cycle) and ΔК in MPa√mm.

Because the position of the fatigue diagram curves at CAL block (Fig. 3b) significantly depends on the asymmetry ratio R, and the position of the curves at VAL, accept asymmetry, is influenced by the irregularity ratio V; therefore, in this article, the approach of the effective SIF ∆Keff is used, which makes it possible to approximate these curves to a single curve specific at R = 0, which is shown in Fig. 3c.

To consider the phenomenon of “crack closure” [17,18,19,20,21], which occurs when the cycle asymmetry ratio R is less than 0.6–0.7, and which reduces the crack growth rate by reducing the range ΔK, the parameter U is introduced to describe crack closure with an asymmetry of 0.1 ≤ R ≤ 0.7 for steels according to the equation in the form of a polynomial dependence:

$$ U = 0.57 + 0.16 \cdot R + 0.53 \cdot R^{2} - 0.43 \cdot R^{3} - 2.58 \cdot R^{4} + 6.15 \cdot R^{5} - 3.13 \cdot R^{6} $$
(3)

Thus, in our calculation, the effective SIF at the crack mouth for different types of loading is taken as:

$$ \Delta K_{eff} = \Delta K \cdot U \cdot V $$
(4)

In this expression, the effect of “crack closure” and the loading cycle asymmetry on the crack growth rate is estimated through the asymmetry ratio R.

4 Results and Discussion

The crack propagation process largely depends on the sequence of loads, which is random. So, the prediction of crack growth is carried out by integration, by the cycle-by-cycle method in the order of the cycles in a random process, starting from the minimum crack length, at which its detection is possible, according to the formula:

$$ N_{CYC} = \int\limits_{{a_{0} }}^{{a_{cr} }} {\frac{1}{{C \cdot \Delta K_{eff}^{n} }}} $$
(5)

where NCYC—estimated of crack growth life under CAL and VAL by the cycle-by-cycle method; ΔKeff—the effective SIF, determined by Eq. (4); a0; acr—initial and critical crack length.

Another approach for estimating of crack growth life at VAL NVAL is based on the proposal to consider the crack growth kinetics on the basis of its growth at CAL NCAL and taking into account the nature of variable loading through the irregularity coefficient V at the same force parameters Pmax and R without taking into account the interaction of amplitudes in the loading spectrum (6):

$$ \begin{gathered} N_{VAL} = N_{CAL} \cdot K \hfill \\ K = 10^{{\left( {1 + A\lg n} \right) \cdot \left( {1 - V} \right)}} \hfill \\ \end{gathered} $$
(6)

where A—scaling parameter equal to 2 for the studied steel, K—the coefficient of increasing the fatigue crack growth life at VAL compared to CAL with the same loading parameters Pmax and R.

The calculation results using the methods describe above are shown in Fig. 4 in the form of the calculated values dependence NCYC or NVAL from the experimental NEXP. Comparison of the durability gives good enough agreement between the calculated and experimental data [9, 10].

Fig. 4
figure 4

Calculation of the crack growth life by the cycle method a and by formula (6) b in comparison with the experiment

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5 Conclusions

Thus, it has been established that the growth rate and the position of curves on fatigue diagram under VAL are influenced not only by the asymmetry ratio R, as in the case of CAL, but also by the irregularity ratio V, which takes into account the irregularity of random loading.

The cycle-by-cycle method for predicting the fatigue crack life in steels, considering the effective SIF, show a high convergence of the calculated and experimental values, as well as the approach based on the use of an equivalent constant amplitude loading to predict irregular loading. The correlation coefficient for the first and second methods is r2 = 0.99 and r2 = 0.91, respectively.