2.1 Self-piercing process with a semi-tubular rivet

Figure 1 depicts the self-piercing process with a semi-tubular rivet. In the first stage, the two (or more) sheets of materials are clamped between the die and the blank holder. In the next stage, the punch pushes the rivet to penetrate the pierced sheet (top sheet). Consequently, the rivet pierces the top sheet and radially expands into the locked (bottom) sheet. Finally, the punch returns to its initial position. The result is both a form-fit and force-fit joint.

2.2 Load spectra and linear damage accumulation

Load spectra are used to characterize and summarize the variable loadings. The load spectrum describes the dependency between the load amplitude, hereafter represented by the force amplitude F_a, and its cumulative frequency H within a load time history. It can be mathematically described by the maximum value of the spectrum , the total frequency H₀ (overall number of cycles), and the shape parameter ν (see Equation 1 according to Hanke⁴). The hardest conceivable spectrum is the rectangular spectrum (ν = ∞), which corresponds to a constant amplitude loading, as shown in Figure 2. For smaller ν, less hard spectra results, namely, a normally distributed spectrum for ν = 2 and a straight-line distributed spectrum for ν = 1.

(1)

Using linear damage accumulation according to Palmgren,⁵ Langer,⁶ and Miner,⁷ it is possible to estimate the service life of a component at variable amplitude loadings from the S-N curve (fatigue life at constant amplitude loading). The linear damage accumulation, also called Miner's rule, assumes that the individual damage of one single load cycle d can be determined from the S-N curve as the fraction of the permissible number of cycles N_i at the same load level. The summation of all these damages from the load spectrum results in the total damage D, Equation 2.

(2)

The premise is that the load spectrum and the S-N curve are valid for the same load ratio R = F_min/F_max or the same mean stress. The theoretical component failure occurs at a damage sum of D = 1. The original Miner's rule was developed for load spectra for which all load levels lie above the endurance limit. This restriction has often been ignored, and the individual damage of each load cycle has been determined from the S-N curve with a pronounced endurance limit, whereby load cycles with an amplitude F_a below the endurance limit F_aE are rated nondamaging. This method is called Miner-original hereafter, Figure 3. However, test results from Gaßner⁸ and Conle⁹ show that stress amplitudes below the endurance limit can also contribute to the damage. This effect has been considered in numerous modifications of Miner's rule of which the following will be discussed in more detail:

However, even if these Miner's rule modifications are applied, discrepancies between the calculated and experimentally determined service life are found (even on average). Therefore, Schütz¹³ introduces the relative Miner rule to compensate for this discrepancy between experiment and calculation. Instead of using the theoretically correct damage sum of D = 1 = 100% to describe component failure, an effective damage sum D_eff ≠ 1 is used. The value of D_eff is determined by the experience for certain groups of metallic components. For analytical strength assessments of components according to the FKM guideline,¹⁴ for example, an effective damage sum D_eff ≤ 1 is used.

2.3 Damage sums for riveted joints

A review of the literature shows that only Meschut et al.¹⁵ published tests under variable amplitude loading on self-pierce riveted specimens. There, three series of tests have been conducted on aluminum components with either one or 10 joining points in different loading conditions. For the individual test results, damage sums with Miner-elementary of D = 0.02–15.8 are calculated. These test series are described in more detail in Section 3 and are taken into account in the evaluations from Section 4 onwards.

For other riveted joints, often only few individual test results are available. These are summarized in Table 1. Lowak and Schütz¹⁶ investigate Huck-Bolt joints. This type of joint is used in aircraft construction and is examined in a two-shear lap specimen with 12 joining points. For these specimens, S-N curves with a constant mean stress are determined. In the subsequent tests under variable amplitude loading, a load spectrum is applied to represent loads from 40,000 flights. The amplitude distribution of this load spectrum corresponds approximately to a straight-line distribution, as shown in Figure 2. The damage sums determined for the high load levels are D > 3 and decrease for smaller maximum values of the spectrum, as shown in Table 1, no. 1. D = 0.81 for the smallest maximum value results.

Buch et al.¹⁹ examine whether the test results determined with different load spectra could be compared with the help of the relative Miner rule. Various notched components are examined, including riveted specimens. Schütz and Lowak¹⁷ also investigate Huck-Bolt rivets in a two-shear lap specimen with 12 joints. The load is applied with the standard load sequence Transport Wing Standard (TWIST) with different maximum values of the spectrum. The resulting damage sums are summarized in Table 1, no. 2, and the following behavior can be seen:

For all further test series, only one or two test results are given for the same load spectrum. The results are summarized in Table 1 without a more detailed description of the load spectrum, since the focus is on the damage sum D. In contrast to the aluminum components considered above, Table 1 also lists damage sums for titanium components. Damage sums of D > 1 also occur for titanium.

Meschut et al.¹⁵ investigate SPR specimens under variable amplitude loading. It is found that damage sums D > 1 occur in multiple cases for SPR joints. Other authors demonstrate the same behavior on other riveted joints. In Section 5, it is shown how the different damage sums can be taken into account for a service life estimation for SPR components using the so called IMAB method for SPR. Section 6 discusses causes that explain damage sums D > 1 for SPR components.

2.4 IMAB method

Hinkelmann²⁰ found a correlation between the service life factor A and the effective damage sum D_eff. The service life factor thereby describes the factor between the fatigue life N_const and the calculated service life N_{calc,Deff = 1}, using D_eff = 1 (Figure 4A):

(3)

The service life factor depends primarily on the spectrum hardness, the slope k of the S-N curve, and the modification of the damage accumulation. For the modification Miner-elementary A can be written as follows:

(4)

For the hardest possible spectrum of a constant amplitude loading (ν = ∞), see Figure 2, the service life factor is A = 1. The less hard the spectrum is, the greater becomes A.

According to Hinkelmann,²⁰ the calculation of the effective damage sum D_eff is performed using the so-called IMAB* method, Equation 5. The relationship between A and D_eff is described by a potency approach with the slope I_IMAB – 1 and the offset I_O for which a double logarithmic plot results in a linear progression. By using the minimum function in Equation 5, D_eff can assume a maximum value of 1. Depending on the modification of Miner's rule used, different values are given for I_O and I_IMAB. In Figure 4B, the course of the effective damage sum D_eff over A is shown qualitatively. Additionally, the shapes of the spectra are assigned to the corresponding service life factors.

(5)

Figure 4C shows the calculated service life, which is determined through linear damage accumulation. The calculated service life N_{calc,Deff = 1} is corrected using the effective damage sum D_eff according to the IMAB method. Using Miner-elementary and a constant cumulated frequency H₀ results in the parallel course between the fatigue life and the service life shown in Figure 4C.

The IMAB method can be used to explain why different damage sums D occur for the same components, such as the riveted components in Section 2.3. By changing the maximum value of the spectrum with a total frequency H₀, a different service life factor A is obtained, which results in a different effective damage sum D_eff; see Equation 5.

2.5 Quality of service life estimations

To describe the quality of service life estimations, the median value m and the deviation range T of the ratio from N_test/N_calc are determined. The deviation range T is defined as the ratio of the 90% quantile to the 10% quantile, as shown in Equation 7.

(6)

(7)

Hinkelmann,²⁰ for example, gives a mean value m = 0.19 and a deviation range T = 9.0 for Miner-original in a database of 293 individual tests for the material group steel. This means that in the examined database, the specimens fail on average at D = 0.19. If the relative Miner rule with D_eff = m = 0.19 is applied, the mean value improves to m = 1.0, but the deviation range remains unchanged. By applying the IMAB method, a mean value of m = 1.0 and a deviation range of T = 7.2 can be achieved. According to Hinkelmann,²⁰ the most accurate service life estimation is achieved for the same database with the Miner-Liu/Zenner. Without correction by the IMAB method, a mean value of m = 0.69 and a deviation range of T = 6.8 are obtained; with the application of the IMAB method, the parameters improve to m = 1.0 and T = 5.8.

2.6 Fretting wear

Under constant and variable amplitude loading, fretting occurs between the upper and lower aluminum joining parts. This is shown by a black powdery debris that emerges from the joint and adheres between the contact surfaces. Chen et al.²¹ investigated this debris on SPR specimens of AA5754. Debris mainly occurs at the beginning of cyclic loading and consists of dissolved aluminum particles that quickly oxidize to Al₂O₃, as shown by EDX analysis. Iyer et al.,²² Han et al.,²³ Li et al.,²⁴ and Zhang et al.²⁵ also proved fretting in SPR joints.

Li et al.²⁴ show that fretting increases the roughness of the surfaces in the fretting area and that this affected area is larger at high amplitudes than at low load amplitudes. Han et al.²³ show that high friction in an SPR joint increases the loading capacity. To reduce friction between the joined parts, Han et al. introduced a PTFE film into the joint. This prevented fretting at small amplitudes and significantly reduced it at large amplitudes. Owing to the reduced friction, the stress distribution in the joint is changed, and the damage location is no longer in the joining sheets but within the rivet. The endurable number of cycles is significantly reduced. This results in the positive influence of a large fretting area on the load capacity of the joint.

However, a large fretting area can also have a negative influence on the load capacity. Chen et al.,²¹ Iyer et al.,²² and Han et al.²³ indicate that the crack initiation sites are located in the fretting area. According to this, a large fretting area statistically leads to a lower strength. Li et al.²⁴ contradict this and demonstrate that cracks can originate in fretting areas; however, this is not always the case. There is no significantly shorter fatigue life for cracks originating in the fretting area as compared to cracks that originate at other locations. It is also highlighted that fretting can also occur only after crack initiation due to the resulting heel.

6.1 Cyclical hardening

Fu and Mallick²⁸ investigated the influence of the setting force during joining on the fatigue behavior under cyclic loading of single-lap specimens made of aluminum 6111-T4 with a shear loading, similar to specimen type 1 investigated in this paper. Among other things, the specimens were loaded with a two-stage spectrum. The applied load spectra can be divided into two categories:

Figure 12 plots the resulting damage D, as determined with Miner-elementary, over the service life factor A. Although the relative Miner rule and the IMAB method are valid only to ideally mixed load sequences, the curve of the effective damage sum D_eff, according to the IMAB method for SPR joints, is shown nevertheless. Because of the two-stage test sequence and the associated high number of high amplitudes in the spectra, a low service life factor A results for all the tested spectra.

For the low–high test sequences, the damage sum is D ≈ 1. The damage sums of the high–low test sequences are greater than 1 and exhibit a progression that follows the IMAB method for SPR. Four of the tests were not tested to failure as the test was aborted when approximately 2.5·10⁶ cycles on the low load horizon were reached. The damage of D ≈ 1.75 achieved up to that point would increase further by testing to failure.

Fu and Mallick²⁸ suggest that the hardening of aluminum under cyclic loading is a possible explanation for the occurrence of damage sums of D > 1 for the high–low test sequences:

In the case of a low–high test sequence, D ≈ 1 occurs:

These findings can be transferred to the test results of the randomly distributed load spectra, as shown above. For this purpose, nonlinear damage accumulation must be assumed during the test under constant amplitude. Cycles at the beginning of the test, in which the cyclic hardening is not yet completed, must show higher damage shares than cycles that are later applied to the hardened material.

Since the focus in this paper is on linear damage accumulation, as this is also used more frequently in practice, the accuracy of a service life estimation with nonlinear damage accumulation is not discussed here. However, an overview of promising approaches to nonlinear damage accumulation is shown by Zhu et al.^{29, 30} Additionally, an interesting approach for combining the linear damage accumulation with the correction by the IMAB method with the nonlinear damage accumulation according to Manson and Halford³¹ is given by Müller.³² The nonlinear damage accumulation according to Manson and Halford also causes a decrease of the effective damage sum D_eff with increasing service life multiple A. Therefore, for the data set examined by Müller³² for the material group steel, a similar accuracy results for the linear and the nonlinear damage accumulation. No statement is made for the aluminum material group because the data set is too small.

For randomly mixed hard load spectra, that is, a small service life factor A, high loads often occur, and cyclic hardening is completed early in the service life. It follows from this that,

In conclusion, it can be stated that the calculated damage D for hard spectra becomes greater than 1 because the partial damage d of the small amplitudes is assumed to be too high. This can also explain the knee point in the course of the IMAB method for SPR. For very small service life factors A < 70, the share of cycles with low amplitudes becomes smaller and smaller, which are responsible for the excessively high damage D.

Large amplitudes rarely occur in less hard spectra with large service life factors. Thus, the end of the cyclical hardening is also shifted further towards the end of the experiment. From this, it follows that

The damage is therefore underestimated, and the specimen fails before reaching the damage sum of 1.

6.2 Fretting wear

The findings from the literature on fretting wear in SPR joints shown in Section 2.6 can be transferred to explain the damage sums D > 1 for small service life multiples A. For hard load spectra, that is, a small service life factor A, many large amplitudes occur. This results in a large fretting area with high roughness as compared to the fretting area with small amplitudes applied with a constant amplitude loading. Consequently, the force of the small amplitudes is transferred over a larger area, leading to smaller stress concentrations in the joining sheets. The partial damage d of the small amplitudes is therefore smaller with variable amplitude than with constant amplitude. This results in damage of D > 1 for the tests under variable amplitude. For less hard load spectra, that is, large service life factors A, high amplitudes rarely occur, and the fretting areas do not become larger as compared to the single-step test for small amplitudes. For low fretting, damage sums of D < 1, which are usual for metallic components, occur.

Form-fit improvement is a further positive effect of fretting on service lives with variable amplitude. On a single-lap specimen with a shear load, see specimen type 1 in Figure 5, a measurement was conducted using the potential probe method. A DC current I flows through the specimen, which for the most part passes through the joining point as the only connection point. At the surface of the locked sheet, the potential drop between two points before and after the joining point is measured. If the size of the contact area in the joint changes, the electrical resistance changes. This change can be made visible by measuring the electrical potential drop. If cyclic loading with constant amplitude is applied to the specimen in the form of a sine, this sine also occurs in the electrical voltage. Therefore, the electrical resistance of the joining point changes under cyclic loading. It can be concluded that the size of the contact surfaces must change during the load. With each further cycle, the amplitude of the voltage signal becomes smaller. Fretting causes the joint to be set and improves the geometric fit of the contact surfaces. If the load amplitude is increased, sine oscillations occur again in the electrical voltage signal, which decay over time.

Thus, hard spectra (small A) are similar to the fretting above: