2.2.2 Performance evaluation
Before testing, bulk density of saturated surface dry specimen was determined (EN 12697-6:2013, Procedure B), and consequently air void content was calculated in accordance with EN 12697-8.
The initial stiffness modulus and phase angle of each testing beam were measured using a 4PBB device in accordance with EN 12697-26, Annex B. The tests were performed at temperature of 20°C and at frequencies of 0.1, 1, 5, 8 and 10 Hz. Samples were subjected to 100 sinusoidal load cycles with a constant strain amplitude of (50 ± 3) µε.
Fatigue resistance of all sets was determined using 4PBB test, at a single temperature of 20°C and at a frequency of 10 Hz, according to EN 12697 − 24/Annex D. Before being loaded, the specimens were conditioned for at least 2h at a testing temperature. Tests were performed in the stress-controlled mode because of the rapid strain increase during the test that causes faster crack propagation, consequently allowing easier determination of the failure occurrence [15]. Tests were carried out at three stress levels, where six beams of each set were tested per each level, so that all fatigue failures occur in the range from 104 to 106 loading cycles.
2.2.3 Data analysis
The results of each individual sample from a certain set (number of loading cycles and initial strain value) were fitted together and presented in the form of a power (fatigue) function:
\(\text{log}\left({N}_{i,j,k}\right)={A}_{0}+{A}_{1}\bullet \text{l}\text{o}\text{g}\left({\epsilon }_{i}\right)\) Eq. 1
where i is the specimen number, j is the chosen failure criteria, k is the set of test conditions (20°C and 10 Hz), \({\epsilon }_{i}\) is the initial strain amplitude measured at the 50th or 100th load cycle (µm/m), depending on the failure criteria, \({A}_{1}\) is the slope of the fatigue function in the log-log plot, and \({A}_{0}\) is the fitting parameter.
In this study, three different failure criteria (approaches) were used to determine fatigue lives: traditional approach (50% reduction in initial stiffness), the Energy Ratio (ER) approach and newly developed approach (SFP). Fatigue laws were obtained using Eq. 1, from which critical strain ε6, that leads to fatigue failure after 106 cycles, was calculated and used for comparison of the fatigue lives among different sets and approaches applied.
2.2.3.1 Conventional approach
Van Dijk and Visser [40] defined the failure under cyclic loading as the point at which the stiffness drops to 50% of its initial value, which is typically regarded as the stiffness at the 100th cycle (Nf,50%, Fig. 5a). Although this method is straightforward, it has some limitations. The estimation of the initial value based on the number of cycles may be affected by nonlinearity [41], irreversible damage and thixotropy [42]. Furthermore, true failure of a testing sample often occurs between 35% and 65% stiffness decrease, although it can occur as low as 20% of initial stiffness for heavily modified materials [43]. Despite its limitations, this approach has often been used in previous studies [44, 45], and it is still a part of European standard for the fatigue resistance EN 12697-24, therefore it was selected as one of the approaches in this study.
2.2.3.2 Energy ratio approach
Hopman et al. [46] proposed the failure in a strain-controlled testing mode as the number of cycles (N1) up to the point at which cracks are considered to initiate and defined the Energy Ratio (ER) as:
\({R}_{E}=\frac{n{W}_{0}}{{W}_{n}}=\frac{n\left[\pi {\sigma }_{0}{\epsilon }_{0}sin{\phi }_{0}\right]}{\pi {\sigma }_{n}{\epsilon }_{n}sin{\phi }_{n}}\) Eq. 2
where n is the number of cycles, 𝑊0 and 𝑊n is the dissipated energy in the first and n-th cycle, respectively, 𝜎0 and 𝜎n are stress levels in the first and n-th cycle, respectively, ε0 and εn are strain levels in the first and n-th cycle, respectively, and 𝜑0 and 𝜑n are phase angles in the first and n-th cycle, respectively.
Rowe [33] stated that the change in sinφ is small compared to the change in the complex modulus (\({E}_{i}^{*}\)) and therefore simplified equation for calculating the ER (Rσ) in a stress-controlled mode:
\({R}_{\sigma }\cong n{E}_{i}^{*}\) Eq. 3
where n is the number of load cycles and \({E}_{i}^{*}\) is the complex modulus in the n-th cycle [MPa].
Slightly modified approach is adopted in standard ASTM D8237-21, where the failure (Nf,ER) is defined as the maximum value of normalized stiffness x normalized cycles versus number of cycles plot (Fig. 5b), that is calculated according to following equation:
\(\widehat{S}\times \widehat{N}=\frac{{S}_{i}\times {N}_{i}}{{S}_{0}\times {N}_{0}}\) Eq. 4
where \(\widehat{S}\times \widehat{N}\) is normalized beam stiffness x normalized cycles, Si is flexural beam stiffness at cycle i (MPa), Ni is cycle i, S0 is initial flexural beam stiffness (MPa), estimated at approximately 50 cycles, and N0 is actual cycle number where initial flexural beam stiffness is estimated.
The calculated normalized stiffness data can be fit to a best-fit six-order polynomial curve or Logit model, ensuring easy determination of the failure. Therefore, it was decided to employ this method as the second approach in this study since it is frequently used to evaluate the fatigue failure of 4PBB samples (ASTM D8237-21).
2.2.3.3 Simplified Flex Point (SFP) approach
The curve \(\epsilon \left(n\right)\) representing the strain amplitude evolution in control load conditions consists of a succession of experimental data in terms of number of loading cycles and corresponding deformation level (Fig. 5c). The acquired loading cycles are numerically very close at the beginning of the test and then become progressively more distant, even irregularly, during the test, mostly because of the machine limitations. This condition is detrimental to the application of a finite difference method (both linear and non-linear).
The experimental curve of strain amplitude evolution under repeated loading cycles (Fig. 6) has three typical stages (primary, secondary and tertiary stage) with completely different experimental trends. In these stages, the strain amplitude rate (slope of the curve) is always positive and in the primary stage, it decreases rapidly. In the second stage, the strain amplitude evolution curve has an inflection, namely flex point (Nf,SFP), which is assumed as a reference for identifying the fatigue resistance of the material tested. In the tertiary stage of the curve, the deformation rate increases rapidly till the physical failure of the specimen is reached.
From these considerations, it can be concluded how difficult is to find a robust and simple interpolation method to identify the flex point of the strain amplitude curve, which is the main outcome of the so-called Simplified Flex Point (SFP) approach. An interpolation method using a non-linear polynomial of suitable order (single or segmented) is not helpful in this case, given the fact that the curve in the secondary stage theoretically could have a waving trend with a certain number of flex points instead of the single flex point suggested by the experimental data (except for their small intrinsic scattering). Furthermore, a high-order polynomial (up to 7th order) would be required to adequately model the primary and tertiary stages. To solve the above computational issues, a dedicated non-linear interpolation method was developed, based on the main characteristics of the experimental strain amplitude evolution curve.
This curve always has the following characteristics:
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the curve is strictly increasing, therefore the slope is always positive;
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the second derivative (related to the curvature) is always negative for the points preceding the flex point and positive for the subsequent ones;
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the third derivative is positive along the whole curve (i.e. the curvature is increasing).
Based on these features, a high-order nonlinear polynomial can be used to set up an interpolation method that complies with the characteristics of the experimental curve by imposing some unilateral and bilateral constraints to its constant parameters.
The theoretical model of the strain amplitude \(\epsilon \left(n\right)\) can be described by the following polynomial of N-order, where \(n\) indicates the generic position along the curve and \({n}_{f}\) denotes the location of the flex point:
\(\epsilon \left(n\right)=\sum _{i=0}^{N}{K}_{i}{(n-{n}_{f})}^{i}\) Eq. 5
where \({K}_{i}\) indicates the constants of the model of \(N\)-order, which are equal to \(N+1\).
The model is completed by adding to Eq. 5 the following conditions:
I. The slope is non-negative for any value of \(n\);
\(\frac{d\epsilon \left(n\right)}{dn}=\sum _{i=1}^{N}{i K}_{i}{(n-{n}_{f})}^{i-1}\ge 0\) Eq. 6
II. The curvature is non-negative for values of \(n\) greater than \({n}_{f}\) and negative for the \(n\) values lower than \({n}_{f}\);
\(\frac{{d}^{2}\epsilon \left(n\right)}{{dn}^{2}}=\sum _{i=2}^{N}{i \left(i-1\right)K}_{i}{(n-{n}_{f})}^{i-2}\ge \le 0 :n\ge \le {n}_{f}\) Eq. 7
III. The third derivative is non-negative for any value of \(n\).
\(\frac{{d}^{3}\epsilon \left(n\right)}{{dn}^{3}}=\sum _{i=3}^{N}{i \left(i-1\right)\left(i-2\right)K}_{i}{(n-{n}_{f})}^{i-3}\ge 0\) Eq. 8
From a theoretical point of view, the problem is solved by determining the constants \({K}_{i}\) in Eq. 5 by using the least squares method and satisfying the unilateral constraints (Equations 6–8). For this purpose, a constrained least squares method is used, so theses constraints are polynomial inequalities of \(N-1\), \(N-2\)and \(N-3\) order, respectively. By calculating them in the flex point \({n}_{f}\) it is obtained:
\(\frac{d\epsilon \left({n}_{f}\right)}{dn}={K}_{1}\ge 0\) Eq. 9
\(\frac{{d}^{2}\epsilon \left({n}_{f}\right)}{{dn}^{2}}={2K}_{2}\ge \le 0 :n\ge \le {n}_{f} \to {K}_{2}=0\) Eq. 10
\(\frac{{d}^{3}\epsilon \left({n}_{f}\right)}{{dn}^{3}}=6{K}_{3}\ge 0\) Eq. 11
Therefore, both \({K}_{1}\) and \({K}_{3}\) must be non-negative, whereas \({K}_{2}\) must be equal to zero.
By replacing Eq. 10 in Eq. 7 and taking the term \((n-{n}_{f})\) out of the summation, constraint from Eq. 7 can be written as:
\(\frac{{d}^{2}\epsilon \left(t\right)}{{dt}^{2}}=(n-{n}_{f})\sum _{i=3}^{N}{i \left(i-1\right)K}_{i}{(n-{n}_{f})}^{i-3}\ge \le 0 :n\ge \le {n}_{f}\) Eq. 12
which can be simplified by dividing by \((n-{n}_{f})\), thus obtaining:
\(\sum _{i=3}^{N}{i \left(i-1\right)K}_{i}{(n-{n}_{f})}^{i-3}\ge 0\) Eq. 13
In summary, the algorithm that solves the problem consists in the search for a constrained minimum. The minimum of the sum of the squared deviations between the values \(\epsilon \left(n\right)\) provided by the model given in Eq. 5 and the corresponding experimental values, ɛ will be sought, in compliance with Eq. 10 and inequalities provided in Equations 6, 8, 9, 11 and 13.