Estimation of Critical Fatigue Conditions Based on the Accelerated Fatigue Locati Method by Mean of Net Damage

Abstract

The increasing utilization of short fiber-reinforced thermoplastics, due to their advantageous mechanical properties and manufacturing convenience, has led to their application in areas traditionally dominated by metals. This shift underscores the importance of understanding the fatigue behavior of these materials. This study evaluates the fatigue behavior of short fiber-reinforced thermoplastics through three characterization methods: continuous fatigue, interrupted fatigue, and the Locati method, with the latter serving as a novel approach for estimating critical fatigue conditions from a single specimen. Continuous fatigue testing provides the baseline for comparison. The effect of load interruption is explored through the interrupted fatigue method. The Locati method, characterized by incrementally increasing load steps until failure, offers a significant benefit by enabling the estimation of critical fatigue conditions efficiently. This research aims to provide a comprehensive understanding of the fatigue behavior of short fiber-reinforced thermoplastics, contributing to the optimization of their use in engineering applications.

1. Introduction

In recent years, short fiber-reinforced thermoplastics (SFRTs) have gained importance, especially as they have begun to replace metal elements in certain applications, primarily due to their ease of fabrication and superior mechanical properties [1]. Polyamide 6 (PA-6) is one of the most widely used materials in this category due to the following characteristics: its ease of manufacture by injection moulding, high strength, stiffness, toughness, translucency, fatigue resistance, and abrasion resistance [2]. Enhancing PA-6 with short glass fibers improves its strength, stiffness, heat distortion temperature, and abrasion resistance, though it may introduce anisotropic properties [3]. Given the common use of this material in engineering components, understanding its fatigue behavior is crucial.

Traditionally, the fatigue limit or endurance—the stress level below which a material can withstand an infinite number of loading cycles without fatigue failure—has been determined from Wöhler curves or S-N curves [4,5]. This approach, however, requires between 16 and 20 specimens to yield reliable results. An alternative, the Staircase method, can achieve similar outcomes with only 11 to 15 specimens [6,7]. This paper proposes using the Locati methodology [8,9,10] to achieve results that are both rapid and cost-effective. The Locati method, an accelerated fatigue test, aims to determine the fatigue limit from a single specimen by applying incrementally increasing stress loads until failure.

Various researchers have employed the Locati method to ascertain the fatigue limits of different materials. Nonetheless, a unified criterion for defining the fatigue limit based on Locati test results has yet to be established. The fatigue effect of laser quenching on the boronized layer of Cr12MoV steel was investigated by Kong et al. [11] through the analysis of fatigue accumulation damage. Maximov et al. [12] studied the high-cycle fatigue of 2024-T3 high-strength aluminum alloy, estimating endurance based on the Palmgren–Miner linear damage theory. Sainz-Aja et al. [13] utilized μCT analysis alongside the Locati method to determine the fatigue limit of recycled aggregate mortar, and in another study, the variation in resonance frequency to define the fatigue limit of recycled aggregate concrete [14]. Casado et al. [15] analyzed the fatigue failure behavior of short glass fiber-reinforced PA 6.6, using strain evolution to determine the critical step. The Locati method was also applied by Thomas et al. [16] to identify the fatigue limit of recycled aggregate concrete, establishing a correlation between the maximum load a specimen can withstand and its fatigue limit.

Traditionally [17,18], damage formulation is based on the concepts of stress and actual strain. Effective stress reflects the material’s response to net section reduction caused by various damage mechanisms, allowing damage (D) to be defined from Equation (1).

$$D=1-\frac{\tilde{E}}{E}$$

(1)

Within this framework, E represents the material’s undamaged elastic modulus, while Ẽ denotes the apparent elastic modulus of the material once damage has occurred. The damage parameter (D) quantifies the extent of material deterioration, ranging from 0 (indicating no damage) to a critical damage level (D_c) corresponding to failure.

Fatigue-induced damage in polymeric composites manifests through the formation and propagation of discontinuities, including micro-cracks, voids, and cracks, which progressively undermine the material’s strength leading to failure [19]. This damage can be conceptualized as the variation in energy relative to the initial state at each moment.

In this study, a model to predict fatigue behavior was developed, aimed at characterizing materials swiftly, economically, and with high precision. This model is anchored in the Locati method and accommodates pauses during fatigue tests and the application of cycles with varying load amplitudes. Through a comprehensive experimental campaign, the damage incurred by each test specimen was evaluated from an energetic perspective. Subsequent analyses incorporated corrections to account for the influence of breaks and fluctuating load intensities.

2. Materials and Methods

The specimens analyzed in this study comprised a composite material consisting of Polyamide 6 (PA-6) reinforced with 25 weight percent (wt.%) short glass fiber (SGFR-PA6). These specimens were fabricated using injection molding techniques and conformed to standardized dimensions and geometry Type 1B as specified in EN ISO 527-2 [20]. The specific dimensions of the specimens are illustrated in Figure 1.

Polyamide 6 (PA 6) is inherently hygroscopic, with moisture absorption having a notable impact on its mechanical properties, particularly its dynamic attributes. To mitigate this and standardize the condition of the specimens, they were subjected to a conditioning process to achieve a moisture content of approximately 2.4%. Under these controlled conditions, the specimens demonstrated a static tensile strength of 100 MPa and a breaking strain exceeding 5%.

2.1. Tensile Tests

Tensile testing of Polyamide 6 (PA6) specimens was conducted to determine the material’s mechanical properties under controlled uniaxial deformation conditions, following the guidelines of EN ISO 527-1 [21] and EN ISO 527-2 [22] standards. These standards specify the procedure for preparing specimens, testing conditions, deformation speed, specimen geometry, and results’ interpretation for plastics. Specimens are typically prepared by injection molding or machining, with a focus on maintaining proper moisture content due to PA6’s hygroscopic nature. The test is performed at a standard temperature of 23 ± 2 °C and 50 ± 5% relative humidity, unless specified otherwise, with data on applied force and specimen deformation collected to calculate elasticity modulus, yield strength, maximum tensile strength, and elongation at break. This process ensures a standardized approach to evaluating PA6’s mechanical performance, facilitating comparison across different conditions or formulations.

2.2. Fatigue Tests

In all fatigue tests performed for this work, a sinusoidal waveform was used, the stress ratio (R) was 0.1, and the frequency (f) was 5 Hz. The experimental campaign was divided into 3 phases: monotonous fatigue, monotonous fatigue with breaks, and Locati tests. The Locati tests were also divided into 4 stages: a reference Locati, analyzing the influence of breaks, analyzing the influence of the number of cycles per step, and analyzing the effect of the load values of each step. These tests were performed in a servo-hydraulic machine with a capacity of 5 kN and a dynamic extensometer with a measuring base of 50 mm and a displacement of 12.5 mm to measure the strain of the specimen.

In the first phase, monotonic fatigue tests were performed until the specimens broke. These tests were performed with maximum load values between 1400 and 2700 N. The load variation between tests was 100 N. These tests were performed in order to define the Wöhler curve and the fatigue limit (Δσe).

In the second phase, monotone fatigue tests with breaks were performed. In these tests, every 25,000 cycles the test was stopped for one hour. This period of time was enough for the specimen temperature to reset to the room temperature value. These interruptions allow the analysis of the behaviour of materials previously damaged by fatigue.

In the third phase, Locati accelerated fatigue tests were performed. This methodology consists of applying a fixed number of cycles at a given frequency. The load values of these steps are increased step by step. Figure 2 shows a diagram of the test and

Table 1. Load values during the Locati test steps.

Step.	Fmax (N)	Fmin (N)	σmax (MPa)	σmin (MPa)	Δσ (MPa)
1	1200	120	30	3	27
2	1300	130	32.5	3.25	29.25
3	1400	140	35	3.5	31.5
4	1500	150	37.5	3.75	33.75
5	1600	160	40	4	36
6	1700	170	42.5	4.25	38.25

shows the variables used.

As previously mentioned, the Locati method tests were carried out in four different stages. The first of these was used as a reference, with 25,000 cycles per step, with a maximum load variation between consecutive steps of 100 N, and without making breaks. The second one was similar to the reference one except that, between steps, a 1 h stop was made. The third was similar to the reference except that the load increase between consecutive steps was reduced to 50 N. Finally, in the fourth stage, the difference from the reference test was that the number of cycles per step was doubled. See

Table 2. Locati test variants.

Stage.	Break? (Yes/No)	Cycles/Step (Cycles)	ΔFmax (N)
Locati 1	No	25,000	100
Locati 2	Yes	25,000	100
Locati 3	No	25,000	50
Locati 4	No	50,000	100

.

2.3. Energy and Damage Quantification

During the fatigue process, the energy in the volume of material consisting of a prismatic base cross-section of the specimen and height being the separation between the reference points of the extensometer, within a cycle and between two instants of time t_i and t_i+1 is obtained from Equation (2).

$$E_{t_i/t_{i+1}}={\displaystyle {\int }_{t_i}^{t_{i+1}}dE}={\displaystyle {\int }_{t_i}^{t_{i+1}}F·d\Delta }$$

(2)

For a whole cycle, this total energy can be decomposed into: dissipated energy represented by the hysteresis loop (E_d) and accumulated energy (E_a), which is recovered by the element at the end of the cycle, as shown in Figure 3.

Therefore, as shown in Equation (3), the total energy per cycle, E_T, is the total of the two previous ones. This E_T will be the energy used to determine the damage generated in the fatigue process.

$$E_T=E_a+E_d$$

(3)

A parameter called total damage, T_D, is defined to quantify the damage suffered by the material during the fatigue process. The definition of this parameter will be made from the variation in energy parameters with respect to their initial value and will be obtained from the expression:

$$T_D=\left(1-\frac{E_{T0}}{E_{Ti}}\right)·100 [\%]$$

(4)

where E_T0 will be the total energy measured in the first of the fatigue cycles and E_Ti the total energy measured in cycle i.

Since the stress level throughout the Locati test is changing, a new parameter is defined, which will be the individual net damage (D_I) and will consider the variation in the total energy with respect to the initial one of each of the steps. See Figure 4.

$$D_{{NI}_j}=D_{I_j}-(D_{0_j}-D_{{NIf}_{j-1}})$$

(5)

where D_NIfj−1 is the individual net damage value of the previous step.

3. Results and Discussion

The main results of the first phase tests, the monotonic fatigue tests and the specimens’ breaks, are shown in Figure 5 which gives the Wöhler curve. The fatigue limit was defined as Δσ = 40.5 MPa (180–1800 N) as all the analysed variables were stabilized after 2 × 10⁶ cycles.

Figure 6 shows the evolution of E_T as a function of the number of cycles for the different stress ranges.

Figure 7 shows that in the thermal fatigue processes, with higher stresses, E_T has a clear increasing trend, while in terms of mechanical fatigue, the energy parameters tend to stabilize or even decrease in some cases.

Figure 8 shows the evolution of total damage for the different stress levels.

It can be seen that for the fatigue limit, Δσ = 40.5 MPa, both E_T and T_D are fully stabilized after 2 × 10⁶ cycles. The damage in all cases followed the same pattern, with an initial rapid growth with a decreasing slope, zone D-I, then a linear growth of the damage, zone D-II, and finally an accelerated growth until break, zone D-III, as can be seen in Figure 8. Both the growth rate, ΔT_D/ΔN, and the orderly one at the origin of the adjustment of the section D-II, D₀, and the critical damage, real and theoretical, D_C and D_CT, are increasing with the stress level.

The second test phase is formed by fatigue tests with breaks and monotonic load values. By analyzing the results of monotonous fatigue with interruptions, see Figure 9, it can be seen that a part of the damage caused during a load step (25,000 cycles) is recovered during the resting period, so that, when the next step is started, the initial damage is less than the damage that ended the previous step. This means that the damage would be composed of an elastic component, which recovers when the cyclic stress is released, and another plastic component, which is the one that remains after the stress and the one that actually causes the material to deteriorate.

The elastic component of the damage will be identified with the one ordered at the origin of the section D-II, D₀, or initial damage, and therefore, a plastic or net damage, D_N, can be defined from Equation (6):

$$D_N=D_T-D_0$$

(6)

Figure 10 shows the evolution of net damage for all levels of monotonous fatigue without breaks. In contrast to the results obtained for total damage, where the critical damage values depended on the stress level, increasing as the stress increased, for net damage, it can be seen that in the case of thermal fatigue, the theoretical critical net damage values, D_NCT, are stabilised at around 15%, regardless of the stress. In the case of mechanical fatigue, the value of critical damage could not be verified because, as mentioned, the breakage took place through the clamp, thus not reaching the number of theoretical breakage cycles.

Figure 11 shows that in the case of fatigue breaks, the number of cycles is higher than in the monotonous one of the same levels, but the theoretical net damage also reaches values close to 15%.

The third experimental phase corresponds with the Locati tests. Analyzing the evolution of the total damage, D_T, in the Locati tests, there was no indication detected that allows it to be established as to which of the steps corresponds to the fatigue limit. Figure 12 shows, on a double-axis of abscissa to facilitate comparison between the different tests, the evolution of total damage, verifying that the results of the four tests are coincidental, reaching very similar values at break.

Similar behaviour can also be observed when analyzing the evolution of the phase angle between the load and strain signal, see Figure 13. It can also be seen in this figure that the value of the critical phase angle is similar in all cases and close to 0.12 radians, so this value could also be a constant in breakage.

Figure 14, Figure 15, Figure 16 and Figure 17 show the D_NI for the four Locati tests carried out. The step corresponding to the fatigue limit (Δσe) measured from the monotonic fatigue tests is indicated on the graphs. It can be seen that the D_NI in all cases remains constant below Δσe, and once exceeded, it begins to grow. Therefore, the fatigue limit can be defined from the individual net damage, D_NI, in a Locati test as that step from which the parameter begins to grow.

It is also observed that the theoretical critical value of individual net damage would be slightly higher than 15%, as already seen in the monotonous and interrupted fatigue tests.

4. Conclusions

This study has provided valuable insights into the fatigue behavior of short glass fiber-reinforced Polyamide 6 (SGFR-PA6), a material of increasing importance in applications traditionally served by metals. Our investigation, utilizing monotonous fatigue tests with and without breaks, alongside the Locati method, has led to several important conclusions regarding the material’s performance under cyclic loading conditions.

Firstly, the differentiation between initial (or elastic) damage and plastic (or net) damage in the total damage observed from monotonous fatigue tests with breaks underscores the material’s capacity for partial recovery between load applications. This phenomenon highlights the composite’s inherent resilience to fatigue, with the elastic component of damage being recoverable upon the release of cyclic stress. The introduction of a model to identify net damage through variations in total energy per cycle relative to initial conditions represents a significant advancement in our understanding of fatigue processes in composite materials. This model provides a robust framework for predicting the onset of critical damage, thereby enhancing the material’s application in engineering designs.

Secondly, our findings confirm that the net critical damage in thermal fatigue processes for SGFR-PA6 stabilizes at values close to 15%, irrespective of the stress level applied. This stabilization indicates a threshold beyond which further stress does not proportionally increase the rate of damage accumulation, suggesting a material property that could be exploited in designing components subjected to high cyclic loads.

Furthermore, the comparative analysis of the four Locati test variants reveals minimal influence from the number of cycles per step, the presence or absence of breaks, and the stress increase value between steps on the overall fatigue behavior of SGFR-PA6. This observation not only validates the efficiency of the Locati method in determining the fatigue limit but also suggests the potential for simplifying the test procedure without compromising the reliability of the results.

The ability to define the fatigue limit from the individual net damage (D_NI) in a Locati test, as demonstrated by the step from which this parameter begins to increase, offers a practical and accurate method for evaluating the fatigue strength of SGFR-PA6. This methodology promises to reduce the time and resources required for fatigue testing, facilitating quicker material evaluation and application in design processes.

In conclusion, the research conducted provides a comprehensive assessment of the fatigue behavior of SGFR-PA6, offering critical insights into its damage recovery capabilities, the stability of its critical damage level under thermal fatigue, and the effectiveness of the Locati method for rapid fatigue limit estimation. These findings contribute significantly to the field of materials science, particularly in the optimization and efficient use of short fiber-reinforced thermoplastics in engineering applications.

Future research will focus on exploring the Locati method under various environmental conditions, such as different temperatures and humidity levels, to better understand their impact on the fatigue behavior of SGFR-PA6. Additionally, we plan to test other types of short fiber-reinforced thermoplastics to determine the generalizability of our findings. Developing more advanced predictive models that incorporate real-time monitoring data and sophisticated error analysis techniques will also be a priority. These efforts will contribute to the broader goal of optimizing the use of short fiber-reinforced thermoplastics in engineering applications.