1. Introduction
Non-monotonic alternation of stress states may lead to permanent changes in the structure of materials and is often the cause of the limited functionality of components. The vast majority of machines and engineering structures are subjected to complex and time-varying operational loads [
1,
2,
3,
4,
5,
6]. Generally, the loads can be dynamic, i.e., caused by forces and moments varying in time. They can also be kinematic loads, caused by time-varying displacements and rotations. In both cases, a complex field of strain and stress arises in the structure. Industrial examples, where such loading scenarios are important, include heavy vehicles used for landscaping, such as diggers or crushers. Operation of such equipment requires completing target kinematic displacement against a uncontrollable and unpredictable dynamic reaction from soil or processed material. A reverse case example can be found in many torsionally loaded shafts, where the kinematic response is stochastic, such as large bore drills in the oil and gas industry.
The fatigue properties of construction materials under uniaxial loads are best described in the literature [
7,
8,
9]. However, due to the fact that in reality complex loading states occur in machines and structures, it is more desirable to test the complex state of loading complexity [
10,
11,
12]. The issues that are currently raised in the literature regarding fatigue of materials are the analysis of the correlation between the material microstructure and fatigue life [
13,
14] as well as issues on the border of fracture mechanics and surface quality [
15,
16]. Under the influence of non-proportional loadings, the principal stress and strain directions change cyclically [
17,
18] and, as proven by various studies [
17,
19], the fatigue life of a given structure can be significantly reduced. A number of studies on a certain group of materials [
20] showed a clear impact of material non-proportional hardening sensitivity on the observed fatigue life.
Most experimental studies conducted under multiaxial fatigue load conditions were performed for independent tension: compression and torsion [
21,
22]. The results of material testing under other loading conditions, for example, for bending with torsion, appear much less frequently in the literature. In most cases, the tests were performed phase-aligned with controlled values of bending and torsional moments amplitudes (dynamic load) [
23,
24,
25,
26,
27]. Testing of materials subjected to bending and torsional loads, with controlled kinematic load, is much less common; however, some studies [
28,
29] analyzed essentially aligned kinematic loads causing bending and torsion. Literature sources on material testing in both possible load control variants (dynamic and kinematic) is lacking. In this context, the knowledge of fatigue of materials subjected to phase-dependent kinematic or dynamic loading seems to be underdeveloped, especially given potential significance to many structures and mechanisms. As a part of a research program, material specimens were tested with controlled, with independent kinematic or dynamic loads. A custom-built fatigue testing machine was implemented to facilitate this project. Similar to typical tensile–compression tests with torsion, fatigue tests were performed for both out-of-phase loads and in-phase loads for bending and torsion alone and combinations thereof.
The main objective of this study was to examine the fatigue properties of S355J2 steel using an energetic description. In this work, fatigue tests were conducted on specimens in kinematic and dynamic scenarios with four paths of loadings. The cyclic stress–strain response of the fatigue-tested specimens were analyzed using finite element method (FEM) analysis with the Chaboche model of cyclic plasticity. We determined the effectiveness of the most popular energy models for many variants of cyclic loads based on the original method of determining specific strain energy components to describe fatigue life. The fatigue lives of these specimens were compared between kinematic and dynamic loading test conditions.
3. Results and Discussion
Table 4 and
Table 5 summarize the results of the numerical calculations of the elastic–plastic strain energy Equation (
11) and the fatigue energy parameters from Equations (
5) to (
10). ASTM E739 standard recommendations were used to assess suitability of the selected fatigue parameter for describing the performed fatigue tests. Following ASTM standard guidelines [
47], a linear regression model was adopted (presented in a double logarithmic system) to assess the suitability of the selected fatigue parameter
where
is durability expressed in cycles,
W is the value of the strain energy or fatigue energy parameter, and
A and
B are parameters calculated for this regression model.
Table 6 summarizes the calculated values of the model parameters in Equation (
16) and the values of the correlation coefficient
.
Values of differ depending on the mode used for specimens loading. For kinematic loads, the values are similar and not less than , with the exception of the Glinka model, where . For dynamic loads, the values of are significantly lower, in the range of to .
All considered energy models achieved good correlation with experimental data regardless of loading path, but only within each loading mode used in the experiments (either kinematic or dynamic).
Figure 13a,b,
Figure 14a,b, and
Figure 15a,b show the calculated values of energy parameters (Equations (
5)–(
9)) and strain energy (Equation (
11)) as a function of fatigue life. For all energy models, we noticed that the values obtained in kinematic loading mode were clearly separated from the results acquired in dynamic loading mode. Both groups formed distinct scatter bands in charts, which seemed to be broadly parallel to each other.
All models described above use the work of stress on strains. However, only the Ellyin model considers the hysteresis loop directly in the calculation of strain energy. The SWT, Liu, Chu, and Glinka models propose the use of certain values describing the state of stress and strain that are relatively easy to calculate. However, according to Liu [
40], their relationship with the plastic strain energy is abstract. It is especially obvious for non-proportional loads, for which these models were not originally intended for use.
Evaluating the results presented in
Table 5 and
Table 6, we observed that the values of Liu’s energy parameters were always greater than strain energy calculated from Ellyin’s models. This was noticeable for both loading modes: kinematic and dynamic. For the former, the SWT, Chu, and Glinka energy parameter values were smaller than strain energy calculated from Ellyin’s equations. For the latter mode, a different situation occurred. In this case, both SWT and Chu’s parameters return values close to Ellyin’s, whereas results calculated according to Glinka are noticeably larger.
Comparing the durability under different load methods (kinematic and dynamic), we found that the value of elastic–plastic strain energy or any energy parameter can be significantly different for the same achieved fatigue life.
Searching for the underlying reasons for the difference in strain energy between kinematic and dynamic loading modes, we directly analyzed the time courses of the moment of force and displacements of bending and torsion levers.
Figure 16 and
Figure 17 depict the total work imputed into the excitation system: levers, holder, and specimen. The rigid design of the test stand ensured that only an insignificant amount of that energy was spent outside of the specimen, dissipated through inter-component clearances, fastened connections, bearings, etc.
Figure 16 presents the curves
showing the relationship between the life
of the specimen, displacement
, and the moment of force
recorded at the levers. Two example specimens were selected, H5 (kinematic mode) and P11 (dynamic mode), which exhibited similar fatigue life. The curve marked in red
and the curve marked in green
correspond to bending and torsion lever in kinematic mode, respectively. This chart also depicts the dynamic mode results, where the black curve
indicates bending and the blue curve
denotes torsion.
Figure 17b illustrates that the tests conducted under kinematic load (lines in red and green) returned a near constant value of amplitudes of moments of force for approximately
of the the fatigue test duration, only to start dropping near the end of the test. The amplitude of displacements (
Figure 17a) did not change throughout the experiment in this mode of loading (kinematic). Throughout the first
of the test, instantaneous strain energy was constant; however, as the moment of force decreased in the final stage of the experiment, so did the strain energy.
A similar effect was observed for tests under dynamic loading. However, in this case, the observed effects were different.
Figure 17a depicts an increase in the amplitude of both lever displacements (black and blue curves) at a constant value of moment loads (
Figure 17b). Notably, in both figures, the black lines correspond to the bending and blue to the torsion. Here, at the conclusion of the tests, the amplitude of displacements increased. Since the values of the moment of force amplitude did not change (
Figure 17b), the value of the strain energy must increase when dynamic load mode is used.
Among the considered energy fatigue parameters, none consider the temporal transiency of strain tensor and the corresponding stress tensor. As a consequence, the values of strain energy (stress work on strains) were calculated with these models for a given time instance. Thus, the results ignore the changes in the values of the strain and stress tensor components during the fatigue process, affecting the strain energy or any energy parameter calculations.
The numerical constitutive cyclic-plasticity model (Chaboche) used in calculations does not consider an important transiency in material properties caused by damage accumulation or crack initiation, propagation, etc. As a result, the obtained strain and stress tensor values are likely impacted.
A second limitation of the considered energy fatigue models is related to their spatial sensitivity (local effect). This is understood as the need to calculate strain energy only at the most stressed point in the smallest cross-section of the specimen. This, in addition to sampling at an arbitrarily determined time instance, cannot provide complete information on the distribution of strain energy in a representative volume of the specimen. This volume can be considered to be subjected to plastic strains.
Figure 18a depict the distribution of mesh nodes at the specimen neck during in-phase kinematic loading. The numerical model expressed the highest levels of stress at the two opposing node regions located on both sides of the neck (marked in red). The plastic stain distribution for this loading case is illustrated in
Figure 18b.
In comparison,
Figure 19a demonstrates the corresponding distribution of mesh nodes during out-of-phase kinematic loading. Here, the highest stress concentration at the nodes form a ring encircling the narrowest cross-section of the neck. This region corresponds directly to the plastic strain distribution region, as presented in
Figure 19b. Similar maximum stress patterns and plastic strain regions were observed for dynamically loaded numerical simulations.
The numerically obtained values of strain energy and energy parameters were expected to be correctly estimated by a single regression model, regardless of loading path and irrespective of loading mode (dynamic or kinematic). Only the former of these assumptions was fulfilled, suggesting that the evaluated common fatigue models might be not entirely applicable in every load case.