1. Introduction
During operation, machine parts are frequently subjected to fatigue loading at elevated temperatures. Such operating conditions have an adverse effect on the materials’ behavior. An elevated temperature accelerates wear processes and contributes to increasing the rate of fatigue crack propagation [
1,
2]. Aluminum alloys, e.g., EN AW 2024T3, are among the materials used to make constructions capable of working at an elevated temperature. It is widely used in the motorization and aviation industries, which place strong emphasis on ensuring high strength and light weight at the same time [
3]. These features make it suitable for the manufacture of parts working in close proximity to combustion and jet engines where elevated temperature zones are present. In the work of Szusta and Seweryn [
4,
5], the influence of elevated temperature on the fatigue life of this material was tested. The authors demonstrated that strength parameters decreased as the test temperature increased. Based on the tests conducted, the authors determined the parameters of the Manson-Coffin fatigue life curve as a function of temperature:
where
Nf is the number of cycles until failure;
E(
T) is Young’s modulus determined at temperature
T;
σf′(
T) and
εf′(
T) are, respectively, coefficients of the elastic fatigue life curve and plastic fatigue life curve for the analyzed temperature
T; and
b(
T) and
c(
T) are, respectively, exponents of the elastic and plastic fatigue life curve for temperature
T. Furthermore, based on observations of the material’s cracking mechanisms at an elevated temperature, the authors proposed a semi-empirical model for estimating the material’s fatigue life. In this model, it was accepted that the material’s fatigue life drops as the test temperature (isothermal) increases. The lower the material’s melting temperature, the more substantial this drop is. Moreover, it was assumed that there exists a limit temperature
Tm at which the given material no longer has any immediate strength. Therefore, knowing the value of this temperature and the function defining the reduction in fatigue life as temperature increases, calculations of the number of cycles until crack initiation can be performed with engineering accuracy for a given temperature level and strain amplitude. For uniaxial tensile-compressive loads, the function of the temperature’s influence on the material’s fatigue life has been defined as follows:
where
A and
B are material constants determined in periodical tensile-compression tests at temperature
T.
In this model, the material constant
B is a function of the material’s recrystallization temperature
Tm (
B = −
ATm). The value of parameter
A is calculated on the basis of the experimentally determined number of cycles to failure (obtained for a given load course at room temperature) related to the value of temperature difference: the material recrystallization and room temperature. Parameter
A is determined from the following equation:
where
is determined from Manson–Coffin–Basquiun equation at room temperature
RT for the set total strain amplitude
.
Modified cast irons and cast steels, which are used in the manufacture of brake disks, are another group of materials that can be exploited at elevated temperatures. In the works of Samec et al. [
6], Peve et al. [
7], and Li et al. [
8], the authors analyzed the fatigue life of the following materials: EN-GJS-500-7, EN-GJL-250, and Cr-Mo-V. Material cracking mechanisms were studied within the LCF range at temperature up to 700 °C, and the influence of temperature on the material’s physical properties was defined.
Alloy steels are another group of materials used to make constructions which are exposed to the action of elevated temperatures. Here, some examples include 8Cr-2WvTa, (RAFM) JLF-1, or 617M steels, which are used to build thermoreactors and machine parts exposed to the action of high temperatures. In the works of Ishii et al. [
9], Mariappan et al. [
10], and Shankar et al. [
11], the characteristics of these steels’ cyclic properties were determined as a function of temperature. Degradation mechanisms of materials subjected to the action of fatigue loads and temperatures were analyzed. In the article by Ishii et al. [
9], an approach to modeling fatigue life of the material was also presented, accounting for the influence of creep on the material’s fatigue life:
where
εeq is the equivalent strain,
εe is the elastic strain,
εpl is the plastic strain, and
εcr is the creep-induced strain.
Works by Nagode and Zingsheim [
12] and Nagode and Hack [
13] present the results of experimental tests performed on steels for work at elevated temperatures: 10 CrMo 9 10 and X22CrMoV121. Fatigue damage accumulation models of the material under non-isothermal conditions at elevated temperatures were developed on the basis of these results. These models were developed for estimation of the fatigue life of materials which undergo changes of the load amplitude and temperature transient over the course of its exploitation. These models do not account for the influence of creep or hardening of the material on its fatigue life. The authors presented a stress and strain approach in which it is assumed that the instant working temperature will be accounted for. In models, stabilized hysteresis loops were described by means of the Ramberg–Osgood equation. The damage parameter
PSWT was identified similarly as in the SWT (Smith–Watson–Topper) model:
where
Te is the actual temperature;
σ′
f(
Te) and
b(
Te) are, respectively, the coefficient and exponent of the fatigue life curve at the tested temperature;
K′(
Te) and
n′(
Te) are, respectively, the coefficient and exponent of the cyclic hardening curve at the tested temperature; and
ε′
f(
Te) and
c(
Te) are, respectively, the coefficient and exponent of the plastic fatigue life curve.
Elements of gas turbines, e.g., rotor blades working under extremely difficult conditions, are also made from Inconel 718 alloy. Test results for this material was given in the work of Schlesinger et al. [
14]. Here, a dispersion-hardening nickel-chromium alloy was analyzed with respect to fatigue loading at temperatures up to 650 °C. Similarly, in a different study [
15], the same authors studied the Inconel 718 alloy. They proposed a model for estimation of the fatigue life of material working at elevated temperatures. The Manson–Coffin equation was used during modeling, in which parameters of the fatigue life curve were made functionally dependent on the structure and grain size in the given material
G, test temperature
T, and strength parameter
S:
Materials adapted for manufacturing tools like casting molds and forging dies play an important role among materials intended for work at elevated temperature. In the study by Tunthawiroon et al. [
16], the single-phase Co-29Cr-6Mo steel alloy, intended for aluminum casting molds, was tested. The study included analysis of the alloy’s cracking mechanisms at temperatures up to 700 °C. A simple fatigue damage accumulation model at elevated temperatures was also proposed based on the Arrhenius equation:
where
Nf is the number of cycles until crack initiation,
T is the test temperature,
R and
A are material constants, and
Q is the activation energy.
Similarly, the works of Gopinath et al. [
17] and He et al. [
18] present the results of experimental tests conducted on 720Li and HAYNES HR-120 superalloys within the low-cycle fatigue range at temperatures up to 980 °C. The material was tested under uniaxial tensile-compressive loads. The mechanisms accompanying the cracking of these materials were defined.
ACI HH50 austenitic stainless steels can also work at elevated temperatures. The results given in the work by Kim and Jang [
19] can serve as evidence of this. Due to its high resistance to elevated temperatures, high fatigue strength, and resistance to pitting and corrosion, the material in question finds applications in parts of combustion engines and power unit structures of nuclear power plants. The paper investigates the material’s degradation mechanisms and presents a model estimating fatigue life in the LCF range. The model assumes that fatigue crack initiation will occur when damage reaches a limit value given by the following function:
where
α is the material constant associated with dislocation movement over the course of the material loading process,
ν is the frequency of load change during the test,
0 and C are material constants independent of test temperature,
Q is activation energy,
R is the gas constant, and
T is the test temperature.
The non-linear creep damage accumulation model under uniaxial loads at elevated temperatures (700 °C) that was also developed on the basis of tests performed on stainless steel X-8-CrNiMoNb-16-16 was presented by Pavlou 2001 [
20]. The model was developed and verified by other researchers. In the work of Grell et al. [
21], attempts have been made to estimate durability under the conditions of uniaxial isothermal loads (constant and incremental) at elevated temperatures using the dependencies proposed by Pavlou 2001 [
20]. The work demonstrates greater accuracy in predicting durability with the use of non-linear models of the accumulation of defects in relation to linear models.
Titanium alloys are the next group of materials used to manufacture components working at elevated temperatures. One study [
22] analyzes the influence of different methods of the surface hardening of the Ti-6Al-4V titanium alloy on the fatigue life of this material under uniaxial tensile-compressive loading conditions at temperatures up to 555 °C. The influence of temperature on the evolution of the cyclic properties of the TNB-V2 titanium alloy is also presented in the work [
23]. Specimens of the material were subjected to constant-amplitude strains within the low-cycle loading range at temperatures within the range of 550–850 °C. Due to its good mechanical properties at elevated temperatures, this alloy finds applications in motorization and aviation and in engine parts. The authors tested the material within the entire safe temperature range, and determined that, in the case of this alloy, the Manson–Coffin fatigue life model does not allow for the prediction of the material’s fatigue life at elevated temperatures due to the length of the phase between fatigue crack initiation and its propagation.
The fatigue damage accumulation model proposed in this paper was designed to predict the fatigue life of the material operated at elevated temperatures. The concept of elevated temperature can be defined here, according to the works by Chen et al. [
24], as the temperature corresponding to two-thirds of the melting point of the bulk material
Tc. In most cases, as temperature increases, a material’s strength and fatigue properties are reduced in a predictable manner until a certain threshold temperature value is reached [
25]. Under these conditions, the operation of technical machinery may be safe when the material’s response to loading conditions and the working environment are known. However, if the conventional upper threshold of elevated temperature is crossed even slightly, the material’s properties cannot be predicted directly. The material’s strength suddenly drops (in a non-linear manner), and use of machinery under such conditions may pose a safety risk.
Uncontrolled loss of durability may lead to costly failures, standstills on process lines, and sometimes even to the death of those nearby when a structural component cracks. This is why information about how long and under what loading and temperature conditions a structural component can work safely is important. The fatigue damage accumulation models of materials working under low-cycle loading conditions at elevated temperatures presented in the literature are modified functions formulated to estimate the material’s fatigue life at room temperature. The equations mentioned earlier can serve as an example: Manson–Coffin–Basquin, Ramberg–Osgood, and Smith–Watson–Topper. In these models, material parameters are determined independently for each of the analyzed temperatures. The process of calculating these parameters is not complicated in itself; however, preparing the data for determining them is very expensive and time-consuming and requires specialized testing apparatus. These models are dedicated for specific loading cases and metal alloys. Due to the low number of functioning fatigue criteria in this scope, there is a justified need to create new models that will provide a more accurate description of the material cracking process, thus enabling calculation of fatigue life with greater accuracy. That is also why it this paper set out to create a new fatigue damage accumulation model that will make it possible to determine the number of cycles until failure of a material working under conditions of fatigue loads of constant amplitude at elevated temperatures with engineering accuracy.