Fatigue analysis of shape memory alloys by self-heating method
Introduction
Shape memory alloys (SMAs) are metallic alloys which can remember their original shape after being deformed. They can transform between two solid phases, austenite and martensite, and three different states: austenite, twinned martensite, and detwinned martensite [1]. SMAs can show two extraordinary behaviors: the shape memory effect (SME) and superelasticity (Pseudoelasticity). For the SME, an SMA in a twinned martensite structure transforms into the detwinned one under loading and shows a comparatively large amount of strain. After being unloaded and then heated, it transforms into the high-temperature austenite phase which results in the recovery of its inelastic deformation. Cooling the alloys under this circumstance does not change its shape although the SMA transforms again from austenite to the low temperature twinned martensite state. An SMA shows its superelasticity when it is austenitic at the beginning of loading. In this situation, loading of an SMA causes it to transform into detwinned martensite directly and to show a rather large strain. Upon unloading, it spontaneously returns to its austenite phase and recovers its inelastic strains. These special behaviors make SMAs suitable for many industrial and medical uses [2]. For instance, superelastic SMAs are used to produce endo-files whose superelasticity allows them to be adapted to anatomy of the root canals [3], [4].
In many of its applications, an SMA experiences fatigue loadings. Thus, it is necessary to conduct a thorough study on the fatigue of SMAs (Mahtabi and Shamsaei [5]; Song et al. [6]). However, it is costly to use conventional fatigue tests to find fatigue properties of these alloys [7]. Also, these tests are time-consuming [8] so there is a need for a faster and, at the same time, cheaper fatigue test for these alloys [9]. In recent years, a fast technique has been developed for determination of fatigue properties of materials. This method is based on measurement of temperature elevations (Poncelet et al. [10], La Rosa and Risitano [9], and Curà et al. [11]) and is done by imposing series of loading-unloading cycles to a specimen. Each series of cycles has a fixed stress amplitude, which is increased when switching to the next series of cycles [12], [9]. This method is based on the fact that temperature of a specimen varies under loading [9]. Change in temperature is more considerable when the amplitude of the imposed stress is higher than the fatigue limit of the specimen [9]. Based on this phenomenon, during each series of cycles, changes in the temperature of the specimen are observed and, when the variations cease, the stable value is plotted versus the amplitude of the corresponding stress [12], [9]. This method was initially empirical, and relation between the predicted and actual fatigue lives of a sample was not completely justified. However, later, modeling approaches permitted to create a link between these life times.
According to the method proposed by La Rosa and Risitano [9], in order to obtain the fatigue limit of specimens, one should record the stable temperature associated with different stress amplitudes (Fig. 1-a) and plot them versus each other (Fig. 1-b). As is shown, intersection of the fitted line with the horizontal axis (with zero temperature change) will give the fatigue limit. To measure the temperature elevations, Curti et al. [13] proposed the so-called thermography method which consists of measuring changes in the surface temperature of a specimen with an infrared camera to suppress possible disturbances of measurement tools [9]. After conducting thermographic tests on steel samples, Luong et al. [14] concluded that, for some materials under loading amplitudes less than fatigue limit, increase in temperature is small but not negligible. Thus, they proposed to fit two lines in order to find the fatigue limit; one for small temperature elevations and the other for large ones. They concluded that intersection of the two lines gives the fatigue limit of the sample [11]. Curà et al. [11] extended this method by expressing that the intersection of the two lines must be found by trial and error. Doudard et al. [15] proposed a probabilistic two-scale model for high cycle fatigue [16], [17]. The main characteristic of their approach was that it could consider both the sample's failure and thermal effects. In their model, they assumed damage under high cycle fatigue to be in microscopic scale and to originate from microplasticity. Based on this assumption, they considered an RVE consisted of a macroscopic elastic matrix inside which there exist microscopic elasto-plastic inclusions whose number increases by increase in stress amplitude according to Poisson distribution. This model was capable of predicting S/N curve of a dual-phase steel for any arbitrary failure probability. They used self-heating tests to identify the parameters of their model. Further, Meneghetti [18] used the volumetric dissipated heat to predict the fatigue strength of stainless steels and claimed that this approach did not depend on geometry of the sample. Moreover, Amiri and Khonsari [19] used the curve of temperature elevation versus time to find the fatigue life of aluminum and stainless steel specimens under bending fatigue. Roué et al. [20] considered a two-scale model and used self-heating method to predict the high cycle fatigue properties of aeronautical alloys (Ti-6Al-4 V in their case) at different temperatures up to 450 °C. Their method is able to measure small temperature elevations in specimens at high temperatures. They concluded that, by increasing the temperature, the high cycle fatigue properties of their tested material decreased.
Although self-heating method has been validated for different metallic materials [12], its use for shape memory alloys is new. Legrand et al. [3] validated the use of self-heating method in NiTi by comparing the classic Wöhler curves with the results obtained from self-heating [21]. They also proposed the first ideas for self-heating modeling in NiTi to create a link between fatigue and self-heating test in this alloy. Casciati et al. [22] employed self-heating method for NiTi wires with different diameters and concluded that the fatigue behavior of NiTi wires with large diameters is different from that of the ones with small diameters. Moreover, Rokbani et al. [23] studied hydrogen effect on high cycle fatigue of NiTi commercial orthodontic wires using self-heating method. Their findings showed that hydrogen has a considerably high impact on thermal properties of NiTi and causes loss of ductility and shape memory properties of the alloy. By using self-heating to empirically predict fatigue properties of NiTi, they showed that hydrogen seriously affects fatigue properties of this alloy and creates heterogeneity in the samples.
In this paper, Doudard et al. [15] two-scale model for high cycle fatigue is used and altered in order to predict S/N curves of an SMA sample for any arbitrary failure probability by using SMA's one-dimensional thermomechanical constitutive equations. In current study, damage in high cycle fatigue is assumed to be originated from micro-transformation i.e., the sample's phase varies in the microscopic scale after being loaded. Parameters of the model are identified with the help of self-heating tests which results in a faster identification procedure. Moreover, the influence of each parameter is investigated using numerical simulations. To validate the numerical results of the model, theoretically-predicted S/N curves are compared with experimental data resulted from classical fatigue tests. Finally, the proposed model is extended in order to investigate specimens with surface treatments made by hydrogen charging. This introduces heterogeneity to the outer layer of samples. The theoretical results are shown to be in a good agreement with experimental findings.
Section snippets
Experimental findings
In this section, experimental tests conducted on both homogeneous and heterogeneous samples will be proposed. Moreover, fatigue limits of these samples will be obtained empirically and will be compared with the ones from classical fatigue tests.
Modeling
In this section, a deterministic two-scale model will be proposed followed by an extension to achieve a probabilistic two-scale model for high cycle fatigue of SMA wires.
Identification and validation by fatigue prediction
In this section, first, identification of the existing parameters in the proposed model will be presented. This process was done using the outcomes of the self-heating test. After that, verification of the theoretical results will be demonstrated with the help of classical fatigue test findings.
Conclusion
In this paper, a two-scale model together with self-heating method was developed to determine fatigue characteristics of shape memory alloy wires. The proposed model is capable of predicting the fatigue curve of these alloys for any arbitrary failure probability. The use of self-heating method in this model speeds up the identification process, leading to a cost-efficient method to determine the whole fatigue curve in SMAs. For instance, conducting a self-heating test with a frequency of 30 Hz
Declaration of conflicting interests
The Authors declare that there is no conflict of interest
Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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