Investigation on the serrated flow behavior of bulk metallic glasses based on machine learning

This study proposed a long short-term memory (LSTM) model for predicting the serrated flow behaviors of bulk metallic glasses (BMGs) under nanoindentation. A series of load-controlled nanoindentation tests were conducted on a Pd40Cu30Ni10P20 BMG. The LSTM model was introduced to establish a neural network for predicting the serrated flow at different loading rates, and was verified by the comparisons of experimental data with predictive results. Further investigation based on the predictive serrated flows under different loading rates showed that the serrations exhibit a significant self-organized critical (SOC) phenomenon at different loading rates. The SOC phenomena of the serrations under a lower loading rate were more obvious than that under a higher loading rate.


Introduction
Bulk metallic glasses (BMGs) have received a great deal of attention due to their excellent properties, such as high yield strength, high elasticity limitation, large wear and corrosion resistance, and a measure of plastic deformation [1][2][3]. Therefore, BMGs are widely used as structural and functional materials in many high technology fields, such as precision machinery, aeronautics and astronautics, and military weapons [4][5][6][7]. Nevertheless, the application of BMGs is still greatly limited by the overall plasticity at room temperature [8]. In order to improve the workability of BMGs, enormous efforts have been made for its plastic deformation mechanism [8][9][10][11][12]. Generally, the plasticity of bulk metallic glasses is strongly related to the activities of shear bands of bulk metallic glasses during the loading process [10,12]. However, it is difficult to obtain the plastic deformation mechanism of BMGs by monitoring the nucleation and expansion of shear bands. In nanoindentation, serration flow behavior has been observed in many BMGs, which is significantly associated with the plasticity of BMGs [13][14][15]. Considering that the serrated flow is accompanied by the plastic deformation of bulk metallic glasses, the plasticity of BMGs can be further explored by analyzing the serrated flow. In literatures, the effect of many factors, including loading rates, ductility, and chemical effects, on the serration flow behavior of BMGs has been investigated [16][17][18][19]. However, the serrated flow behavior of BMGs at a large indentation depth cannot be fully investigated due to the limitation of experimental techniques. In order to resolve this problem, Long Short-Term Memory (LSTM) neural network can be adopted as a suitable approach due to its long-term ability in self-studying and forecasting [20][21][22].
A series of load-controlled nanoindentation tests were carried out on a Pd-based BMG (Pd 40 Cu 30 Ni 10 P 20 ). The serrations were identified and separated by using the proposed approach. Furthermore, long short-term memory was introduced to establish a neural network for predicting the serrated flow at different loading rates. The proposed LSTM model was verified by the comparisons of experimental and predictive results. Furthermore, the self-organized critical phenomena are further investigated based on the predictive serrated flow under different loading rates. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Experimental
The experimental material used in the study is Pd 40 Cu 30 Ni 10 P 20 bulk metallic glass (abbreviated as Pd-based BMG) [23,24]. The sample is a rod approximately 3 mm in diameter and 40 mm in length, which was cut into 5mm-thick cylinders. Load-controlled nanoindentation tests were performed on TI900 TriboIndenter system (Hysitron Inc.) with a Berkovich indenter. The resolution of load and displacement are 3 nN and 0.2 nm respectively. The peak load, P max , was maintained at 100 mN, and the loading rates, dP/dt, were selected at 0.5, 5 and 10 mN/s. The holding time is 10 s at P max . At least three measurements for each loading condition were performed to ensure the reliability of the experiment results. The experimental data used in this paper gained from earlier experiments, and the data have also been used in our previous studies [25].

The LSTM model
Machine learning methods have been widely used in the field of materials science [26,27]. Recurrent Neural Networks (RNN) is one of machine learning methods which have been widely adopted to investigate variablelength sequence [26][27][28]. However, the vanishing gradient and exploding gradient may encounter while longterm dependencies are optimized by using RNN [29]. To resolve these problems, a lot of variations of RNNs such as BiRNNs, EMI-FastGRNN and MSC-RNN have been proposed [28,30,31]. The Long Short-Term Memory is one of the most effective variations having ability to deal with long-term dependencies with a special gated architecture [20][21][22]. A representative LSTM model consists of a forget gate, an input gate, an output gate, and a memory cell, as shown in figure 1.
In LSTM model, the forget gate can select the memory cell state at the previous step. Then, the input gate is adopted to control the component of memory cell state affected by the inputting vectors x . t Finally, the output gate can influence the targeted outputs by the memory cell state. For a special input time series , where x t ( ) is the time-dependent serration sequence in the current study, the numerical calculation cycle in the LSTM model is updated as follows: wherex t 1 refers to the previous input value. -

Experimental results
The representative curves of load versus displacement (P-h) at various loading rates in nanoindentation were plotted in figure 2. In this figure, the origin of each curve has been offset, so that the multiple curves can be distinguished. As shown in figure 2, an obvious serrated flow on the P-h curves during loading periods was observed. Generally, the serrated flow of BMGs is strongly related to the activity of shear band [32,33]. In nanoindentation, the nucleation of a shear band in BMGs is intermittent during loading process, leading a serrated flow in P-h curves. In addition, it is also clear that the serrations significantly depend on loading rates, which is consistent with the experimental observations reported by Schuh et al [25,34]. The number of recorded points in the serration process is much fewer than that in process, indicating the higher velocity of serrated flow. An acceptable prediction on serrated flow acquired by machine learning method needs a typical serration dataset which has significant features. However, it is difficult to directly capture the details of serrations from the  P-h curves under nanoindentation. The curves of displacement versus time (h-t) were taken as a breakthrough to analyze [35]. Firstly, obtained dh/dt-t curves by differentiating the h-t curves using two-point forwardderivative algorithm, as shown in figure 3. The serration events shown as sharp bursts in the curve of dh/dt-t, and many fluctuations with higher frequencies but smaller amplitudes from the instrument and environment noise were also observed. Then eliminated the instrument and environment noise using a method of curvefitting and statistical analysis, but preserve the original shape of the deformation curve. Lastly, the clear dh/dt-t curves at different loading rates can be obtained, as shown in figure 4. It can be observed that the h-t curves obtained by integrating the dh/dt-t curves are very close to the experimental results, which indicates that these results are capable of supporting the identification and separation of serrations during nanoindentation.

LSTM model prediction results
The processed experimental results were used as datasets for the LSTM model. The proportion of training, testing and validation datasets were chosen as 70%, 15% and 15%, respectively. In addition, the number of hidden layers and nodes can significantly affect the performance of the LSTM model. After numerous experiment sessions, the single hidden layer with 200 hidden nodes was employed in the model due to the minimum absolute percentage error. The learning rate set as 0.0001, epochs as 300, the full batch learning was adopted then the mini-batch size was set as 8000. The dh/dt-t responses of Pd-based BMG under nanoindentation at various loading rates was simulated by the proposed LSTM model. The comparisons between the predicted and experimental results were illustrated in figure 5. As can be seen, the predicted results are very close to the experimental results, which implies that the proposed LSTM model can effectively inherit the intrinsic features of the original dataset obtained from laboratory experiments, and precisely characterize the serrated flow of Pd-based BMG under nanoindentation.
The h-t plot can be obtained by integrating the dh/dt-t plot, as shown in figure 6. The coefficient of determination (R 2 ) of the predicted and the experimental results for three various loading rates 0.5 mN s −1 , 5 mN s −1 and 10 mN s −1 are 0.925, 0.906, and 0.912, respectively. That indicates the proposed LSTM model can accurately predict the serrated flow of Pd-based BMG under nanoindentation. The predicted results obtained by multi-step predictive output of the LSTM model present a similar tendency as the earlier period. The serrations can be observed at a large indentation depth either.

Discussions
The distribution of serration size (Δh) with time obtained from LSTM model predicted results were illustrated in figure 7. Statistical distributions of Δh were illustrated in figure 8 as bar charts, the cumulative distributions of Δh were calculated and inseted in figure 8. The cumulative distribution curve can be well fitted by the empirical relation [36][37][38]: where A and a represent the normalization parameter and scaling parameter, respectively, and Dh c is the critical size of serrations [39]. From the equation (2), it can clearly notice that the serrations size which is smaller than Dh c obeys the power-law distribution well, while the serrations size which is larger than Dh c is constrained by the exponential decay.
The power-law relation is usually indicative of the self-organized critical (SOC) phenomena in dynamics. As seen in figure 8, the fitted value of Dh c means that the serration size smaller than Dh c follow the SOC behavior. The ratio of I=n soc /n serration is further introduced to quantitatively characterize the intensity of SOC phenomena at different loading rates. As n soc refers to the number of serrations following the SOC behavior, n serration refers to the total number of serrations. It is clear that the serrations exhibit a significant SOC behavior at different loading rates. In addition, the value of I index for three different loading rates (0.5 mN s −1 , 5 mN s −1 , 10 mN s −1 ) are 0.953, 0.744, and 0.731, respectively, thus implying that the SOC phenomena under a lower loading rate is more obvious than that under a higher loading rate. SOC reflects the stability of the deformation process, larger I value means the better stability and plastic deformation capacity of BMG deformation process under nanoindentation. It is reasonable to assume that the loading rate affecting the formation and evolution of flow units in BMGs.

Conclusion
In this study, a series of load-controlled nanoindentation tests were carried out on a Pd-based BMG. The serrations of experimental results were identified and separated. The experimental results were used to training and validating a LSTM neural network model for predicting the serrated flow of BMGs at a large indentation depth during nanoindentation. The self-organized critical phenomena are further investigated based on the predictive serrated flow under different loading rates. It is found that the serrations exhibit a significant selforganized critical phenomenon at different loading rates, and the self-organized critical phenomenon of the serrations under a lower loading rate is more obvious than that under a higher loading rate. Although we only present a preliminary analysis and discussion of the serrated flow of BMGs in this paper, it could also provide reference and help for further study of the mechanical properties of these alloys.