A comparison study at the flow stress prediction of Ti-5Al-5Mo-5V-3Cr-1Zr alloy based on BP-ANN and Arrhenius model

The experimental obtained flow stress curves are shown in figure 2. It is obvious to be seen that the flow stress rises with the increasing strain rate and reduces with the increasing. Besides, at the initial stage, working-hardening (WH) predominates, which results in the true stress increasing dramatically to a peak value with the rising strain. After reaching peak values, the curves show two different features, which indicates different microstructural evolution mechanisms: (1) at 973 K and 0.001, 0.01, 0.1 and 1 s⁻¹, the decrease of stresses is relative obvious, which was believed that dynamic recrystallization (DRX) or dynamic globularization predominates by Xia et al [19]; (2) at 1123 K and 0.001 and 0.01 s⁻¹, the stresses are relative steady, which was believed that dynamic recovery (DRV) predominates by Wang et al [20].

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The BP-ANN has been used to compute the constitutive relationships between experimental data and predicted data with high precision [21]. As shown in figure 3, the BP-ANN has a structure consisting of input layer, hidden layer and output layer. The outside signals (strain, strain rate and temperature) were received by input layer and delivered to hidden layer which can provide complex network to train and mimic the non-linear relationship between input layer and output layer [22]. Subsequently, processed signals (stress) were generated and outputted by output layer. For the simple and quick computation, the input and output signals should be normalized and non-normalized respectively. The normalized values were chosen from 0 to 1 in previous report [23]. In this work, to make the calculation more convenient, the normalized values were determined from −1 to 1 using the default settings of ANN. The values of input signals were normalized by the following equation:

$$\begin{eqnarray}&&X^{\prime} =\displaystyle \frac{2\times (X-{X}_{\min })}{({X}_{\max }-{X}_{\min })}-1\end{eqnarray} \tag{ 1 }$$

where X is original data, $X^{\prime}$ is the normalized result of X, X_min is the minimum value of original data and X_min is the maximum value of original data.

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To achieve optimal accuracy, the number of hidden layers was determined to be two according to previous report [24] which choose two hidden layers after comparing with single hidden layer. Besides, it is very important to chose suitable neurons for the hidden layers. Too many or too few neurons may both lead the training processes insufficient. Thus, a trial-and-error procedure was provided to determine the neuron number: two neurons were taken into the training at the beginning and more neurons were added one-by-one. The sum square error (SSE) [25] (expressed as equation (2)) between experimental and predicted values was introduced to evaluate the accuracy of the trained ANN model [22]. Generally, lower SSE-value in ln scale means high accuracy of the model. As shown in figure 4, the SSE-value reaches a bottom value at 12 neurons, which indicates that the best accuracy in the trial-and-error procedure is achieved.

$$\begin{eqnarray}&&SSE=\displaystyle \sum _{i=1}^{N}{({E}_{i}-{P}_{i})}^{2}\end{eqnarray} \tag{ 2 }$$

where E_i is the value of experiment, P_i is the value of prediction, and N is the total number of values.

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Figure 5 shows the comparisons between the experiment data of Ti-55531 alloy and the predicted data based on BP-ANN at different conditions. In figure 5, the hollow dots are predicted data based on the developed model and the red points are the limited input training data. It can be seen that, for most curves, the training data were picked from the strain value of 0.1 to 0.9 with the interval of 0.1 and the predicted data were chosen from the strain value of 0.05 to 0.85 with the interval of 0.1. As an exception, no data from 0.001 s⁻¹ and 973 K and 0.1 s⁻¹ and 1073 K was taken into training and these two curves were totally predicted. Most of all, the prediction points are nearly on the curves of experimental results, which means that the developed model is capable to simulate the curve types corresponding to different microstructural mechanism including working hardening, DRX, DRV and their interactions with each other. Based on the good agreement of experimental results and predicted data, it can be concluded that this developed model is capable to predict the flow stress of Ti-55531 alloy accurately and efficiently.

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As a very classical constitutive model, the Arrhenius equation (as show in equation (3)) is widely applied to express the relationships of flow stress, temperature and strain rate of alloys. [13, 26, 27]

$$\begin{eqnarray}&&\dot{\varepsilon }=F(\sigma )\exp (-Q/RT)\end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray*}F(\sigma )=\left\{\begin{array}{l}\begin{array}{c}\begin{array}{cc}{A}_{1}{{\sigma }_{p}}^{{n}_{1}} & \alpha \sigma \lt 0.8\end{array}\end{array}\\ \begin{array}{cc}{A}_{2}\exp (\beta {\sigma }_{p}) & \alpha \sigma \gt 1.2\end{array}\\ \begin{array}{cc}A{[\sinh (\alpha {\sigma }_{p})]}^{n} & \begin{array}{ccc}for & all & \alpha \sigma \end{array}\end{array}\end{array}\right.\end{eqnarray*}$$

and where $\dot{\varepsilon },$ Q, R, T, σ are the strain rate, activation energy, gas constant, absolute temperature and flow stress, respectively. A, A₁, A₂, α, β, n and n₁ are material constants, and α = β/n₁.

Besides, the Zener-Hollom parameter (Z) is introduced to present the effects of deformation parameters on deformation behaviors. It is expressed in an exponent-type equation as following [28]:

$$\begin{eqnarray}&&Z=\dot{\varepsilon }\exp (Q/RT)\end{eqnarray} \tag{ 4 }$$

Based on equations (3) and (4), the function could be denoted as:

$$\begin{eqnarray}&&\sigma =\displaystyle \frac{1}{\alpha }\,\mathrm{ln}\left\{{(Z/A)}^{\displaystyle \frac{1}{n}}+{[{(Z/A)}^{\displaystyle \frac{2}{n}}+1]}^{\displaystyle \frac{1}{2}}\right\}\end{eqnarray} \tag{ 5 }$$

On the basis of the solution processes of the material constants used by Xia et al [29]. The values of α, Q, lnA, n from the strain of 0.05 to 0.9 with the interval of 0.05 have been determined. Then, the determined values were fitted by the sixth order polynomial functions in equation (6). The relationships between true strain and constants including α, lnA, n, Q by polynomial fit were shown in figure 6. In figure 6, the adjusted coefficients of determination between the fitted curves and determined α, lnA, n, Q are 0.9996, 0.9994, 0.9729 and 0.9993 respectively. The corresponding coefficients are listed in table 2.

$$\begin{eqnarray}&&\left\{\begin{array}{l}\alpha ={B}_{0}+{B}_{1}\varepsilon +{B}_{2}{\varepsilon }^{2}+{B}_{3}{\varepsilon }^{3}+{B}_{4}{\varepsilon }^{4}+{B}_{5}{\varepsilon }^{5}+{B}_{6}{\varepsilon }^{6}\\ \mathrm{ln}\,A={C}_{0}+{C}_{1}\varepsilon +{C}_{2}{\varepsilon }^{2}+{C}_{3}{\varepsilon }^{3}+{C}_{4}{\varepsilon }^{4}+{C}_{5}{\varepsilon }^{5}+{C}_{6}{\varepsilon }^{6}\\ n={D}_{0}+{D}_{1}\varepsilon +{D}_{2}{\varepsilon }^{2}+{D}_{3}{\varepsilon }^{3}+{D}_{4}{\varepsilon }^{4}+{D}_{5}{\varepsilon }^{5}+{D}_{6}{\varepsilon }^{6}\\ \alpha ={E}_{0}+{E}_{1}\varepsilon +{E}_{2}{\varepsilon }^{2}+{E}_{3}{\varepsilon }^{3}+{E}_{4}{\varepsilon }^{4}+{E}_{5}{\varepsilon }^{5}+{E}_{6}{\varepsilon }^{6}\end{array}\right.\end{eqnarray} \tag{ 6 }$$

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According to equations (4) and (5), the solution expression for stress can be expressed as:

$$\begin{eqnarray}&&\sigma =\displaystyle \frac{1}{g(\varepsilon )}\,\mathrm{ln}\left\{{\left(\displaystyle \frac{\dot{\varepsilon }\exp [j(\varepsilon )/8.314T]}{f(\varepsilon )}\right)}^{\displaystyle \frac{1}{h(\varepsilon )}}\right.\\ &&\,\,+\,\left.{\left[{\left(\displaystyle \frac{\dot{\varepsilon }\exp [j(\varepsilon )/8.314T]}{f(\varepsilon )}\right)}^{\displaystyle \frac{2}{h(\varepsilon )}}+1\right]}^{\displaystyle \frac{1}{2}}\right\}\end{eqnarray} \tag{ 7 }$$

where g(ε), f(ε), j(ε) and h(ε) are the projection functions of α, A, Q and n, respectively.

So far, the stress predicted values based on developed Arrhenius model can be obtained by applying the material constants into equation (7). The predicted values based on developed Arrhenius model at different deformation conditions are shown in figure 7. It can be seen that, to some extent, the predicted values are accurate and the share the similar features with experimental results.

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As mentioned in Part 3.2, no data from 0.001 s⁻¹ and 973 K and 0.1 s⁻¹ and 1073 K was taken into training and 18 points for each of these deformation conditions were predicted. Thus, to assess the prediction accuracy of the two models, the predicted results at these two deformation conditions predicted by the two models were taken into comparison using the relative error calculated by equation (8).

$$\begin{eqnarray}&&\delta ( \% )=\displaystyle \frac{{P}_{i}-{E}_{i}}{{E}_{i}}\times 100 \% \end{eqnarray} \tag{ 8 }$$

where E_i is the experimental result and P_i is the predicted result.

The relative errors of the predictions of the two models under the condition of 973 K and 0.001 s⁻¹ and 1073 K and 0.1 s⁻¹ are listed in table 3. In table 3, the relative errors of predicted results by Arrhenius model vary from −10.87% to 11.84%. As distinct comparisons, the relative errors of BP-ANN model vary from −3.63% to 6.87%. To analyze statistically, the relative errors distribution histograms of the two models were plotted as shown in figure 8. Meanwhile, the mean value (μ) and the standard deviation (ω) were introduced to evaluate the concentration degree of relative errors distribution. Statistically speaking, smaller values of μ and ω close to zero mean better errors distribution, which indicate better prediction accuracy. For BP-ANN model, the μ-value and the ω-value were calculated to be 1.4714% and 2.2271% according to equation (9) and equation (10), respectively, while for Arrhenius model, the corresponding values were −1.2213% and 5.3641%, respectively. It is obvious that the concentration degree in the low relative error values field (near zero) of BP-ANN model is higher, which further indicates the better prediction capability of this model.

$$\begin{eqnarray}&&\mu =\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}{\delta }_{i}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\omega =\sqrt{\displaystyle \frac{1}{N-1}\displaystyle \sum _{i=1}^{N}{({\delta }_{i}-\mu )}^{2}}\end{eqnarray} \tag{ 10 }$$

Where δ_i is the relative error.

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For further evaluation, as two important statistical indicators, correlation coefficient (R) (expressed as equation (11)) and average absolute relative error (ARRE) (expressed as equation (12)) were introduced to assess the predication capability of the two models [30]. It is known that the R-values close to −1, 0 and 1 are corresponding to strong negative relationship, no relationship and strong positive relationship, respectively. Also, smaller absolute AARE-value indicates the smaller difference between predicted values and experimental values.

$$\begin{eqnarray}&&R=\displaystyle \frac{\displaystyle {\sum }_{i=1}^{N}({E}_{i}-\bar{E})({P}_{i}-\bar{P})}{\sqrt{\displaystyle {\sum }_{i=1}^{N}{({E}_{i}-\bar{E})}^{2}\cdot \displaystyle {\sum }_{i=1}^{N}{({P}_{i}-\bar{P})}^{2}}}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&AARE( \% )=\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}\left|\displaystyle \frac{{P}_{i}-{E}_{i}}{{E}_{i}}\right|\times 100 \% \end{eqnarray} \tag{ 12 }$$

Where E_i is the value of experiment and P_i is the value of prediction. $\bar{E}$ and $\bar{P}$ are the corresponding mean values, and N is the number of data.

Figure 9 shows the correlation relationships between the values of experiment and prediction by the two models. The R-values of the BP-ANN model and the Arrhenius model are 0.9949 and 0.9761 respectively and the AARE-values of BP-ANN model and Arrhenius model are 2.2836% and 23.4527% respectively. So far, the prediction capability of BP-ANN model has been proved to be better than Arrhenius model.

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According to the results and discussion above, the reliability and accuracy of the developed BP-ANN model has been proved. As shown in figure 10, the flow stress of Ti-55531 alloy at the temperatures of 998, 1058, 1098, 1158 K and the strain rates of 0.001, 0.01, 0.1, 1 s⁻¹ were predicted by the developed BP-ANN model. In figure 10, the red curves represent the 3D curves of experimental results and the blue square dots represent the predicted points. Also, among the extended curves, some curve types corresponding to different microstructural mechanism such as DRX (0.1 s⁻¹ and 998 K), WH (0.001 s⁻¹ and 998 K) and DRV (0.01 s⁻¹ and 1158 K) can be observed, which validates the applicability of BP-ANN model in predicting the flow curves of Ti-55531 alloy outside the experimental conditions. It is well known that stress-strain curves are the fundamental data used in the finite element simulation during hot deformation. With high accuracy predicted expanded data and basic experimental data, more accurate finite element simulation of a certain forming process could be achieved.

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