As shown in Figure 2, the flow stress as well as the shape of the flow curves of as-forged Ti-10V-2Fe-3Al alloy is sensitively dependent on temperature and strain rate. The stress–strain curve articulates the intrinsic characteristics of flow stress with thermodynamic softening behaviors. It can be summarized from all the true stress–strain curves that there are three distinct stages in the stress evolution with strain. At the first stage where WH predominates, the flow stress increases rapidly to a critical value. Then the flow stress increases lesser and lesser until a peak value or an inflection of work-hardening rate at the second stage, indicating that the work softening due to DRX and DRV becomes more and more predominant, then it exceeds WH. At the final stage, it can be summarized that two types of curve variation tendency exist: in the parameter domains of 948–1,023 K where $\mathrm{\alpha }$-phase and $\mathrm{\beta }$-phase coexist, the flow stress decreases continuously with strain due to obvious DRX softening; however, in the parameter domains of 1,048–1,123 K where $\mathrm{\beta }$-phase predominates, the variation of flow stress with strain becomes almost smooth and appears to be a steady-state characteristic with significant DRV softening.

During hot deformation process, it is very widely known that the constitutive relationships between the steady-state flow stress, strain rate and deformation temperature are represented by Zener–Hollomon parameter, in an exponent-type equation, and the hyperbolic law in the Arrhenius-type equation gives better approximations between $Z$ parameter and stress, as expressed in eq. (1) [15–18]: $Z=\dot{\mathrm{\epsilon }}exp\left(\frac{Q}{RT}\right)=AF(\mathrm{\sigma })$(1)where $F(\mathrm{\sigma })=\{\begin{array}{cc}{\mathrm{\sigma }}^m& \mathrm{\alpha }\mathrm{\sigma }<0.8\\ exp(\mathrm{\beta }\mathrm{\sigma })& \mathrm{\alpha }\mathrm{\sigma }<1.2\\ [\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathrm{\alpha }\mathrm{\sigma }){]}^n& \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{\sigma }\end{array}$in which $\dot{\mathrm{\epsilon }}$ is the strain rate (s⁻¹), $R$ is the universal gas constant (8.31 J mol⁻¹ K⁻¹), $T$ is the absolute temperature (K), $Q$ is the apparent activation energy for hot deformation (kJ mol⁻¹), $\mathrm{\sigma }$ is the flow stress (MPa) for a given stain, $A$, $\mathrm{\alpha }$, $m$ and $n$ are the material constants, $\alpha =\beta /n$.

In the current study, the experimental datasets involving deformation temperature, strain rate and peak stress (${\mathrm{\sigma }}_P$) at peak strain were fitted into eq. (1). The relationships between $ln\dot{\mathrm{\epsilon }}-ln{\mathrm{\sigma }}_P$, $ln\dot{\mathrm{\epsilon }}-{\mathrm{\sigma }}_P$, $ln\dot{\mathrm{\epsilon }}-\mathrm{l}\mathrm{n}[\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathrm{\alpha }{\mathrm{\sigma }}_P)]$ and $\text{ln}[\text{sinh}(\alpha {\sigma }_{\text{P}})]-10,000/\text{T}$ in $\mathrm{\alpha }+\mathrm{\beta }$-phase temperature range are shown in Figure 3 The value of $m$ and $\mathrm{\beta }$ was obtained through taking average slope of the lines in Figure 3(a) and 3(b), respectively. Thus the $\mathrm{\alpha }$-value in $\mathrm{\alpha }+\mathrm{\beta }$-phase temperature range was calculated to be 0.005195. Subsequently, the calculation of $n$ and $Q$ has been performed according to the following relationships: $Q=Rns$(2) $n=\frac{dln\dot{\mathrm{\epsilon }}}{dln[sinh(\mathrm{\alpha }\mathrm{\sigma })]}$(3) $s=\frac{dln[sinh(\mathrm{\alpha }\mathrm{\sigma })]}{d(1/T)}$(4)Hence the value of $n$ and $Q$ in $\mathrm{\alpha }+\mathrm{\beta }$-phase temperature range was calculated to be 4.1972 and 463.09 kJ/mol, respectively, from the slope of the lines in Figure 3(c) and 3(d). In the same way, the value of $\mathrm{\alpha }$, $n$ and $Q$ in $\mathrm{\beta }$-phase temperature range was calculated to be 0.010199, 3.2812 and 210.45 kJ/mol, respectively, from the plots of $ln\dot{\mathrm{\epsilon }}-ln{\mathrm{\sigma }}_P$, $ln\dot{\mathrm{\epsilon }}-{\mathrm{\sigma }}_P$, $ln\dot{\mathrm{\epsilon }}-\mathrm{l}\mathrm{n}[\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathrm{\alpha }{\mathrm{\sigma }}_P)]$ and $\mathrm{l}\mathrm{n}[\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathrm{\alpha }{\mathrm{\sigma }}_P)]-$ $10,000/T$ in $\mathrm{\beta }$-phase temperature range, as shown in Figure 4.

Figure 3

Relationships between (a) $ln\dot{\mathrm{\epsilon }}-ln{\mathrm{\sigma }}_P$, (b) $ln\dot{\mathrm{\epsilon }}-{\mathrm{\sigma }}_P$, (c) $ln\dot{\mathrm{\epsilon }}-\mathrm{l}\mathrm{n}(\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathrm{\alpha }{\mathrm{\sigma }}_P))$, (d) $\text{ln}(\text{sinh}(\alpha {\sigma }_P))-10,000/T$ in $\mathrm{\alpha }+\mathrm{\beta }$-phase temperature range

Figure 4

Relationship between (a) $ln\dot{\mathrm{\epsilon }}-ln{\mathrm{\sigma }}_P$, (b) $ln\dot{\mathrm{\epsilon }}-{\mathrm{\sigma }}_P$, (c) $ln\dot{\mathrm{\epsilon }}-\mathrm{l}\mathrm{n}(\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathrm{\alpha }{\mathrm{\sigma }}_P))$, (d) $\text{ln}(\text{sinh}(\alpha {\sigma }_P))-10,000/T$ in $\mathrm{\beta }$-phase temperature range

The hot deformation behaviors are influenced by the process of thermal activation, and the softening mechanism is related to the thermal activation energy ($Q$), whose value reflects the difficulty of the dislocation moving, DRV and DRX. Many researchers have revealed that the self-diffusion activation energy of $\mathrm{\alpha }‐\mathrm{T}\mathrm{i}$ in pure titanium is in the range of 169–242 kJ/mol, while the self-diffusion activation energy of $\mathrm{\beta }\mathrm{-T}\mathrm{i}$ is 153–166 kJ/mol. As mentioned earlier, the activation energy for hot deformation in $\mathrm{\alpha }+\mathrm{\beta }$-phase temperature range is determined to be 463.09 kJ/mol, which is much larger than the activation energy of self-diffusion. However, the calculated $Q$-value (210.45 kJ/mol) in $\mathrm{\beta }$-phase temperature range is almost equal to the self-diffusion activation energy of $\mathrm{\beta }\mathrm{-T}\mathrm{i}$. In general, the DRV predominates in the work softening process of hot deformation when the activation energy for hot deformation did not appear to be much different from the self-diffusion activation energy, while the DRX is the main softening mechanism when the activation energy for hot deformation is much larger than the self-diffusion activation energy.

The variation of activation energies for as-forged Ti-10V-2Fe-3Al alloy in hot deformation along with the increase of strain from 0.02 to 0.8 is shown in Figure 5. It is obviously seen from Figure 5 that the activation energy for hot deformation decreases with strain increases in $\mathrm{\beta }$-phase temperature range, reflecting that the energy for deformation decreases with the increase of the plastic deformation. The average activation energy for hot deformation in $\mathrm{\beta }$-phase temperature range was calculated from Figure 4 to be 188.71 kJ/mol. It is almost equal to the self-diffusion activation energy of 166 kJ/mol for $\mathrm{\beta }\mathrm{-T}\mathrm{i}$, indicating that the DRV predominates in the work softening process of as-forged Ti-10V-2Fe-3Al alloy in $\mathrm{\beta }$-phase temperature range. This can be proved in Figure 2, as the variation of flow stress with strain in $\mathrm{\beta }$-phase temperature range appears to be a steady-state characteristic with significant DRV softening.

Figure 5

The relationship between $Q$ and strain

Figure 6

The microstructure of as-forged Ti-10V-2Fe-3Al alloy at different deformation conditions

However, in $\mathrm{\alpha }+\mathrm{\beta }$-phase temperature range, when the strain increases, the activation energy for hot deformation increases rapidly to a peak value and then decreases gradually to a relatively low level, as shown in Figure 5. Meanwhile, it can also be seen in Figure 5 that the activation energy for hot deformation in $\mathrm{\alpha }+\mathrm{\beta }$-phase temperature range is much larger than the self-diffusion activation energies of $\mathrm{\alpha }\mathrm{-T}\mathrm{i}$ (242 kJ/mol) and $\mathrm{\beta }\mathrm{-T}\mathrm{i}$ (166 kJ/mol), which means that the DRX predominates in softening mechanism during hot deformation of as-forged Ti-10V-2Fe-3Al alloy. Figure 6 shows the typical microstructure of as-forged Ti-10V-2Fe-3Al alloy at different deformation conditions in $\mathrm{\alpha }+\mathrm{\beta }$-phase temperature range. It can be seen from Figure 6 that the $\mathrm{\alpha }$-phase grains are fined by globularization which can be regarded as a type of DRX, and the matrix of $\mathrm{\beta }$-phase is surrounded by equiaxed $\mathrm{\alpha }$-phase grains. In fact, when the specimen is compressed, it is significantly difficult to identify the grain boundaries of $\mathrm{\beta }$-phase due to the crush, globularization and confusion of aciculate $\mathrm{\alpha }$-phase grains in the grain boundaries of $\mathrm{\beta }$-phase.

It indicates that for as-forged Ti-10V-2Fe-3Al the softening mechanism transforms from DRX to DRV with increasing temperature, which is due to an allotropic phase transformation from $\mathrm{\alpha }$-phase to $\mathrm{\beta }$-phase. As a result, the extremely complex flow behaviors of as-forged Ti-10V-2Fe-3Al in a wide temperature range make it difficult to predict the flow stresses during hot deformation with constitutive models by regression method, whereas the ANN is able to deal with the highly nonlinear relationships well.

ANN, as a kind of computational model of biological neuron, is able to generalize the deformation behavior characteristics by simulating the working process of biological neurons with network structures, making it possible to build a constitutive model with high precision. The typical structure of an ANN consists of one input layer, one output layer and one or more hidden layers. These layers are connected by the fundamental units of ANN named artificial neurons with a function of taking a weighted sum of the inputs. The hidden layer works as a complex network architecture to simulate the complex nonlinear relationships between outside information received in input layer and output information generated from output layer [6, 9, 19, 20]. Among the various kinds of ANNs available, the multilayer feed-forward ANN with BP algorithm is becoming increasingly popular in materials modeling. The training of feed-forward ANN in this work is undertaken using BP algorithm to get a better understanding of the constitutive relationships between the inputs and outputs since it is a typical means of adjusting the weights and biases by utilizing gradient descent to minimize the target error [21] and has a great representational power for dealing with highly nonlinear and strongly coupled relationship [22].

In this investigation, the input variables of ANN contain deformation temperature ($T$), strain ($\mathrm{\epsilon }$) and strain rate ($\dot{\mathrm{\epsilon }}$), while the output variable is flow stress ($\mathrm{\sigma }$). Thus, a feed-forward network with BP algorithm was developed, as shown in Figure 7. In the present study, 1,600 normalized datasets have been selected from the 40 stress–strain curves. When developing the ANN model, the datasets at a strain of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 and 0.8 were chosen as the testing sets, while the available datasets remaining were used to train the model. The experimental data containing temperature, strain, strain rate and stress were measured in different units, leading to the decrease of the convergence speed and accuracy of the model. As a result, the input and output variables should be normalized into dimensionless units before training to reduce the effects caused by different units. From the stress–strain curves shown in Figure 2, it can be seen that the input strain data vary from 0.02 to 0.8, strain rate data vary from 0.001 s⁻¹ to 10 s⁻¹, temperature data vary from 948 K to1,123 K and the output flow stress data vary from 23.93 MPa to 587.45 MPa. The temperature data, strain data and the flow stress data were normalized within the range from 0.05 to 0.3 using the relation given by eq. (5). Meanwhile, the strain rate data, after taking logarithm of the values, were normalized using the relation given by E. (6), since the range of the data is much too wide: $y_{\mathrm{n}}=0.05+{0.25}^{\ast }\frac{y-0.95y_{min}}{1.05y_{max}-0.95y_{min}}$(5) $y_{\mathrm{n}}=0.05+{0.25}^{\ast }\frac{(4+y)-0.95(4+y_{min})}{1.05(4+y_{max})-0.95(4+y_{min})}$(6)where $y_{\mathrm{n}}$ is the normalized value of $y$, y is the experimental data, $y_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $y_{\mathrm{m}\mathrm{i}\mathrm{n}}$ are the maximum and minimum value of $y$, respectively.

Figure 7

Schematic illustration of the neural network architecture

The structural parameters, involving hidden layer number, transfer function, training function and neuron number for each hidden layer, are of great significance for an excellent ANN model. Firstly, two hidden layers were adopted in the current investigation to ensure high accuracy. Subsequently, an associated transfer function, which can represent how the weighted sum of its inputs is transferred to the results into outputs, was set for each layer. The selected transfer function is “tan sigmoid” for each hidden layer, and “pure linear” for output layer. In addition, the training function is “trainbr.” Finally, the neuron number for each hidden layer was set by means of trail-and-error method according to the experience of designers and the training sample size. If the neuron number of each hidden layer in the ANN model is too small, the model may be insufficient to learn the process correctly when in training. On the contrary, too many neurons may slow the convergence rates or over-fit the data. The ANN model in the current investigation was trained firstly with three neurons in each hidden layer, and then the neuron number was adjusted continually (from 3 to 21) for the purpose of approaching the expected accuracy.

The value of mean square error (MSE), expressed by eq. (7). [23], is introduced to evaluate the ability of the ANN training work and determine the neuron number for each hidden layer: $\mathrm{M}\mathrm{S}\mathrm{E}=\frac{1}{N}\sum _{i=1}^N{(E_i-P_i)}^2$(7)where $E$ is the sample of experimental value, $P$ is the sample of predicted value by ANN model and $N$ is the number of strain–stress samples.

The MSE-value of each actual training work was calculated as the training work of ANN models with different neuron numbers was accomplished. It is obviously seen in Figure 8 that the MSE decreases to the minimum value when the neuron number in each hidden layer is 18, showing that the ANN model with 18 neurons in each hidden layer exhibits optimal performance.

Figure 8

Performance of the network at different hidden neuron levels

The generalization performance of the well-trained ANN model is measured in terms of R and the AARE [23–26], as expressed by eqs (8) and (9), respectively. R is a widely used evaluator to measure the strength of linear relationships between experimental and predicted values, while the AARE indicates the accuracy of the prediction. $R=\frac{{\sum }_{i=1}^N(E_i-\bar{E})(P_i-\bar{P})}{\sqrt{{\sum }_{i=1}^N(E_i-\bar{E}{)}^2{\sum }_{i=1}^N{(P_i-\bar{P})}^2}}$(8) $\mathrm{A}\mathrm{A}\mathrm{R}\mathrm{E}(\mathrm{\%})=\frac{1}{N}\sum _{i=1}^N\left|\frac{E_i-P_i}{E_i}\right|\times 100$(9)where $E$ is the sample of experimental value, $P$ is the sample of predicted value by ANN model, $\bar{E}$ and $\bar{P}$ are the mean value of $E$ and $P$, respectively, $N$ is the number of strain–stress samples.

High levels of R-values and low levels of AARE-values indicate that the predicted flow stress values agree very well with the experimental value. As shown in Figure 9, R was linearly fitted with a value of 0.9999, and the AARE is calculated to be a determined percent of 0.402%, showing a good correlation between experimental and predicted flow stress values by the ANN model with 1,280 training datasets in total. As a result, the well-trained ANN model has excellent accuracy and is able to predict the flow behaviors of as-forged Ti-10V-2Fe-3Al alloy well.

Figure 9

Correlation between experimental and predicted flow stress for the training datasets

In order to check the prediction ability, the ANN model is tested by the testing datasets involving datasets at strain of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 and 0.8, which were not used in the training work. Figure 10(a) and 10(b) shows the correlation relationships between predicted flow stress values by ANN model and experimental values in testing procedure for the $\mathrm{\alpha }+\mathrm{\beta }$-phase temperature range and $\mathrm{\beta }$-phase temperature range, respectively. As shown in Figure 10, in the $\mathrm{\alpha }+\mathrm{\beta }$-phase temperature range where temperature is lower than 1,068 K, the R-value and the AARE-value are calculated to be 0.99981 and 0.576%, respectively, as well as the 0.99984 and the 0.691% for the $\mathrm{\beta }$-phase temperature range where temperature is higher than 1,068 K, illustrating that the accuracy has been achieved to the ideal level.

Figure 10

Correlations between experimental and predicted flow stress in testing procedure for (a) $\mathrm{\alpha }+\mathrm{\beta }$-phase temperature range, (b) $\mathrm{\beta }$-phase temperature range

As the well-trained ANN model has achieved excellent accuracy, it can be adopted to predict the flow stress in a wide range of temperature, strain and strain rate. Figure 11 shows the comparisons between the datasets predicted by ANN model and the strain–stress curves obtained from compression tests. It is obviously seen in Figure 11 that the predicted datasets fit the experimental curves well in a wide temperature range of 948–1,123 K, a wide strain rate range of 0.001–10 s⁻¹ and a wide strain range of 0.1–0.8, no matter what the softening mechanism is and whether or not the phase transformation occurs.

Figure 11

Comparison between the experimental and predicted flow stress by ANN model at different strain rates and temperatures: (a) 948 K, (b) 973 K, (c) 998 K, (d) 1,023 K, (e) 1,048 K, (f) 1,073 K, (g) 1,098 K, (h) 1,123 K

The constitutive model for as-forged Ti-10V-2Fe-3Al alloy was obtained in the form of eq. (1) through nonlinear multivariate regression analysis similar to the method in Chapter 3 with datasets involving deformation temperature, strain rate and stress at the strain of 0.5.

The relationships between $ln\dot{\mathrm{\epsilon }}-ln\mathrm{\sigma }$, $ln\dot{\mathrm{\epsilon }}-\mathrm{\sigma }$, $ln\dot{\mathrm{\epsilon }}-\mathrm{l}\mathrm{n}[\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathrm{\alpha }\mathrm{\sigma })]$ and $\text{ln}[\text{sinh}(\alpha \sigma )]\text{-10},\text{000/T}$ are shown in Figure 12. Hence the value of $\mathrm{\alpha }$, $n$ and $Q$ was calculated to be 0.008065, 3.759 and 214.16 kJ/mol, respectively. The value of $A$ was then obtained to be 2.295 × 10⁹ from the intercept of the fitted line in Figure 13. Finally, the constitutive equation by regression method can be presented as eq. (10): $Z=\dot{\mathrm{\epsilon }}exp\left(\frac{214.16}{RT}\right)=2.295\times {10}^9[sinh(0.008065\mathrm{\sigma }){]}^{3.759}$(10)Nevertheless, it is very difficult to calculate a set of equation parameters for the whole strain range to deal with the specific deformation characteristics. On the contrary, the ANN model is able to figure out this problem.

Figure 12

Relationships between (a) $ln\dot{\mathrm{\epsilon }}-ln\mathrm{\sigma }$, (b) $ln\dot{\mathrm{\epsilon }}-\mathrm{\sigma }$, (c) $ln\dot{\mathrm{\epsilon }}-\mathrm{l}\mathrm{n}(\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathrm{\alpha }\mathrm{\sigma }))$, (d) $\text{ln}(\text{sinh}(\alpha \sigma ))-10,000/T$ at strain of 0.5

Figure 13

Variation of the Z parameter with flow stress (at strain of 0.5)

Figures 14 and 15 show the correlation relationships between the experimental and predicted data at the strain of 0.5 from the well-trained ANN model and the regression model, respectively. The R-value and the AARE-value calculated from the ANN model are 0.9998 and 0.572%, respectively, better than the 0.9902 and 6.583% from the regression method, respectively, indicating that the ANN model has better capacity to predict the flow behaviors of as-forged Ti-10V-2Fe-3Al alloy.

Additionally, as expressed by eq. (11), the relative percentage error ($\mathrm{\eta }$) is introduced to compare various measurements of the relative difference at strain of 0.5, besides R and the AARE. $\mathrm{\eta }(\mathrm{\%})=\frac{P_i-E_i}{E_i}\times 100\mathrm{\%}$(11)where E is the sample of experimental value, P is the sample of predicted value by ANN model and N is the number of strain–stress samples.

Figure 14

Correlation between experimental and predicted flow stress by ANN model at strain of 0.5

Figure 15

Correlation between experimental and calculated flow stress by regression model at strain of 0.5

The comparison of $\mathrm{\eta }$-values of the ANN model and the regression model is shown in Figure 16. It can be demonstrated that the $\mathrm{\eta }$-values obtained from the ANN model vary from −2.54% to 2.34%, whereas for the regression model, the $\mathrm{\eta }$-values are in the range from −17.75% to 18.33%. Subsequently, the distribution of $\mathrm{\eta }$-values was analyzed further since the larger fluctuation range of $\mathrm{\eta }$-values does not mean poorer predictability. Figure 17(a) and 17(b) shows the distribution of $\mathrm{\eta }$-values corresponding to the well-trained ANN model and regression model, respectively, in which the height of each histogram expresses the relative frequency of each $\mathrm{\eta }$-level. The distribution diagrams of $\mathrm{\eta }$-values appear to be typical Gaussian distributions after being nonlinearly fitted. The two indicators, namely mean value of $\mathrm{\eta }$-values ($\mathrm{\mu }$) and standard deviation ($w$), are of great significance for evaluating the two Gaussian distributions. Mean value is a evaluator obtained by dividing the sum of observed values by the number of observations to measure the magnitude of the datasets, while standard deviation ($w$) represents the degree of dispersion and gives an idea of how close the entire set of data is to the average value, as expressed by eqs (12) and (13) [7, 23, 27]: $\mathrm{\mu }=\frac{1}{N}\sum _{i=1}^N{\mathrm{\eta }}_i$(12) $\mathrm{w}=\sqrt{\frac{1}{N-1}\sum _{i=1}^N({\mathrm{\eta }}_i-\mathrm{\mu }){}^2}$(13)where $\mathrm{\eta }$ is the sample of relative percentage error, $\mathrm{\mu }$ is the mean value of $\mathrm{\eta }$-values and $N$ is the sample number of relative percentage errors.

Figure 16

Comparison of relative percent error of predicted value by ANN and calculated value by regression model with experimental value at strain of 0.5

Figure 17

Distribution on relative percentage error by (a) ANN model and (b) regression model

Small values of $\mathrm{\mu }$ and $w$ indicate tightly grouped, precise data. As shown in Figure 17(a) and 17(b), $\mathrm{\mu }$-value and $w$-value of the ANN model are −0.1264% and 0.8745%, respectively, compared with the 1.3146% and 8.2033% from the regression model. The smaller $\mathrm{\mu }$-value of the ANN model indicates that the predicted stress data are closer to the experimental stress data. And the smaller standard deviation ($w$) of the ANN model is related to a more centralized distribution of $\mathrm{\eta }$-values, that is, it is of greater possibility for more samples in the predicted stress data to be close to the experimental stress data.