To forecast the flow stress of metals accurately, individuals proposed various physical-based, phenomenological models and artificial neural networks (ANN) [
28]. The Arrhenius model given by Sellars et al. [
29] is usually used to depict the effects of strain rate and temperature on flow stress at high deformed temperatures. The Johnson–Cook (J–C) model could consider the effects of temperature, stain rate, and strain independently [
30]. Therefore, in the following sections, the J–C model was utilized to predict the flow behavior at a strain rate of 10
−3 s
−1, and the dynamic hot compression process was predicted by the Arrhenius model. Furthermore, an ANN model was also developed for the dynamic compression process. Before mathematical modeling, the stress-strain data with abnormal hardening were discarded.
3.3.2. Dynamic Modeling
The detailed mathematical expressions of the Arrhenius model are given in Equations (7)–(10):
where
,
,
,
,
,
, and
are material-related constants,
,
is the strain rate (s
-1),
is the universal gas constant (8.314 J/mol/K),
is the absolute temperature (K),
is the activation energy of deformation (J/mol), and
represents the Zener–Hollomon parameter. Thus, the flow stress
could be also represented as follows:
where the material constants could be determined by substituting Equation (7) into Equations (8)–(10) and taking the logarithm.
At a given temperature, the constants
and
were obtained by averaging the reciprocal of slopes of curves
vs.
and
vs.
according to Equations (12) and (13), respectively. The activation energy
is the average of the slope of curves
vs.
and
was determined by the intercepts of these curves for a given strain rate according to Equation (14). For accurate prediction, the strain compensation should be considered [
31]. Subsequently, the related constants were computed in the strain range from 0.05–0.4 at the interval of 0.025 in the same manner. The relationship between the strain and material constants was fitted using a fifth-order polynomial according to Equation (15). The resulting fitted curves are shown in
Figure 8. The fitted polynomial coefficients are listed in
Table 4.
After the material constants were determined, the flow stress could be predicted using Equations (7), (11), and (15). The predicted values by the Arrhenius model and experimental results are described in
Figure 9. Despite a similar trend, large relative deviations still exist under certain processing conditions. Consequently, necessary improvements to the Arrhenius model are required.
From
Figure 9, it could be found that the measured stress is lower than the predicted value when the strain rate is greater or equal to 1 s
−1 and the deformed temperature is less than or equal to 750 °C. This overestimation could be attributed to the adiabatic heating during deformation. Large amounts of deformation heat would be generated during the rapid compression of the Ti–6Al–4V alloy, which cannot be dissipated in time because of its poor thermal conductivity, resulting in localized stress softening [
32]. After trial and error, an adiabatic temperature rise of Δ
T = 10 °C is added to the
T in Z as displayed in Equation (16). Secondly, the underestimation is observed in
Figure 9c,d for the strain rate greater than 10 s
−1 and temperatures of 800–850 °C. The compensation of strain rate is considered by regulating the exponent of strain rate in
Z as shown in Equation (17) according to that in Peng et al. [
33]. While the temperature reaches 900 °C, the model could basically predict the change of flow stress except when the strain rate is 10
−2 s
−1 and 1 s
−1. For the strain rate of 1 s
−1, the deviation gradually increases with the strain if the true strain is greater than 0.2. It is difficult to directly determine the specific cause, or it may be from the test error. For the strain rate of 10
−2 s
−1, the measured value is lower than that of the predicted value in the whole strain range. The deformed temperature of 900 °C is much higher than the final decomposition temperature of Ti–6Al–4V and the strain rate is minimal. The martensite decomposition during compression contributes to a share of stress softening. Hence, the compensation of both strain rate and the temperature was carried out at 900 °C–10
−2 s
−1 as shown in Equation (18). The prediction by the modified Arrhenius model and measured stresses are depicted in
Figure 10.
ANN is a powerful tool that provides individuals an approach to bridge the complex connection between the input variables and output responses for any complex system [
34]. In this section, an ANN model with a backpropagation (BP) algorithm was employed to describe the stress-strain relationship of SLM-manufactured Ti–6Al–4V in dynamic compression processes.
Figure 11 displays the structure of a typical three-layer BP artificial neural network (BP-ANN). The input data consist of three variables such as strain, strain rate, and temperature. The output is flow stress. The hyperbolic sigmoid function is the transfer function in the hidden layer, and the linear function is the transfer function in the output layer. The gradient descent optimization algorithm is adapted to update the weights and biases. The Bayesian regularization algorithm is used to train the network. After several attempts, the number of neurons in the hidden layer was selected as 10 so that the network could perform best. A total of 285 experimental data were picked from a strain range of 0.05–0.4 in steps of 0.025. Before the network training, the input and output data were normalized according to the Equation (19) for more efficient training:
where,
is the standardized input data,
is the maximum experimental value, and
is the minimum. Additionally, it is noteworthy here that 60% of input and output variables are selected randomly as training data, and the rest is used for testing. The outputs are converted into the original values by Equation (20):
As shown in
Figure 12, it could be found that the predicted stresses by the BP-ANN model are well consistent with the measured stresses. The value of the average absolute relative error between prediction and measurement is only about 0.7%.
As shown in
Table 5, the values of R for all the models are above 0.98 and the values of AARE are 7.6%, 4.0%, and 0.7% for the original Arrhenius model, the modified model, and the BP-ANN model, respectively. After modification, the relative error of the Arrhenius-type is reduced by 47%. Among these models, the ANN model has the highest accuracy. This is mainly because of its intelligent algorithm, which can approach the target value infinitely. With respect to the BP-ANN model, the modified Arrhenius-type model has a relatively low precision but definite physical significance, which considers the heat activated softening mechanism and gives out detailed expressions. In addition, for the stage where the strain should be less than 0.05, it can be regarded as the ideal plasticity and exhibits no obvious softening.
Lastly, the proposed models were further cross-validated according to that in Mosleh et al. [
35]. In each verification, one experimental curve was excluded from the model data. Three typical experimental curves were picked out as shown in
Table 6. The deviation between the predicted and experimental values in the cross-validation was evaluated using Equation (21):
where
is the maximum strain value in the model data. The comparison of experimental and predicted curves for No. 1 verification is depicted in
Figure 13a. The deviations between the measurement and prediction for the proposed models in all the excluded conditions are shown in
Figure 13b. After cross-validation, it could be clearly observed that the modified Arrhenius model has stronger predictability for the flow stress than the BP-ANN model in excluded conditions. The ANN could well fit and approach the model data. However, it requires a large amount of data to train the network for the accurate prediction of untested data.