Modeling of the Hot Flow Behaviors for Ti-6Al-4V-0.1Ru Alloy by GA-BPNN Model and Its Application

Introduction

Ti-6Al-4V-0.1Ru, a α+β titanium alloy, exhibits various properties that are advantageous for chemical processing and other application, such as high mechanical strength to weight ratio, ductility, and ability to withstand high temperatures and resist corrosion [1]. It is well recognized that the mechanical characterization of a specific material is of great complexity and strongly sensitive to the processing parameters, i. e. strain rate and temperature [2]. Consequently, it is definitely beneficial to investigate the flow behaviors of materials over a wide range of strain rates and temperatures. Since deformation behaviors of a material during hot deformation are highly non-linear, modeling and predicting the constitutive stress–strain relationships with a high precision are quite complex in nature. Therefore, it is critical to model the stress–strain relationships accurately.

Phenomenological constitutive functions are developed to implement material flow behaviors in numerical models, particularly in finite element models used for process simulation. Like Arrhenius-type equation, Johnson-Cook (JC), and Khan-Huang-Liang (KHL), etc., these models were selected due to their common usage in simulation of metal forming and frequent citation in the literature. Many researchers have used such rheological models to predict the behaviors of different alloys such as titanium alloy [3, 4, 5, 6], nickel-based superalloy [7], and Al-Zn-Mg-Cu alloy [8]. However, constitutive equations representing non-linear relations between empirical parameters such as σ, ε, ε˙, and T, they are usually limited to specific deformation mechanisms and break down when deformation domains change [5, 9, 10]. Consequently, to overcome these limitations, different equation parameters or separate equations are required to represent data over large domains. Many researchers have also used neural networks to determine the rheological behaviors of materials [11, 12, 13]. A feed-forward back-propagation neural network (BPNN) was generally selected because it has previously shown to work well on the prediction of the flow stress for different materials subject to both warm and hot forming conditions in a range of strain rates [14, 15, 16] even when the materials’ dominant deformation mechanisms were changed [17]. However, back-propagation has drawbacks due to its use of gradient descent. As the dimensionality and/or complexity of data increased, the performance of back-propagation falls off rapidly. It is prone to get trapped at local minima and has the possibility of over-fitting data as reported in Ref. [18]. Moreover, a back-propagation neural network with another difficulties, like that the initialization of the connection weight matrix and optimization of the neural network architectures as mentioned in Ref. [16], blocks its application for some practical problems. Therefore, a BPNN needs to run many times for an optimal network topology and training parameters to achieve higher accuracy, which will consume much time.

Genetic algorithms (GAs) are search and optimization procedures that are motivated by the principles of natural genetics and natural selection [19]. GAs are widely applied to a variety of problem domains due to their ability to reach the global minimum. Particularly, in the field of metal forming, GAs have been utilized to predict the forming behaviors of different materials under a variety of conditions. To exemplify, Lin and Yang [20] determined the constitutive equation of Ti-6Al-4V super-plastic alloy using GA. Yang et al. [21] exploited GA to optimize the parameters for stress-strain constitutive equations during plane strain compression. It is worth to pointing out that GAs should not have the same problem with scaling as back-propagation. One reason for this is that GAs generally improve the current best candidate monotonically by keeping the current best individual as the part of their population while they search for better candidates. In addition, GAs are generally not bothered by local minima. The mutation and crossover operators can step from a valley across a hill to an even lower valley with no more difficulty than descending directly into a valley [22]. There has been a great interest in combining genetic algorithms with artificial neural networks (ANNs) in recent years. Consequently, GAs have been successfully introduced into ANNs at roughly three different levels: connection weights; architectures; and learning rules [23]. These hybrid evolutionary approaches have been significantly used to determine the behaviors of different alloys. For instance, Aguir et al. [24] used GA-ANN hybrid model to calculated material parameters of an anisotropic material. Fu et al. [18] used ANN in conjunction with GA to successfully design an air-bending punch.

In this study, to overcome the gradient-descent-based training algorithm’s shortcomings, GA will be introduced into BPNN, i. e. to formulate the training process as the evolution of connection weights. GA will be used effectively in the evolution to find a near-optimal set of connection weights globally without computing gradient information. The fitness of a BPNN will be defined according to the mean square error (MSE) between targets and actual outputs. Then the accuracy of GA-BPNN model will be assessed with conventional linear regression calculations using well-known statistical errors, such as average relative error (AARE) and correlation coefficient (R) etc. The accurate results will be used to process simulation for the mechanical response of Ti-6Al-4V-0.1Ru using explicit finite element software.

Acquisition of experimental stress-strain data

Material and experimental procedures

The isothermal compression tests were performed on Ti-6Al-4V-0.1Ru titanium alloy. The as-received material was a cold-rolled bar with chemical concentrations (in mass %) of Fe≤0.25, C≤0.08, N≤0.03, H≤0.015, O≤0.13, Al: 5.5~6.5, V: 3.5~4.5, Ru: 0.08~0.14, and the balance of Ti. To eliminate the microstructural characteristics, the raw bar was first annealed at 750 °C for 60s, and cooled to 595 °C in the furnace, followed by air cooling. After that, the homogenized bar was machined into thirty-six cylindrical specimens by wire-electrode cutting with 8 mm in diameter and 12 mm height. Figure 1 shows the optical microstructure of the homogenized specimen after hot treatment, where α-phase grains dispersedly distribute in the matrix of β-phase grains. The transformation temperature from α+β phase to β phase is about 915 °C. The thermo-mechanical simulator, Gleeble-3500, was used for all compression tests over four strain rates (0.01s⁻¹, 0.1s⁻¹, 1s⁻¹, 10s⁻¹) and nine temperatures (750 °C, 800 °C, 850 °C, 900 °C, 950 °C, 1000 °C, 1050 °C, 1100 °C, 1150 °C). All specimens were heated by 10 °C/s and held for 180s, to obtain uniform deformation temperature and simultaneously decrease the anisotropy of material. A special high temperature lubricant was used to prevent cementation. Ultimately, the deformed specimens were quenched immediately in water to retain the elevated-temperature microstructures. During the compression processes, the load-displacement data with respect to the nominal stress-strain relationships were recoded and converted to true stress and true strain according to the following formula: σT=−σN1+εN, σT=−ln1+εN, where σT is true stress, σN is nominal stress, εT is true strain and εN is nominal strain [25].

Figure 1:

Optical photograph of Ti-6Al-4V-0.1Ru after hot treatment.

Flow behaviors of Ti-6Al-4V-0.1Ru

The true compressive stress-strain curves for Ti-6Al-4V-0.1Ru over a wide range of temperatures and strain rates are illustrated in Figure 2. As shown in Figure 2, both deformation temperature and strain rate have considerable influences on the flow behaviors of Ti-6Al-4V-0.1Ru. The flow stress decreases markedly with the increasing of temperature at a certain strain rate, but increases with the strain rate as temperatures fixed. It is owing to the variation of the dislocation multiplication rate and dislocation density [26]. The present flow stress evolution with strain can be roughly summarized into three distinct deformation stages observed by researchers empirically [13, 26, 27, 28]. At the first stage, the flow stress sharply increases to a critical value. Primarily, the work hardening (WH) predominates, and the dynamic recrystallization activation energy accumulates from the storage energy at the grain boundaries. Once the critical driving force achieved, the recrystallization nucleation occurs, that can soften and restore the ductility of the material. The second stage displays a slow growth of flow stress to a peak. Coexisting with dynamic recrystallization (DRX) and dynamic recovery (DRV) simultaneously, work hardening yet predominates. At the third stage, the flow curves can be divided into two types based on the variation tendency. At temperatures of 750–900 °C where α+β phase dominates, the flow stress descends monotonously as evidenced by DRX softening. However, at 950–1150 °C where β phase dominates, the stress approximately keeps a steady state with significant DRV softening.

Figure 2:

The true stress-strain curves of Ti-6Al-4V-0.1Ru titanium alloy under different temperatures with strain rates (a) 0.01s⁻¹; (b) 0.1s⁻¹; (c) 1s⁻¹; (d) 10s⁻¹.

The hybrid GA-BPNN prediction model for the hot flow behaviors of Ti-6Al-4V-0.1Ru alloy

The working principles of genetic algorithm

Genetic algorithms (GAs) are search and optimization procedures that are motivated by the principles of natural genetics and natural selection [19]. GAs encode parameter values or solutions which satisfy constraints and minimize to a specific problem on simple chromosome string. Coding parameters in a binary string is primarily used. Once a string (or a solution) is created by genetic operators, it is necessary to evaluate the solution. Commonly, the fitness of a string is assigned a value which is a solution of a special function with regard to the objective function value. In most cases, however, the fitness is made equal to the objective function value. Since the objective of the optimization is to minimize the objective function, it is to be noted that a solution with a smaller fitness value is better compared to another solution. Figure 5 shows a flowchart of a GA process used in this study. Unlike the method of artificial neural networks (ANNs), a GA begins its search with random set of solutions, instead of just one solution. Once a population of solutions is created at random, each solution is evaluated. A termination criterion is then checked. If the termination criterion is not satisfied, the population of solutions is modified by three main operators and a new (and hopefully better) population is created. The generation counter is incremented to indicate that one generation of GA is completed.

Figure 3:

Variation of MSE with neurons in each hidden layer.

Figure 4:

Schematic illustration of the neural network architecture.

Figure 5:

Schematic of GA-BPNN structure used in this study.

The specified operators (such as reproduction, crossover and mutation operators) are applied so as to preserve important information and to create a new set of populations. The primary objective of the reproduction operator is to emphasize good solutions and eliminate bad solutions in a population, while keeping the population size constant. The reproduction operator cannot create any new solutions in the population. Creation of new solutions is performed in crossover and mutation operators. Usually, a single-point crossover operator is recommended to use. It is performed by randomly choosing a crossing site along the string and by exchanging all bits on the right side of the crossing site. To preserve some good strings selected during the reproduction operator, a crossover probability of pc is used. As a result, 100 pc% strings in the population are used in the crossover operation and 1001−pc% of the population are simply copied to the new population. Furthermore, the mutation operator changes a 1 to a 0 and vice versa with a small mutation probability, pm. The need for mutation is to keep diversity in the population. Readers are referred to Ref. [19] for further details on GA concepts and theory.

The establishment of GA-BPNN model

In this section, a prediction model of the deformation behaviors of Ti-6Al-4V-0.1Ru alloy was established, namely GA-BPNN. It was set up to do a back-propagation neural network weight optimization in conjunction with genetic algorithm. The specific flowchart of GA-BPNN is illustrated in Figure 5. The neural network process is exhibited in Figure 4. More complete description of GA-BPNN is described in subsections.

The preparation of BPNN topological structure and datasets

Several structures of neural networks with different transfer functions, layers, and numbers of neurons were tested and a network with reasonable performance was selected (as shown in Figures 3 and 4). The network was trained with the Bayesian regulation back-propagation training algorithm. It consisted of four layers including two hidden layers with equal neurons number, an input layer with the reciprocal temperatures, the strain rate and the true plastic strain both in logarithmic scale as input values and the output layer with logarithmic stress as the target. Table. 1 summarized the structure of the network used in this study. It is worth pointing out that all data ran in present process were normalized into the dimensionless units in order to ensure training efficiency and prevent a special factor which from the training process [13, 29].

Table 1:

Neural networks structure used.

Number of layers	Neural network
	4
Inputs	I=logε;logε˙;1T
Input range	0.036:0.3010.041:0.2960.068:0.272
Output	logσ
Output range	0.057:0.282
Input layer	3
First hidden layer	15
Second hidden layer	15
Target SSE	1×10−4
Training algorithm	Bayesian regulation

As stated, there is a probability of over-fitting data by using back-propagation algorithm. To avoid this difficulty, the experimental data of Ti-6Al-4V-0.1Ru were divided into two sets, a training dataset and a testing dataset [30]. The neural network was produced only using the training data which were over a strain range from 0.08 to 0.88 and five temperatures with a range between 750 and 1150 °C. The testing data over all temperatures as experiments in a strain regime of 0.08 to 0.88 with 0.04 intervals were then used to check that the network behaves itself.

The preparation of hybrid prediction GA-BPNN model

Given the back-propagation neural network topological structure as described above, the GA-BPNN model operated based on following components [22]:

(1). Chromosome coding: The connection weights (and thresholds) in the neural network were encoded into binary strings with 20 bits, as recommended in Ref. [31].

(2). Evaluation function: Assigned the decoded weights (and thresholds) to the links in the network of the given architecture, ran the network over training sets of samples, and returned the mean square error (MSE) which as the fitness function of GA-BPNN model,

(1)MSE=1N∑i=1N(Ei−Pi)2

where Ei represents the experimental true stress and Pi is the predicted stress. N is the total number of training data employed in this study.

(3). Initialization procedure: The connection weights of the initial members of the population were usually chosen at random with a uniform probability distribution between −0.5 and 0.5. The probability distribution significantly influences the training process.

(4). Operators: The three basic operators are reproduction, crossover and mutation, respectively. In this study, the reproduction operator was implemented by using the stochastic universal sampling method (‘sus’) to determine which population members were chosen as parent that would create offspring for next generation. A single-point crossover operator was used with a crossover probability, pc=0.7. Moreover, to converge more efficiently and maintain the diversity in the population, the mutation operator was employed with a small probability, pm=0.01.

(5). Parameter settings: There were a number of parameters whose values could greatly influence the performance of the algorithm. Except where stated otherwise, these constants were kept across runs. Such as, the population size and the fraction of parent-to-children populations were set to 100 and 0.98, respectively. Also, the maximum number of generations was equal to 100.

To avoid premature termination of the GA-BPNN process and to obtain superior results, loops were designed to re-run the optimization several times; each loop ended if the value of the fitness function (0.0001) or the maximum generation of 100 was reached. The results of each generation were saved and the one leading to minimum MSE was chosen for further network training process (Figure 6 shows the variation of performance of fitness function with generation for GA-BPNN established.).

Figure 6:

Variation of performance of fitness function with generation for GA-BPNN.

Results and discussion

Determination of accuracy

The predicted flow curves compared to the experimental data are shown in Figure 7. It can be seen that the predicted results are in good agreement with the experimental values. GA-BPNN exhibits its ability to determine the flow behaviors of Ti-6Al-4V-0.1Ru over a wide range of temperatures and strain rates. In order to evaluate the prediction performance of GA-BPNN in modelling the flow behaviors, relative error (δ) and average absolute relative error (AARE) were introduced, as expressed by eqs (2) and (3).

(2)δ%=Ei−PiEi×100

(3)AARE(%)=1N∑i=1N|Ei−PiEi|×100

Figure 7:

Comparisons between the experimental values and predicted values by GA-BPNN model at different temperatures and strain rates.

where Ei and Pi represent the experimental and predicted stress values. N is the total number of testing samples.

The relative error calculated is ranging from −5.63 to 7.94 % and the AARE value is 1.202. They are found to be smaller than −12.99~9.33 % and 3.2 that reported in Ref. [32]. In addition, the mean value (μ) and standard deviation (w) given in eqs (4) and (5) are used as the important indexes to reflect the central tendency and discrete degree of a set of data [13]. The smaller the absolute values of w and μ are, the better errors distribution is. As shown in Figure 8, the mean value (μ) and the standard deviation (w) is 0.0490 and 1.6152 respectively. They are obviously better than the values of Ref. [13]. It suggests that the distribution of relative percentage errors obtained by GA-BPNN is more centralized and the GA-BPNN model has good generalization capability.

(4)μ=1N∑i=1Nδi

(5)w=1N−1∑i=1N(δi−μ)2

Figure 8:

The distribution of relative error on the true stress points predicted by GA-BPNN.

where δi represents a value of the relative error, μ and w are the mean value and the standard deviation, respectively.

Moreover, the correlation coefficient (R) was applied to check the accuracy of the GA-BPNN model. It can be calculated using eq. (6). R is a numerical value between −1 and 1 that expresses the strength of the linear relationships between two variables. When R-value is close to 1, it illustrates that the predicted values agree well with the experimental values; When R-value is 0, there is no relationship; and when R-value is close to −1, the strong negative relationships are obtained [13].

(6)R=∑i=1NEi−EˉPi−Pˉ∑i=1NEi−Eˉ2∑i=1NPi−Pˉ

where Eˉ and Pˉ are the mean values of experimental and predicted values.

Figure 9 shows the training/testing true stress values determined by GA-BPNN against the experimental values of all specimens. It can be seen that the GA-BPNN model was able to predict the flow behaviors of Ti-6Al-4V-0.1Ru sufficiently well since the results lie mostly on the line of exact fit. Meanwhile, the R-values shown in Figure 9 for training and testing are 0.99998 and 0.99974 respectively. Compared to the data in Ref. [32], the GA-BPNN model exhibits higher accuracy in modelling flow stress.

Figure 9:

Comparison of predicted true stress values with experimental values for the GA-BPNN model.

Prediction potentiality of GA-BPNN model

There is no doubt that the hybrid GA-BPNN model successfully predicts the flow behaviors of Ti-6Al-4V-0.1Ru in any temperature and strain rate tested. With the well-trained GA-BPNN model, the stress-strain curves at temperatures of 800 °C, 900 °C, 1000 °C, 1100 °C were predicted and confirmed. In this section, based on all sets of predicted data from GA-BPNN model, an interpolation method was implemented to densely insert stress-strain data for Ti-6Al-4V-0.1Ru [33]. Additionally, a 3-D continuous response space was set up by a surface fitting process in MATLAB (2015a). As Figure 10 shown, the 3-D continuous response space with orthogonal slice planes was constructed through volumetric data (V, sx, sy, sz) comprised of the values of temperature, strain rate, true strain and true stress. Figure 10(b–d) was obtained, to slice the object in Figure 10(a) along the x, y, z directions in the volume V. V was a volume array containing true stress values at the default location X=750:1150, Y=0.01:10, Z=0.08:0.88. Each element in the vectors sx, sy, and sz (which respectively means temperature, strain rate and true strain) defined a slice plane in the x-, y-, or z-axis direction. The colour at each point was determined by 3-D interpolation into the volume V. The overall flow stress in three different directions is shown in Figure 10(a). And Figure 10(b–d) respectively exhibit the slice planes corresponding to arbitrary temperatures, strain rates and true strains. It can be seen that the 3-D continuous response space leads to visually reveal the successive relationships among temperature, strain rate, true strain and true stress confirmed with GA-BPNN excellent prediction performance. Obviously, a large number of stress-strain data that are outside of experimental conditions can enrich the flow stress and narrow the interpolation intervals, and make a perfection of 3-D continuous response space. Therefore, it can be summarized that the usage of 3-D continuous response space in conjunction with GA-BPNN model can enrich the database which corresponding to the flow behaviors of Ti-6Al-4V-0.1Ru at arbitrary temperature, strain rate and strain, that is especially useful for further FEM simulation application.

Figure 10:

The 3-D relationships among temperature, strain rate, strain and stress: (a) 3-D continuous interaction space, 3-D continuous mapping relationships under different (b) strains; (c) strain rates and (d) temperatures.

Application of GA-BPNN in FEM simulation

To further study the accuracy of GA-BPNN model and verify the mechanical response of Ti-6Al-4V-0.1Ru based on the predicted stress-strain data, the hot isothermal compression tests were simulated using DEFORM dynamic explicit finite element software. The simulation processes were carried on with a height reduction ratio of 60 % at 800 °C and 1s⁻¹. All of the initial conditions were identical except for stress-strain curves of Ti-6Al-4V-0.1Ru. Stress-strain curves measured experimentally at temperatures of 750 °C, 850 °C, 900 °C, 950 °C, 1000 °C and strain rates of 0.01s⁻¹, 0.1s⁻¹, 10s⁻¹, were applied to Scheme-A, where flow curves at 800 °C and 1s⁻¹ would be automatically interpolated by DEFORM. However, in Scheme-B, the flow curves at 800 °C and 1s⁻¹ predicted by GA-BPNN were totally applied in simulation. It is worth pointing out that all stress-stain curves used were interrupted at the strain of 0.08 as plastic deformation taken into account only. The thermal radiation and heat exchange among objects were ignored and corresponded with the actual isothermal experimental conditions. Besides, the shear friction coefficient of 0.1 was chosen to imitate the real graphite lubrication. The velocities of anvils measured experimentally were adopted to ensure that the specimen would deform at the same strain rates in the numerical simulation. Mesh sensitivity was considered since the absolute mesh generation method was used to enhance simulation accuracy by fixing mesh size. Table. 2 displays the details of time-stroke and time-force data that are equivalent to the experimental values.

Figure 11 shows contours of effective strain in compressed specimens at the final plastic deformation simulated by FEM simulations using the different stress-strain curves. Figure 11(a) corresponds to Scheme-A and Figure 11(b) represents Scheme-B. In both cases, specimens display a typical drum-type. The main area deformed uniformly but the edges experienced larger strains. Furthermore, it can be seen from Figure 11 that the average strains of present intersection are similar, which are 0.868 and 0.881 respectively. However, there is a great difference between the maximal equivalent strain values. Namely, the specimen was strained less in Scheme-A (the maximum is 2.55) than Scheme-B (the maximum is 3.79).

Figure 11:

Contours of effective strain values predicted by (a) Scheme-A and (b) Scheme-B at final deformation for specimens deformed at 800 °C and 1s⁻¹.

In order to quantify the accuracy of each scheme, the load-time data predicted by finite element analysis at 800 °C and 1s⁻¹ were compared with the experimental values, and the overall relative errors (δ) were calculated in each case. It can be seen from Figure 12 that both the Scheme-A and the Scheme-B are able to predict the mechanical response of this billet material with acceptable accuracy. However, the GA-BPNN model leads to more accurate simulation results. It is worth pointing out that both two schemes loosed their precision before one point, 0.07s, as evidenced by the two flow curves used disregarding the elastic deformation for Ti-6Al-4V-0.1Ru. Moreover, as shown in Table 2, the relative error (δ) of the simulation results from DEFORM ranges from 4.21 to 11.77 %, but that with respect to GA-BPNN model is smaller from −3.62 to 9.46 %. It indicates that the simulation results based on GA-BPNN model are closer to the experimental values. Consequently, it can be summarized that GA-BPNN model can predict more reasonable flow curves than the interpolation method of DEFORM itself, and the GA-BPNN would be a superior approach to improve simulation precision in complex forming process.

Figure 12:

The corresponding relationships between time and the loading force of top die for two scehmes compared with experimental values.

Conclusions

In this study, the flow behaviors of Ti-6Al-4V-0.1Ru over a strain-rate range from 0.01 to 10s^-1and a temperature range of 750–1150 °C were investigated using a GA-BPNN hybrid prediction model. The predicted flow curves were then compared to experimental data and the accuracy was evaluated with linear regression method calculations. The following conclusions can be made:

(1). The hot deformation behaviors of Ti-6Al-4V-0.1Ru show a highly non-linear intrinsic relationships with temperature, strain and strain rate, following work hardening (WH), dynamic recrystallization (DRX) and dynamic recovery (DRV).

(2). It is well-known that genetic algorithms have been introduced into ANNs at roughly three different levels: connection weights; architectures; and learning rules. GA-BPNN model was set up to overcome gradient-descent-based training algorithm’s (back-propagation’s) shortcomings by formulating the training process as the evolution of connection weights. GA-BPNN model shows an excellent performance on modeling flow behaviors of Ti-6Al-4V-0.1Ru.

(3). A series of statistical indexes, involving the relative error (δ), average absolute relative error (AARE), mean value (μ), standard deviation (w), and correlation coefficient (R), were introduced to evaluate the prediction accuracy of GA-BPNN model. The relative error calculated is ranging from −5.63 to 7.94 % and the AARE value is 1.202. The mean value (μ) and standard deviation (w) are 0.0490 and 1.6152 respectively. Meanwhile, the R-values for training and testing are 0.99998 and 0.99974 respectively. These results indicate that GA-BPNN prediction model has a great capability on calculating stress-strain values of Ti-6Al-4V-0.1Ru against experimental data.

(4). The 3-D continuous response space in conjunction with GA-BPNN leads to visually reveal the successive relationships among temperature, strain rate, strain and stress. Flow stress values predicted by GA-BPNN model exhibit better accuracy when compared analytically with the values interpolated by DEFORM. More accurate load-time curves were achieved.