# Prediction of the Hot Compressive Deformation Behavior for Superalloy Nimonic 80A by BP-ANN Model

^{*}

## Abstract

**:**

^{−1}. A comparison on a BP-ANN model and modified Arrhenius-type constitutive equation has been implemented in terms of statistical parameters, involving mean value of relative (μ), standard deviation ($w$), correlation coefficient (R) and average absolute relative error (AARE). The $\mathsf{\mu}$-value and $w$-value of the improved Arrhenius-type model are 3.0012% and 2.0533%, respectively, while their values of the BP-ANN model are 0.0714% and 0.2564%, respectively. Meanwhile, the R-value and ARRE-value for the improved Arrhenius-type model are 0.9899 and 3.06%, while their values for the BP-ANN model are 0.9998 and 1.20%. The results indicate that the BP-ANN model can accurately track the experimental data and show a good generalization capability to predict complex flow behavior. Then, a 3D continuous interaction space for temperature, strain rate, strain and stress was constructed based on the expanded data predicted by a well-trained BP-ANN model. The developed 3D continuous space for hot working parameters articulates the intrinsic relationships of superalloy nimonic 80A.

## 1. Introduction

^{−1}on a Gleeble 3500 thermo-mechanical simulator (Dynamic Systems Inc., New York, NY, United States). A BP-ANN model which takes temperature ($T$), strain rate ($\dot{\mathsf{\epsilon}}$) and strain ($\mathsf{\epsilon}$) as the input variables, and true stress ($\mathsf{\sigma}$) as the output variable was established by determining proper network structure and parameters to predict the non-linear complex flow behaviors. Meanwhile, a strain-dependent Arrhenius-type constitutive model was constructed to predict the flow stress of nimonic 80A. Subsequently, a comparative analysis on the performance of two such models has been carried out by a series of evaluators such as relative error (δ), average absolute relative error (AARE) and correlation coefficient (R), which predictably indicates that the former has higher prediction accuracy. In the following, as described previously, a 3D continuous interaction space within the temperature range of 950–1250 °C, strain rate range of 0.01–10 s

^{−1}, and strain range of 0.1–0.9 was constructed.

## 2. Materials and Experimental Procedure

^{−1}, 0.1 s

^{−1}, 1 s

^{−1}and 10 s

^{−1}[25]. After each compression, the deformed specimen was immediately quenched into water to retain the high temperature microstructures.

_{T}= σ

_{N}(1 + ε

_{N}), ε

_{T}= ln(1 + ε

_{N}), where σ

_{T}is true stress, σ

_{N}is nominal stress, ε

_{T}is true strain and ε

_{N}is nominal strain.

## 3. Flow Behavior Characteristics of Superalloy Nimonic 80A

^{−1}and 1050–1250 °C, 0.1 s

^{−1}and 1050–1200 °C and 1–10 s

^{−1}and 1050–1200 °C. However, in the parameter domains of 0.1–10 s

^{−1}and 1250 °C, the stress approximately keeps a steady state with significant DRV softening. From the previous descriptions, the typical form of flow curve with DRX softening involved a single peak followed by a flow of steady state. The reason lies in the fact that the highter rate of work hardening slows down the DRX softening rate with lower temperatures and higher strain rates, therefore, the onset of steady state flow is shifted to higher levels [4].

## 4. Development of Constitutive Relationship for Superalloy Nimonic 80A

#### 4.1. BP-ANN Model

^{−1}and 1100 °C and 1 s

^{−1}and 1200 °C. Among such two curves, the stress values of 36 points picked out from 0.05 to 0.9 with a strain interval of 0.05, and 162 points from the other eighteen training curves in a strain range of 0.05 to 0.9 with a strain interval of 0.1 were considered as the test data for the BP-ANN work performance. The BP-ANN model was trained based on the training dataset, and generalization property of the trained network was assessed by the test dataset selected with a fixed strain rate.

^{−1}, and temperature data varies from 1050 to 1250 °C, the output flow stress data varies from 25.93 MPa to 387.63 MPa. Therefore, before training the network, the input and output datasets have been normalized to avoid value concentrating on weights and some neurons when the iterative calculation of BP-ANN. The main reason for normalizing the data matrix is to recast them into the dimensionless units to remove the arbitrary effect of similarity between the different data. In this research, the normalization processing was realized by Equation (1) [6,17]. The coefficients of 0.05 and 0.25 in Equation (1) are regulating parameters for the sake of narrowing the magnitude of the normalized data within 0.05 to 0.3. Furthermore, it should be noted that the initial numerical values of true stain rates exhibit great magnitude distinction, thereby a logarithm was taken for transforming the true stain rate data before normalization processing.

_{min}is the minimum value of x and x

_{max}is the maximum; x

_{n}is the value of x after normalization processing.

^{−1}and 1100 °C and 1 s

^{−1}and 1200 °C, are used to assess the generalization property of the BP-ANN model. The result of comparisons shows that the true stresses predicted by BP-ANN model has good agreement with experimental stress-strain curves, which indicates the high generalization property of the BP-ANN model.

#### 4.2. Arrhenius-Type Constitutive Model

^{−1}), $R$ is the universal gas constant (8.31 J·mol

^{−1}·K

^{−1}), $T$ is the absolute temperature (K), $Q$ is the activation energy of deformation (kJ·mol

^{−1}), $\mathsf{\sigma}$ is the flow stress (MPa) for a given stain, $A$, $\mathsf{\alpha}$, ${n}^{\prime}$ and $n$ are the material constants, $\mathsf{\alpha}={\mathsf{\beta}/n}^{\prime}$.

^{−1}and 0.0336 MPa

^{−1}, respectively. Furthermore, the value of another material constant $\mathsf{\alpha}=\mathsf{\beta}/{n}^{\prime}=0.0063\text{\hspace{0.05em}}$MPa

^{−1}was also obtained.

^{−1}. In addition, the material constant A can be calculated as 4.5496 × 10

^{14}s

^{−1}.

^{−1}, 0.1 s

^{−1}, 1 s

^{−1}and 10 s

^{−1}. It can be seen that the proposed constitutive equation gives an accurate estimation on the flow stress of superalloy nimonic 80A in most of the experimental conditions.

## 5. Prediction Capability Comparison between the BP-ANN Model and Arrhenius Type Constitutive Equation

^{−1}and 1200 °C and 1 s

^{−1}at a strain range of 0.05 to 0.9 with a strain interval of 0.05 were calculated to compare with the true stress predicted by BP-ANN model and obtained from the isothermal compression tests. For the sake of the contrast of prediction accuracy between these two models, the relative error (δ) is introduced, which is expressed by Equation (14).

_{i}is a value of the relative error; μ, w, and y are the mean value, standard deviation and probability density of δ respectively; y

_{0}and A are constants, and N is the number of relative errors, here N = 36.

^{−1}and 1200 °C and 1 s

^{−1}did not participate. However, when establishing the constitutive equation, they were involved. However, even on this premise, the BP-ANN model still shows smaller errors, giving the full proof that the present BP-ANN model has better prediction capability than the constitutive equation in the flow characteristics of superalloy nimonic 80A.

## 6. Prediction Potentiality of BP-ANN Model

^{−1}, 0.1 s

^{−1}, 1 s

^{−1}and 10 s

^{−1}, an interpolation method was implemented to densely insert stress-strain data into these data; furthermore, a 3D continuous response space (illustrated in Figure 10) with flow stress along the V-axis and deformation temperature, logarithm of strain rate and strain and along the X, Y and Z axes, respectively, was constructed by a surface fitting process. The values of V-axis are represented by different colors. Figure 10a shows the 3D continuous interaction space, which reveals the continuous response relationship between stress and strain, strain rate and temperature of superalloy nimonic 80A. Figure 10b–d respectively exhibit the cutting slices of 3D continuous response mapping at diverse parameters, involving temperature, strain rate and strain. In the 3D continuous interaction space, all the stress-strain points are digital and can be determined, since the surface fitting step has transformed the discrete stress-strain points into continuous stress-strain surface and space. The accuracy of such a 3D continuous interaction space is strongly guaranteed by the excellent prediction performance of an optimally-constructed and well-trained BP-ANN model. As is known, the stress-strain data are the most fundamental data to predict the deformation behaviors of the superalloy nimonic 80A during electric upsetting with finite element model It is realizable to pick out dense stress-strain data from the 3D continuous interaction space and insert such continuous mapping relationships into commercial software such as Marc, etc. by program codes. In this way, the accurate simulation of one certain forming process is able to perform.

## 7. Conclusions

- (1)
- A BP-ANN model taking the deformation temperature (T), strain rate ($\dot{\mathsf{\epsilon}}$) and strain ($\mathsf{\epsilon}$) as input variables and the true stress ($\mathsf{\sigma}$) as output variable was constructed for the compression flow behaviors of superalloy, nimonic 80A, which presents desired precision and reliability.
- (2)
- A strain-dependent Arrhenius-type model is developed to predict the flow behavior of superalloy nimonic 80A under the specific deformation conditions. A sixth order polynomial is adopted to reveal the relationships between variable coefficients (including activation energy $Q$, material constants $n$, $\mathsf{\alpha}$, and $A$) and strain with good correlations.
- (3)
- A series of statistical indexes, involving the relative error (δ), mean value ($\mathsf{\mu}$), standard deviation ($w$), correlation coefficient (R) and average absolute relative error (ARRE), were introduced to contrast the prediction accuracy between the improved Arrhenius type constitutive equation and BP-ANN model. The mean value ($\mathsf{\mu}$) and standard deviation ($w$) of the improved Arrhenius-type model are 3.0012% and 2.0533%, respectively, while their values of the BP-ANN model are 0.0714% and 0.2564%, respectively. Meanwhile, the correlation coefficient (R) and average absolute relative error (ARRE) for the improved Arrhenius-type model are 0.9899 and 3.06%, while their values for the BP-ANN model are 0.9998 and 1.20%, which indicate that the BP-ANN model has a good generalization capability.
- (4)
- The true stress data within the temperature range of 950–1250 °C, the strain rate range of 0.01–10 s
^{−1}, and the strain range of 0.1–0.9 were predicted densely. According to these abundant data, a 3D continuous interaction space was constructed by interpolation and surface fitting methods. It significantly contributes to all the research requesting abundant and accurate stress-strain data of superalloy nimonic 80A.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**The true stress-strain curves of superalloy nimonic 80A under the different temperatures with strain rates (

**a**) 0.01 s

^{−1}; (

**b**) 0.1 s

^{−1}; (

**c**) 1 s

^{−1}; (

**d**) 10 s

^{−1}.

**Figure 3.**The comparison of the BP-ANN prediction with experimental values at different temperatures and strain rates (

**a**) 0.01 s

^{−1}; (

**b**) 0.1 s

^{−1}; (

**c**) 1 s

^{−1}; (

**d**) 10 s

^{−1}.

**Figure 4.**Relationship between (

**a**) $\text{ln}\mathsf{\sigma}$ and $\text{ln}\dot{\mathsf{\epsilon}}$ and (

**b**) $\mathsf{\sigma}$ and $\text{ln}\dot{\mathsf{\epsilon}}$.

**Figure 5.**Relationships between: (

**a**) $\text{ln}(\text{sinh}(\mathsf{\alpha}\mathsf{\sigma}))$ and $\text{ln}\dot{\mathsf{\epsilon}}$ (

**b**) $\text{ln}(\text{sinh}(\mathsf{\alpha}\mathsf{\sigma}))$ and $1/T$ .

**Figure 6.**Relationships between: (

**a**) $\mathsf{\alpha}$; (

**b**) lnA; (

**c**) n; and (

**d**) Q and true strain $\mathsf{\epsilon}$ by polynomial fit.

**Figure 7.**Comparisons between predicted and measured under different deformation temperatures with strain rates of (

**a**) 0.01 s

^{−1}; (

**b**) 0.1 s

^{−1}; (

**c**) 1 s

^{−1}and (

**d**) 10 s

^{−1}.

**Figure 8.**The relative errors distribution on the true stress points predicted by (

**a**) the BP-ANN model and (

**b**) the Arrhenius type constitutive equation relative to the experimental ones.

**Figure 9.**The correlation relationships between the predicted and experimental true stress for the (

**a**) BP-ANN model and (

**b**) Arrhenius-type model.

**Figure 10.**The 3D relationships among temperature, strain rate, strain and stress: (

**a**) 3D continuous interaction space, 3D continuous mapping relationships under different (

**b**) temperatures; (

**c**) strain rates and (

**d**) strains.

α | lnA | n | Q | ||||
---|---|---|---|---|---|---|---|

B_{0} | 0.01006 | C_{0} | 40.29126 | D_{0} | 7.23447 | E_{0} | 467.28232 |

B_{1} | −0.05783 | C_{1} | −144.12464 | D_{1} | −44.03632 | E_{1} | −1502.21622 |

B_{2} | 0.33009 | C_{2} | 982.95072 | D_{2} | 234.75515 | E_{2} | 10438.62694 |

B_{3} | −0.95142 | C_{3} | −3062.36904 | D_{3} | −653.05658 | E_{3} | −32768.83516 |

B_{4} | 1.47519 | C_{4} | 4851.02761 | D_{4} | 984.16398 | E_{4} | 52125.14469 |

B_{5} | −1.16554 | C_{5} | −3813.23640 | D_{5} | −756.88970 | E_{5} | −41.091.74522 |

B_{6} | 0.36783 | C_{6} | 1183.77777 | D_{6} | 232.91924 | E_{6} | 12784.92195 |

**Table 2.**Relative errors of the predicted results by the back-propagational artificial neural network (BP-ANN) model and constitutive equation to experimental results under the condition of 1100 °C and 0.01 s

^{−1}and 1200 °C and 1 s

^{−1}.

Strain Rate (s^{−1}) | Temperature (°C) | Strain | True Stress (MPa) | Equation | Relative Error (%) | ||
---|---|---|---|---|---|---|---|

Experimental | BP-ANN | BP-ANN | Equation | ||||

0.01 | 1100 | 0.05 | 65.08 | 63.59 | 64.37 | −2.29 | −1.08 |

0.10 | 71.80 | 71.58 | 71.18 | −0.31 | −0.86 | ||

0.15 | 75.24 | 74.75 | 72.79 | −0.64 | −3.25 | ||

0.20 | 76.09 | 75.31 | 72.42 | −1.03 | −4.83 | ||

0.25 | 75.35 | 74.99 | 72.30 | −0.47 | −4.04 | ||

0.30 | 73.81 | 74.05 | 71.59 | 0.33 | −3.01 | ||

0.35 | 72.06 | 72.75 | 71.23 | 0.96 | −1.15 | ||

0.40 | 70.57 | 71.31 | 70.43 | 1.04 | −0.20 | ||

0.45 | 69.47 | 69.93 | 71.33 | 0.67 | 2.68 | ||

0.50 | 68.73 | 68.77 | 70.15 | 0.05 | 2.06 | ||

0.55 | 68.32 | 67.90 | 70.40 | −0.62 | 3.04 | ||

0.60 | 68.17 | 67.36 | 70.40 | −1.18 | 3.28 | ||

0.65 | 68.20 | 67.13 | 69.34 | −1.56 | 1.68 | ||

0.70 | 68.34 | 67.13 | 70.24 | −1.78 | 2.77 | ||

0.75 | 68.54 | 67.26 | 69.70 | −1.87 | 1.69 | ||

0.80 | 68.72 | 67.39 | 68.91 | −1.93 | 0.28 | ||

0.85 | 68.80 | 67.37 | 69.76 | −2.08 | 1.39 | ||

0.90 | 68.73 | 67.08 | 67.71 | −2.41 | −1.49 | ||

1 | 1200 | 0.05 | 91.67 | 87.76 | 90.72 | −4.26 | −1.04 |

0.10 | 102.36 | 105.69 | 111.12 | 3.24 | 8.55 | ||

0.15 | 106.88 | 108.91 | 118.77 | 1.90 | 11.13 | ||

0.20 | 109.66 | 110.69 | 121.14 | 0.94 | 10.47 | ||

0.25 | 111.91 | 112.48 | 121.60 | 0.51 | 8.66 | ||

0.30 | 113.69 | 114.08 | 120.83 | 0.35 | 6.28 | ||

0.35 | 114.99 | 115.35 | 120.39 | 0.31 | 4.69 | ||

0.40 | 115.82 | 116.21 | 119.02 | 0.34 | 2.76 | ||

0.45 | 116.23 | 116.63 | 120.40 | 0.34 | 3.59 | ||

0.50 | 116.27 | 116.61 | 118.44 | 0.30 | 1.87 | ||

0.55 | 115.99 | 116.25 | 118.69 | 0.22 | 2.33 | ||

0.60 | 115.45 | 115.62 | 118.07 | 0.14 | 2.26 | ||

0.65 | 114.71 | 114.75 | 115.69 | 0.04 | 0.86 | ||

0.70 | 113.82 | 113.58 | 116.16 | −0.20 | 2.06 | ||

0.75 | 112.83 | 112.03 | 114.25 | −0.71 | 1.26 | ||

0.80 | 111.81 | 110.07 | 111.94 | −1.55 | 0.12 | ||

0.85 | 110.80 | 107.84 | 111.96 | −2.67 | 1.05 | ||

0.90 | 109.86 | 105.50 | 107.05 | −3.97 | −2.56 |

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**MDPI and ACS Style**

Quan, G.-z.; Pan, J.; Wang, X. Prediction of the Hot Compressive Deformation Behavior for Superalloy Nimonic 80A by BP-ANN Model. *Appl. Sci.* **2016**, *6*, 66.
https://doi.org/10.3390/app6030066

**AMA Style**

Quan G-z, Pan J, Wang X. Prediction of the Hot Compressive Deformation Behavior for Superalloy Nimonic 80A by BP-ANN Model. *Applied Sciences*. 2016; 6(3):66.
https://doi.org/10.3390/app6030066

**Chicago/Turabian Style**

Quan, Guo-zheng, Jia Pan, and Xuan Wang. 2016. "Prediction of the Hot Compressive Deformation Behavior for Superalloy Nimonic 80A by BP-ANN Model" *Applied Sciences* 6, no. 3: 66.
https://doi.org/10.3390/app6030066