Arrhenius constitutive equation and artificial neural network model of flow stress in hot deformation of offshore steel with high strength and toughness

ABSTRACT

In this study, the thermal deformation behaviour of a high strength offshore steel at different temperatures and rates was investigated through thermal compression experiments.An Arrhenius constitutive model and a back propagation artificial neural network (BP-ANN) were established to address more complex deformation characteristics. The performances of both models were was evaluated using standard statistical parameters such as the correlation coefficient (R) and average absolute relative error (AARE). The results showed that both models can accurately predict the rheological stresses generated during deformation.The BP-ANN outperforms the Arrhenius equation model with correlation coefficients of fit greater than 99.9% and less than 0.8% relative error. At a strain rate of 0.01 s⁻¹ and 10 s⁻¹, the accuracy of the ANN decreases slightly due to the fact that it exceeds the strain rate range of the training set, as compared to the Arrhenius constitutive equations as these are more accurately predicted.

KEYWORDS:

• High strength offshore steel
• compression experiment
• Arrhenius constitutive equation
• back-propagation artificial neural network
• flow stress

Introduction

The thermal deformation behaviour of metals is significantly influenced by the deformation parameters (temperature, deformation rate, etc.). During plastic deformation under different conditions, the phenomena of work hardening (WH), dynamic recovery (DRV), and dynamic recrystallisation (DRX) occur [1,2]. In WH the plastic properties of the material are reduced, and the rheological stress is increased. However, softening phenomena such as DRV and DRX reduce the rheological stress and restore plasticity [3,4]. Therefore, the interaction under thermal deformation conditions is an important reason for complex deformation behaviour [5,6].

In recent years, researchers have proposed various intrinsic constitutive equations to describe the high-temperature deformation behaviour of metallic materials. One of the most typical constitutive equations is the Arrhenius intrinsic constitutive equation, which considers the deformation temperature and strain rate [7]. However, the traditional Arrhenius equation does not consider the effect of degree of deformation. Shi et al. [8] combined the strain degree and temperature in the equation to obtain the flow stress constitutive equation of material strain parameters, which is widely used to predict the flow stress of various metals. Examples include P91 steel [9], Ti-deformed austenitic stainless steel [10], 800 H high-temperature alloy [11], TC4-DT titanium alloy [12], AZ81 magnesium alloy [13], and 2124 aluminium alloy [14]. In these studies, the average relative error (AARE) between the calculated and experimental values was mostly concentrated within the range of 5–9%, and artificial neural network (ANN) models that did not require repeated regression calculations were introduced to address the problem of low prediction accuracy [15–19].

ANNs, a newer artificial intelligence technique, are more effective in solving highly complex problems compared to regression methods. The typical structure of a BP-ANN consists of an input layer, output layer, and one or more hidden layers with artificial neurons that select and weigh the inputs. The input layer receives the input data, and after processing, sends the data to the hidden layer. The hidden layer acts as a complex network structure to model the nonlinear relationship between the input and output layers. It processes the data computations, and sends the responses to the output layer, which produces outputs [20,21].

In recent years, an increasing number of researchers have used ANN models to dissect the relationship between material properties. Lin et al. [22] established optimal hot-forming process parameters for 42CrMo steel based on ANN models. Zhao et al. [23] characterised the thermal deformation of Ti600 titanium alloy using an instantaneous equation and ANN. Quan et al. [24] developed an ANN model for cast Ti-6Al-2Zr-1Mo-1 V alloy over a wide temperature range involving phase transformation, and predicted the high-temperature flow behaviour of 20 MnNiMo alloys using an ANN [25].

Intrinsic constitutive equations and ANNs have been applied to characterise the rheological behaviour of materials. Examples include titanium alloys [26,27], stainless steels [28,29], high-temperature alloys [30], aluminium [31,32] and magnesium alloys [33,34]. The relationship between the process, microstructure, and properties can be accurately characterised after considering the effects of the parameters together. For the offshore steels, the thermal deformation behaviour also has a strong influence on the combination of strength and toughness [35–43], which is of great significance for applications in the offshore engineering environments. In this study, the hot deformation stresses of high-strength offshore steels under different models were predicted separately using an Arrhenius equation-based model and a BP-ANN. Error analysis was performed to evaluate the performance of these models in predicting rheological stresses.

Material and experiments

Experiments were conducted using forged ingots with a mass of approximately 65 kg, and the smelting process was conducted under complete vacuum (Table 1). From the ingots, Φ8 mm × 12 mm sized specimens were machined and introduced in a Gleeble-3800 testing machine for single pass hot compression tests. The specimens were heated at a rate of 5°C/s to 1200°C for 5 min, cooled at a rate of 5°C/s to the corresponding deformation temperatures (800°C, 900°C, 1000°C, 1100°C), held for 30 s, compressed at the corresponding strain rates (0.1 s⁻¹, 0.5 s⁻¹, 1 s⁻¹, 5 s⁻¹) with a 55% depression, and then water quenched after compression to retain the deformation morphology. Hot-compression tests were performed under Ar gas to avoid metal oxidation during heating and deformation.

Table 1. Chemical composition of high-strength offshore steel (in wt%).

Element	C	Cr	Ni	Mo	Si	Mn	Al	Fe
Mass fraction	0.29	0.7	0.6	0.45	0.3	1.3	0.02	Balance

The true stress-true strain curves of the experimental specimens deformed at different rates and temperatures for a single pass of compression, as shown in Figure 1. As the deformation temperature decreases and the strain rate increases, the rheological stress of the material shows an increasing trend. The slow decrease in the rheological stress after the strain-hardening stage is due to dynamic softening caused by the occurrence of DRV and DRX. The rheological stress gradually tends to a steady state at 1100°C and 0.1 s⁻¹, indicating a more balanced state between DRV, DRX and WH. These findings show that the stresses in high-strength offshore steel exhibit highly nonlinear behaviour in response to the three deformation parameters (temperature, strain, and strain rate)

Figure 1. True stress-strain curves of the experimental steel at the temperatures of 1073 ~ 1373 K and at the strain rates of (a) 0.1 s⁻¹, (b) 0.5 s⁻¹, (c) 1 s⁻¹, and (d) 5 s⁻¹.

Comparison between improved Arrhenius-type constitutive equation and BP-ANN model

Improved arrheniue constitutive equation

The Arrhenius-type constitutive equation is widely used as an image-only model to describe the intrinsic constitutive relationship of materials. However, strain, which has an important influence on the flow behaviour, has been neglected in equations proposed in previous studies. Thus, the previous Arrhenius-type constitutive equations could not predict flow behaviour accurately. Lin [44] et al. first proposed a modified model considering the effect of strain, and used it to accurately describe the deformation behaviour of 42CrMo steel at high temperatures. Owing to the superiority of predictions of the revised model, the modified Arrhenius-type intrinsic constitutive equation has been successfully applied to describe the intrinsic constitutive relationships of various materials [45–47].

(1) $\dot{\epsilon }=A_1{\sigma }^{n1}exp[-Q/\left(RT\right)]\left(\alpha \sigma <0.8\right)$(1)

(2) $\dot{\epsilon }=A_{2}exp\left(\beta \sigma \right)\cdot exp\left[-Q/\left(RT\right)\right](\alpha \sigma >1.2)$(2)

(3) $\dot{\epsilon }=A[sinh\left(\alpha \sigma \right){]}^n\cdot exp\left[-Q/\left(RT\right)\right]\left(\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\right)$(3)

In this study, the coefficients of the intrinsic constitutive equations are solved at a strain of 0.5. The natural logarithms of Equations 1(1) $\dot{\epsilon }=A_1{\sigma }^{n1}exp[-Q/\left(RT\right)]\left(\alpha \sigma <0.8\right)$(1) and 2(2) $\dot{\epsilon }=A_{2}exp\left(\beta \sigma \right)\cdot exp\left[-Q/\left(RT\right)\right](\alpha \sigma >1.2)$(2) can be expressed as

(4) $ln\dot{\epsilon }=ln{A}_1+n_{1}ln\sigma -Q/\left(RT\right)$(4)

(5) $\mathrm{l}\mathrm{n}\dot{\epsilon }=\mathrm{l}\mathrm{n}{A}_2+\beta \sigma -Q/\left(RT\right)$(5)

As shown in Figure 2, the linear regression of the data under different deformation temperatures using the least-squares method can yield the value of $n_1$ at different temperatures, and the average value is 17.299. Similarly, $\beta$ is 0.131 MPa⁻¹, and $\alpha =\beta /n_1$, which is obtained as 0.00757 MPa⁻¹.

Figure 2. The relationship between flow stress and strain rate for the experimental steel: (a) $\sigma -ln\dot{\epsilon }$ plot; (b) $ln\sigma -ln\dot{\epsilon }$ plot.

Equation 6(6) $Q=R\cdot \left\{\mathrm{\partial }ln\dot{\epsilon }/\mathrm{\partial }ln\left[sinh\left(\alpha \sigma \right)\right]\right\}{|}_T\cdot \left\{\mathrm{\partial }ln\left[sinh\left(\alpha \sigma \right)\right]/\mathrm{\partial }\left(1/T\right)\right\}{|}_{\dot{\epsilon }}$(6) is obtained by taking the natural logarithm of both sides of Equation 3(3) $\dot{\epsilon }=A[sinh\left(\alpha \sigma \right){]}^n\cdot exp\left[-Q/\left(RT\right)\right]\left(\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\right)$(3) and applying a partial differential.

(6) $Q=R\cdot \left\{\mathrm{\partial }ln\dot{\epsilon }/\mathrm{\partial }ln\left[sinh\left(\alpha \sigma \right)\right]\right\}{|}_T\cdot \left\{\mathrm{\partial }ln\left[sinh\left(\alpha \sigma \right)\right]/\mathrm{\partial }\left(1/T\right)\right\}{|}_{\dot{\epsilon }}$(6)

As shown in Figure 3, the slope reciprocal of $ln\left[sinh\left(\alpha \sigma \right)\right]-ln\dot{\epsilon }$ is evaluated as 6.001, and the slope of $ln\left[sinh\left(\alpha \sigma \right)\right]-1000/T$ is evaluated as 7.385, respectively.

Figure 3. Experimental rheological stress as a function of strain rate and deformationtemperature: (a) $ln\left[sinh\left(\alpha \sigma \right)\right]-ln\dot{\epsilon }$ plot; (b) $ln\left[sinh\left(\alpha \sigma \right)\right]-1000/T$ plot.

The regression coefficient obtained from Figure 3 is introduced into Equation 6(6) $Q=R\cdot \left\{\mathrm{\partial }ln\dot{\epsilon }/\mathrm{\partial }ln\left[sinh\left(\alpha \sigma \right)\right]\right\}{|}_T\cdot \left\{\mathrm{\partial }ln\left[sinh\left(\alpha \sigma \right)\right]/\mathrm{\partial }\left(1/T\right)\right\}{|}_{\dot{\epsilon }}$(6) , and the heat deformation activation energy $Q$ is calculated to be 368.469 kJ·mol⁻¹ for the experimental steel at a strain level of 0.5. By inserting $Q$ into the above Equation 3(3) $\dot{\epsilon }=A[sinh\left(\alpha \sigma \right){]}^n\cdot exp\left[-Q/\left(RT\right)\right]\left(\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\right)$(3) the following relationships of Zener-Hollomon parameter (Z parameter) are obtained as Equations 7(7) $Z=\dot{\epsilon }\cdot \mathrm{e}\mathrm{x}\mathrm{p}[368469.736/(RT)]$(7) ~9(9) $\mathrm{l}\mathrm{n}Z=\mathrm{l}\mathrm{n}A+n\mathrm{l}\mathrm{n}[\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\alpha \sigma )]$(9) . In Equation 9(9) $\mathrm{l}\mathrm{n}Z=\mathrm{l}\mathrm{n}A+n\mathrm{l}\mathrm{n}[\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\alpha \sigma )]$(9) , the constants are the intercept and the slope of the regression line, which are 35.1464 and 5.9051, respectively, as shown in Figure 4.

(7) $Z=\dot{\epsilon }\cdot \mathrm{e}\mathrm{x}\mathrm{p}[368469.736/(RT)]$(7)

(8) $Z=\dot{\epsilon }\cdot \mathrm{e}\mathrm{x}\mathrm{p}\left[Q/\left(RT\right)\right]=A[sinh\left(\alpha \sigma \right){]}^n$(8)

(9) $\mathrm{l}\mathrm{n}Z=\mathrm{l}\mathrm{n}A+n\mathrm{l}\mathrm{n}[\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\alpha \sigma )]$(9)

Figure 4. Z-parameter versus rheological stress of the experimental steel.

Repeating the method of solving the coefficients of each principal equation for a strain of 0.5, the coefficients of each stress-strain principal equation (α, β, n, Q and lnA) can be obtained for different strains (0.1, 0.2, 0.3, 0.4, 0.45, 0.55, 0.6, 0.65, 0.7), and the results are shown in Table 2.

Table 2. Coefficients of the equation at different degrees of deformation.

True strain	$\alpha$, MPa^−1	β, MPa^−1	$n$	Q, J·mol^−1	$\mathrm{l}\mathrm{n}A$
0.1	0.00868	0.0888	8.34682	492.169	46.492
0.2	0.00759	0.0893	8.29395	487.920	45.789
0.3	0.00740	0.0995	7.40676	450.600	42.024
0.4	0.00740	0.1140	6.42610	402.502	37.292
0.45	0.00756	0.1256	5.97244	380.486	35.136
0.5	0.00757	0.1305	5.90510	368.469	35.146
0.55	0.00753	0.1347	5.32672	363.609	33.339
0.6	0.00756	0.1375	5.27868	355.703	33.032
0.65	0.00763	0.1407	5.07219	347.493	32.422
0.7	0.00766	0.1411	5.17656	348.285	32.315

Therefore, in order to describe the intrinsic structure of the experimental steel under the aforementioned deformation conditions more precisely, the relationship between the equation coefficients (α, β, n, Q and lnA) and strain is established by polynomial equations of order6 considering the degree of deformation. The fitting curve of polynomial coefficients is shown in Figure 5. The constitutive Equation 11(11) $\left\{\begin{array}{c}\sigma =\frac{1}{\alpha }\{(\frac{Z}{A}{)}^{\frac{1}{n}}+[(\frac{Z}{A}{)}^{\frac{2}{n}}+1{]}^{\frac{1}{2}}\}\\ Z=\dot{\epsilon }exp(\frac{Q}{RT})\\ \alpha =0.01120-0.03357\epsilon +0.08547{\epsilon }^2+0.01818{\epsilon }^3-0.38097{\epsilon }^4+0.54783{\epsilon }^5-0.24440{\epsilon }^6\\ \beta =0.06052+0.71381\epsilon -6.59852{\epsilon }^2+27.88814{\epsilon }^3-55.25173{\epsilon }^4+52.03427{\epsilon }^5-18.96014{\epsilon }^6\\ n=7.10303+16.52829\epsilon -6.34175{\epsilon }^2-500.30071{\epsilon }^3+1765.65152{\epsilon }^4-2348.20496{\epsilon }^5+1121.59614{\epsilon }^6\\ Q=626.04636-3346.82229\epsilon +33006.85263{\epsilon }^2-148222.46232{\epsilon }^3+322246.31304{\epsilon }^4-336998.50511{\epsilon }^5\\ +1365900.81012{\epsilon }^6\\ InA=55.2217+237.13491\epsilon +2411.83505{\epsilon }^2-11449.40994{\epsilon }^3+25777.18778{\epsilon }^4-27617.26188{\epsilon }^5+11404.34674{\epsilon }^6\end{array}\right.$(11) is obtained from Table 3 and Equation 10(10) $\left\{\begin{array}{c}\alpha =B_0+B_1{\epsilon }^1+B_2{\epsilon }^2+B_3{\epsilon }^3+B_4{\epsilon }^4+B_5{\epsilon }^5\\ \beta =C_0+C_1{\epsilon }^1+C_2{\epsilon }^2+C_3{\epsilon }^3+C_4{\epsilon }^4+C_5{\epsilon }^5\\ n=D_0+D_1{\epsilon }^1+D_2{\epsilon }^2+D_3{\epsilon }^3+D_4{\epsilon }^4+D_5{\epsilon }^5\\ Q=E_0+E_1{\epsilon }^1+E_2{\epsilon }^2+E_3{\epsilon }^3+E_4{\epsilon }^4+E_5{\epsilon }^5\\ InA=F_0+F_1{\epsilon }^1+F_2{\epsilon }^2+F_3{\epsilon }^3+F_4{\epsilon }^4+F_5{\epsilon }^5\end{array}\right.$(10) . Figures 6 and 7 show the high correlation between the predicted and actual stress values.

Table 3. Polynomial coefficients of α, β, n, Q and lnA.

α	β	n	Q	lnA
B0=0.0112	C0=0.06052	D0=7.10303	E0=626.04636	F0=55.2217
B1=-0.03357	C1=0.71381	D1=16.52829	E1=-3346.82229	F1=237.13491
B2=0.08547	C2=-6.59852	D2=-6.34715	E2=33006.85263	F2=2411.83505
B3=0.01818	C3=27.88814	D3=-500.30071	E3=-148222.42632	F3=-11449.40994
B4=-0.38097	C4=-55.25173	D4=1765.65152	E4=322246.31304	F4=25777.18778
B5=0.5478	C5=52.03427	D5=-2348.24096	E5=-336998.50511	F5=-27617.26188
B6=-0.2444	C6=-18.96104	D6=1121.59614	E6=136590.81012	F6=11404.34674

(10) $\left\{\begin{array}{c}\alpha =B_0+B_1{\epsilon }^1+B_2{\epsilon }^2+B_3{\epsilon }^3+B_4{\epsilon }^4+B_5{\epsilon }^5\\ \beta =C_0+C_1{\epsilon }^1+C_2{\epsilon }^2+C_3{\epsilon }^3+C_4{\epsilon }^4+C_5{\epsilon }^5\\ n=D_0+D_1{\epsilon }^1+D_2{\epsilon }^2+D_3{\epsilon }^3+D_4{\epsilon }^4+D_5{\epsilon }^5\\ Q=E_0+E_1{\epsilon }^1+E_2{\epsilon }^2+E_3{\epsilon }^3+E_4{\epsilon }^4+E_5{\epsilon }^5\\ InA=F_0+F_1{\epsilon }^1+F_2{\epsilon }^2+F_3{\epsilon }^3+F_4{\epsilon }^4+F_5{\epsilon }^5\end{array}\right.$(10)

(11) $\left\{\begin{array}{c}\sigma =\frac{1}{\alpha }\{(\frac{Z}{A}{)}^{\frac{1}{n}}+[(\frac{Z}{A}{)}^{\frac{2}{n}}+1{]}^{\frac{1}{2}}\}\\ Z=\dot{\epsilon }exp(\frac{Q}{RT})\\ \alpha =0.01120-0.03357\epsilon +0.08547{\epsilon }^2+0.01818{\epsilon }^3-0.38097{\epsilon }^4+0.54783{\epsilon }^5-0.24440{\epsilon }^6\\ \beta =0.06052+0.71381\epsilon -6.59852{\epsilon }^2+27.88814{\epsilon }^3-55.25173{\epsilon }^4+52.03427{\epsilon }^5-18.96014{\epsilon }^6\\ n=7.10303+16.52829\epsilon -6.34175{\epsilon }^2-500.30071{\epsilon }^3+1765.65152{\epsilon }^4-2348.20496{\epsilon }^5+1121.59614{\epsilon }^6\\ Q=626.04636-3346.82229\epsilon +33006.85263{\epsilon }^2-148222.46232{\epsilon }^3+322246.31304{\epsilon }^4-336998.50511{\epsilon }^5\\ +1365900.81012{\epsilon }^6\\ InA=55.2217+237.13491\epsilon +2411.83505{\epsilon }^2-11449.40994{\epsilon }^3+25777.18778{\epsilon }^4-27617.26188{\epsilon }^5+11404.34674{\epsilon }^6\end{array}\right.$(11)

Figure 5. The variation of material constants for the constitutive equation with true strains: (a) α; (b) β; (c) n; (d) Q; (e) lnA.

Figure 6. Comparison between the experimental and predicted flow stress from the deformation resistance mathematics model of the experimental steel at the temperatures of 1073 ~ 1373 K and at the strain rates of (a) 0.1 s⁻¹, (b) 0.5 s⁻¹, (c) 1 s⁻¹, and (d) 5 s⁻¹.

Figure 7. Correlation between the rheological stress calculated by the Arrhenius constitutive model and the experimentally measured rheological stress.

Development of BP-ANN model

In this study, a BP-ANN model for the correlation of hot compression rheological stress with strain, strain rate and temperature for high-strength offshore steel is developed. The deformation temperature (T), strain rate ($\dot{\epsilon }$), and strain ($\epsilon$) are the input variables and the actual stress ($\sigma$) is the only output. A schematic of the structure is shown in Figure 8. To develop an accurate BP-ANN model, it is crucial to determine the appropriate number of hidden layers and the number of neurons in each hidden layer. As experimental data, such as temperature, strain, strain rate, and stress, are measured in different units, these data must be normalised to dimensionless units before training, which reduces the convergence speed and accuracy of the model. Assuming that one and two hidden layers are used for testing, the appropriate number of hidden layers is determined by evaluating the predicted data from the training and testing using the experimental data with appropriate tolerances. In addition, an empirical equation is proposed to determine the range of E-values, i.e. the range of number of neurons per hidden layer, and the range is calculated accordingly as 3–12. Finally, the accuracy of the neural network model is guaranteed using a two-layer hidden layer structure.

Figure 8. Schematic diagram of the BP-ANN structure.

(12) $e=\sqrt{n+m}+a$(12)

where $e$ is the number of neurons in each hidden layer, $n$ and $m$ are the number of neurons in the input and output layers of the network, respectively. $n$ = 3, $m$ = 1. and $a$ is a ranging from 1 to 10.

The values of the input and output variables are distributed over different ranges and uniform dimensions, which leads to poor convergence speed and prediction accuracy of ANN models. Therefore, normalisation of the initial experimental stress-strain data is essential to ensure dimensionless and approximately similar quantitative values of the input and output variables. In this study, the magnitude of the normalised data is narrowed to adjust the parameters to within 0–0.3, with coefficients of 0.05 and 0.25 in the regression equation. The pilot algorithm demonstrates that such magnitudes can improve convergence speed and forecast accuracy. Excessive errors caused by a wide range are avoided. In addition, it should be noted that the initial value of the true strain rate has a large magnitude. Thus, we take the logarithm to convert the true strain rate data prior to normalisation:

(13) $y_n=0.05+0.25\times \frac{y-0.95y_{min}}{1.05y_{max}-{0.95}_{min}}$(13)

(14) $y_n=0.05+0.25\times \frac{\left(3+y\right)-0.95\left(3+y_{min}\right)}{1.05\left(3+y_{max}^{\hspace{0.17em}}\right)-0.95\left(3+y_{min}\right)}$(14)

where $y_n$ is the normalised value of $y$, $y$ is the experimental data, $y_{max}$ and $y_{min}$ are the maximum and minimum values of, respectively.

In this study, R and AARE were used as statistical indicators to comprehensively evaluate the predictive ability of the ANN model, and are expressed as Equations 15 and 16, respectively. A high R value close to 1 indicates that the predicted value agrees well with the experimental value, while a low AARE value close to 0 indicates that the sum of the errors between the predicted and experimental values tends to zero.

(15) $R=\frac{{\sum }_{i=1}^N(Ei-\bar{E})(Pi-\bar{P})}{\sqrt{{\sum }_{i=1}^N{(Ei-\bar{E})}^2}{\sum }_{i=1}^N{(Pi-\bar{P})}^2}$(15)

(16) $AARE(\mathrm{\%})=\frac{1}{N}{\sum }_{i=1}^N\left|\frac{Pi-Ei}{Ei}\right|\times 100\mathrm{\%}$(16)

where E and P are the experimental and predicted values of the true stress, respectively, $\bar{E}$ and $\bar{P}$ are the average values of $E$ and $P$, respectively, $P_i$ is the predicted stress value, $E_i$ is the corresponding experimental stress value, and N is the number of predicted points.

The relative error ($\delta$) in the equation represents the percentage error of each stress-strain prediction relative to the corresponding experimental value, and is introduced for a more in-depth and detailed evaluation of the ANN model. The parameters $\mu$ and $w$ are the mean and standard deviation, respectively. For a smaller $\delta$ value, the precision is higher. Information on the values is provided by calculating and counting the $\delta$ values of all prediction points, including the training and testing points. Figure 9 shows the correlation between predicted and true stress values for the training and test data. Figure 10 shows a histogram of the relative errors in the training and testing parts. The relative errors are within a narrow range of 0–6% for both the training and testing parts. More notably, most of the $\delta$ values are concentrated around the ideal value of 0. In the training part, 79.03% of the points with ~ 6% error are within the [−1%, 1%] interval. In the test part, 78.47% of the points are concentrated within the [−1%, 1%] interval. These results, generated using statistical data, provide direct evidence that the ANN model achieves high accuracy in both the training and testing phases.

(17) $\delta (\mathrm{\%})=\frac{Pi-Ei}{Ei}\times 100\mathrm{\%}$(17)

(18) $\mu =\frac{1}{N}\sum _{i=1}^N{\delta }_i$(18)

(19) $w=\sqrt{\frac{1}{N-1}\sum _{i=1}^N{({\delta }_i-\mu )}^2}$(19)

where $P_i$ is the predicted stress value and $E_i$ is the corresponding experimental stress value. ${\delta }_i$ is the relative error, $\mu$ and $\omega$ are the mean and standard deviation of $\delta$, respectively, and $N$ is the number of relative errors.

Figure 9. Correlation between predicted and true stress values for the (a) training and (b) test data.

Figure 10. Relative error distribution of predicted true stress values for the (a) training and (b) test points.

As shown in Figure 9, the data points with the true stress values as the horizontal axis and the predicted stress as the vertical axis are consistent with the best linear fit line, indicating excellent prediction performance. The correlation coefficients, R, for the training and test points are 0.99974 and 0.9998, respectively. In addition, the AARE values calculated from the training and test parts are 0.74575% and 0.70758%, respectively. The small errors in both cases fully illustrate the high-accuracy prediction performance of the BP-ANN in training and testing.

Based on the well-trained ANN model, the actual stress values under experimental conditions are included to predict the deformation conditions corresponding to the previously trained and tested points. Figure 11 compares the stress values predicted by the BP-ANN model with the experimental stress values. The predicted values follow the same trend as the true values, and increase with decreasing temperature or increasing strain rate. This indicates that the BP-ANN model is more accurate compared to the Arrhenius-type constitutive equation in determining the variation law of the rheological stress.

Figure 11. Comparison of predicted stress values from the BP-ANN model with experimental values at temperatures of 1073 ~ 1373 K and at the strain rates of (a) 0.1 s⁻¹, (b) 0.5 s⁻¹, (c) 1 s⁻¹, and (d) 5 s⁻¹.

The experimental stress-strain curves, predictions from the BP-ANN model and Arrhenius constitutive equation at 1123 K and strain rates 0.01 s⁻¹, 0.03 s⁻¹ and 10 s⁻¹, are shown in Figure 12. The new deformation temperature and deformation rates are introduced. Among the three curves predicted by BP-ANN, when the strain rate is 3 s⁻¹, it shows good predictive tracking ability in the range of 0.1 s⁻¹ to 5 s⁻¹ of the training set. However, as the rates of 0.01 s⁻¹and 10 s⁻¹exceeded the training range of the training set, although the trend was consistent, the accuracy of the ANN model is lower than that of the Arrhenius constitutive mode.

Figure 12. Comparison of predicted stress values of the Arrhenius constitutive model and BP-ANN model with experimental values at 1123 K and different strain rates.

Conclusions

1. A BP-ANN model of high-strength offshore steel is developed based on isothermal compression test data from a Gleeble 3800 thermal simulator. The established neural network model can effectively simulate complex thermal deformation behaviour and has good generalisation ability over a wide range of temperatures and strain rates.

2. In the BP-ANN model, the deformation temperature ($T$), strain rate ($\dot{\epsilon }$), and strain ($\epsilon$) are the input variables, and true stress ($\sigma$) is the output variable. The higher R values, lower AARE values, and stable percentage error distribution results indicate that the BP-ANN model has a better predictive performance than the Arrhenius-type intrinsic constitutive equation under limited experimental conditions.

3. The ANN model has excellent potential for application in investigating the thermal deformation processes. The accuracy of the predicted rheological stress provides strong theoretical support for the determination of parameters for processes such as rolling and for the heat treatment of high-strength offshore steel.

Acknowledgments

This research was financially supported by Science and Technology Research Projects of the CNPC (No.2021ZG14 and LHT-2023-02).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Science and Technology Research Projects of CNPC (No.2021ZG14 and LHT-2023-02).