In recent years, the increased performance and safety requirements as well as a more rational and cost-effective manufacturing have broadened the interest in high-strength and ultra-high-strength steels in many industrial sectors such as commercial vehicles, railcars, containers, cranes, machinery, etc. The application of ultra-high-strength steels makes it possible to reduce material thicknesses and thus the weight by up to 70 %. Successful light-weight design improves the working efficiency, decreases fuel consumption and even provides cost savings from procurement to manipulation and processing. Nevertheless, it is known that the formability of a material with higher strength deteriorates, and even its springback degree grows rapidly at room temperature. A hot stamping process [1] in the austenite temperature region followed by room temperature die quenching is employed for fabricating structural components having strength of 1,500 MPa or more, by which the sheet is formed, and its strength is improved by the transformation from austenite to martensite [2]; meanwhile, the springback is eliminated. Hot stamping is a thermo-mechanical forming process with large deformation and intended phase transformation. So a realistic finite element model (FEM) for such process must consider the interaction among mechanical, thermal and microstructural fields. This requires several main process characteristics such as heat transfer coefficient, material flow behaviors and phase transformation kinetics under process-relevant conditions. Consequently, as the work shown here, it is valuable to deeply understand, accurately characterize and predict the flow behaviors of ultra-high-strength sheet material, which provides the basis stress–strain data for numerical simulation [3].

Over the past decades, great attentions have been paid to the characterization of the complex nonlinear relationships between true stress and deformation parameters such as strain, strain rate and temperature. Numerous efforts have been made to three types of constitutive models involving analytical, phenomenological [4–6] and empirical or semi-empirical ones [7]. The analytical constitutive model is considered as the simplest and most computationally efficient model for predicting metal flow behaviors. But each analytical model is based on certain assumptions, and if these assumptions are not met, the model may fail to predict correctly. The empirical or semi-empirical constitutive model is proposed on a range of assumptions, and it couples the effect of strain rate and temperature but treats strain in an exclusive way. Usually the effects of strain, strain rate and temperature are statistically analyzed in a mutually exclusive manner from the data collected in measurements or observations. The accuracy of this model mainly depends on the experimental data amount, regression tolerance and the authenticity of these assumptions. The phenomenological constitutive model is an accurate mathematical model and can have relatively many coefficients that need to be calibrated with experimental data. This constitutive model requires many computational and experimental efforts, and its accuracy mainly depends on the regression tolerance and the formula applicability.

Relative to the previous three types of constitutive models, an easier and more adaptable modeling approach, the artificial neural network (ANN), has been rapidly developed in recent years and popularly applied in material flow behaviors [8]. ANN is a model emulating some functions of biological neural networks with a data-driven black box structure [9]; thus, it merely needs a collection of some typical examples from the anticipant mapping functions for training regardless of explicit professional knowledge of deformation mechanisms. Most ANNs contain some different “learning rules” to modify the weights of the connections on the basis of the input patterns that it is presented with. Some different sorts of learning rules such as the Hebbian rule, delta rule, the competitive learning rule and the anti-Hebbian rule are utilized by neural networks, in which the delta rule is often used by the most widely and successfully used ANN called “backpropagational artificial neural network” (BP-ANN). With BP, “learning” is a supervised process that follows with each cycle through the feed-forward computation of activations and the backward propagation of error signals for weight adjustments via the generalized delta rule. Recently, a lot of efforts have been spent on the hot deformation behaviors and constitutive description of several alloys by BP-ANN model. Phaniraj et al. [10], Mandal et al. [11] and Lin et al. [12] conducted BP-ANN models to predict the flow behaviors of carbon steels, type 304 L stainless steel, 42CrMo steel, and so on. These reports about BP-ANN revealed that BP-ANN is an effective tool to predict the complex nonlinear hot flow behaviors by self-training to be adaptable to the material characteristics.

As for ultra-high-strength steel, there are few reports of the constitutive modeling of the hot flow behaviors by BP-ANN. As is known, the stress–strain data amount and accuracy input into the FEM has a significant influence on the simulation accuracy of the hot stamping process of ultra-high-strength steel sheet. The BP-ANN meets this request just right since it has not only a high accuracy of characterization but also a high predictability. In this work, the stress–strain data for ultra-high-strength steel sheet, here BR1500HS, beyond experimental conditions were predicted by BP-ANN; then the predicted data were interpolated densely. The most important is that a three-dimensional (3D) continuous interaction space representing the continuous response of stress to strain, strain rate and temperature has been constructed by the surface fitting of limited dense data. In the 3D continuous interaction space, all the stress–strain points are digital and can be determined, which means the stress under any temperatures, strain rates and strains is known as long as the deformation conditions are within the scope of such a 3D continuous interaction space. Thus, it provides continuous stress–strain data for a series of studies of ultra-high-strength steel sheet, such as processing maps, dynamic recrystallization (DRX) kinetics, ductile damage evolution, even more important, FEM simulations of the hot stamping processes. The accuracy of such 3D continuous mapping relationships among temperature, strain rate and strain is strongly guaranteed by the high accuracy of BP-ANN model, which undoubtedly induces the high accuracy of relative studies needing stress–strain data. This 3D continuous interaction space was put forward for the first time in the characterization field of material flow behaviors.

In this work, the stress–strain data of BR1500HS sheet were collected from a series of isothermal hot tensile tests carried out in a wide temperature range of 973–1,123 K and a strain rate range of 0.01–10 s^{−1} on a Gleeble 3800 thermo-mechanical simulator. A BP-ANN model that takes temperature (*T*), strain rate ($\dot{\mathrm{\epsilon}}$) and strain ($\mathrm{\epsilon}$) as the input variables and true stress ($\mathrm{\sigma}$) as the output variable was established by determining proper network structure and parameters to predict the nonlinear complex flow behaviors of BR1500HS sheet. The predictability and adaptability of this BP-ANN model were evaluated, having admirable performance by a series of evaluators such as correlation coefficient (*R*), average absolute relative error (AARE) and relative error (*δ*). Meanwhile, a comparison between BP-ANN model and a phenomenological constitutive model, i. e. improved Arrhenius-type constitutive model of BR1500HS sheet, was implemented, 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 973–1,223 K, strain rate range of 0.01–10 s^{−1} and strain range of 0–0.16 was constructed.

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