Mechanism and data dual-driven multimodal deep learning approach for constitutive relationship modeling of bearing steels

Abstract

Accurate modeling of material constitutive relationships under compositional fluctuations poses significant challenges. Traditional mechanism-driven methods struggle to capture the complex nonlinear behavior of material properties as composition varies, while data-driven deep learning approaches, despite their high accuracy and robustness, lack strict constraints from physical metallurgical mechanisms, often leading to substantial prediction deviations. To address this critical issue, this study proposes a multimodal deep learning model based on an encoder-decoder framework, integrating physical metallurgical theory with machine learning to achieve high-precision prediction of material constitutive relationships under complex loading and compositional fluctuations. Firstly, the performance of three encoder architectures, long short-term memory (LSTM), gated recurrent unit (GRU), and temporal convolutional network (TCN), was systematically compared, with the TCN encoder-based model demonstrating the best performance, achieving mean absolute error (MAE), root mean square error (RMSE), mean absolute percentage error (MAPE), and correlation coefficient (R) values of 1.84 MPa, 2.75 MPa, 4.48%, and 0.9918, respectively. By comparing with purely data-driven models, the critical guiding role of physical metallurgy in the deep learning process was validated. Furthermore, the accuracy and generalizability of the proposed model in material processing scenarios were verified by embedding it into a numerical simulation framework and applying it to out-of-domain data. Finally, the influence of elemental content on material mechanical properties was analyzed using the developed model. This study provides an efficient and reliable alternative method for obtaining material constitutive relationships, avoiding the high costs associated with traditional experimental approaches, and offering potential for computer-aided material design and process optimization for material processing.

1. Introduction

Constitutive models play a critical role in the field of material processing by describing the essential functional relationship between the deformation behaviour of materials at high temperature and the thermal processing parameters, forming the basis of plastic deformation theory [1–3]. Over recent decades, numerous phenomenological models have been proposed to describe the constitutive behavior of metals, among which the Johnson–Cook model [4], Zerilli–Armstrong model [5], and Arrhenius model [6] have been widely applied in thermomechanical processing simulations. Although these models are characterized by their simplicity and few parameters, their prediction accuracy often falls short of practical requirements. Moreover, it often requires adopting specific models or introducing additional correction parameters for different steel grades, which typically consumes substantial resources and time [7–11].

In recent years, artificial intelligence methods have been introduced to model constitutive relationship [12–16]. Early research primarily focused on data-driven methods. In 2019, Li et al. [17] introduced neural networks into the Johnson-Cook model, relying on plastic strain, strain rate, and temperature to describe the nonmonotonic temperature dependence. In 2020, Ahmadi et al. [18] developed a back propagation artificial neural network model to predict the constitutive relationship of 5CT-L80 steel. In the same year, Gorji et al. [19] proposed a neural network with a gated recurrent unit (GRU) and a fully connected layer for plasticity stress modeling under arbitrary loading paths. Starting in 2021, researchers shifted their focus to the innovation of physics-enhanced neural network architectures. Linka et al. [20] designed a constitutive artificial neural network (CANN) to learn the complex nonlinear and constitutive behavior of anisotropic materials by combining stress‒strain data, theoretical knowledge, and material information. Around the same time, Masi et al. [21] encoded the two fundamental laws of thermodynamics into the network structure to generate thermodynamically consistent predictions, significantly enhancing the robustness of the model.

To address issues related to data quality and collection costs [22–24], researchers began exploring methods to learn underlying constitutive relationships from indirectly measured data. In 2021, Xu et al. [25] developed a calibrated parameter and learned neural-network-based constitutive relationship of viscoelastic materials from indirect displacement data in the context of multiple-physics systems. A year later, Zhang et al. [26] proposed a hybrid FEM-NN framework for learning constitutive relationship from full-field data, enabling efficient training of neural network. Most studies consider only materials with specific chemical compositions. In practice, the influence of heat deformation parameters and chemical composition on material...