Unsupervised identification of constitutive laws using neural networks in a thermodynamically consistent framework guided by the CRE and mCRE methods

Mechanical structures are made of materials with diverse and often complex behaviors. To ensure reliable assessment, simulations of these structures must accurately capture this behavior, which strongly depends on the integration of physically consistent constitutive models. To this end, a common approach involves collecting experimental data from instrumented structures and using inverse methods to calibrate these models. Traditionally, this calibration involves choosing an analytical form for the constitutive law and fitting its parameters to the data. While these models are interpretable, they may struggle to capture complex material responses. To overcome this limitation, data-driven methods have emerged, aiming to identify the law directly from data, without prior assumptions [1,2]. Some of these approaches can be embedded within a thermodynamically consistent framework based on convex energy potentials, which helps ensure physical reliability. However, a limitation of many of such methods is that they assume the form of these energy potentials, as well as the number and type of internal variables, which limits their applicability in real-world cases where such information is unknown. To overcome this limitation, we propose a neural network-based approach [1] that simultaneously identifies both energy potentials and internal variables without predefined structural assumptions [1,2]. The proposed method remains thermodynamically consistent and is guided by the Constitutive Relation Error (CRE), which provides a physically meaningful error indicator based on full-field measurements, and its modified version (mCRE), which accounts for measurement noise. First, we focus on identifying the viscoelastic behavior of materials used in vascular surgical simulators. The experimental data is obtained through Digital Image Correlation during uniaxial and biaxial tests conducted on a dedicated testing machine. The objective is to extend the mCRE concept coupled with neural networks [1] to learn the underlying thermodynamic potentials governing the constitutive law [1,2]. To validate the method in a controlled setting, we also study an academic example of a plate undergoing elasto-visco-plastic deformation described by the Marquis–Chaboche model. The analytical formulation, developed by Ladevèze et al. [3] serves as a reference to assess the ability of our framework to recover known constitutive components. These two studies illustrate the robustness and versatility of the proposed framework, which combines physics-based modeling, error indicators, and machine learning for unsupervised identification of constitutive laws. References : [1] Rosenkranz, M., Kalina, K. A., Brummund, J., Sun, W., & Kästner, M. (2024). Viscoelasticty with physics-augmented neural networks: Model formulation and training methods without prescribed internal variables. Computational Mechanics, 74(6), 1279-1301. [2] Benady, A., Baranger, E., & Chamoin, L. (2024). NN‐mCRE: A modified constitutive relation error framework for unsupervised learning of nonlinear state laws with physics‐augmented neural networks. International Journal for Numerical Methods in Engineering, 125(8), e7439. [3] Ladevèze, P., & Chamoin, L. (2024). Data-driven material modeling based on the Constitutive Relation Error. Advanced Modeling and Simulation in Engineering Sciences, 11(1), 23. This project has received funding from the European Research Council (ERC) under the under the European Union’s Horizon 2020 research and innovation program under grant agreement No. 101002857.