[Paper Review] Modular machine learning-based elastoplasticity: generalization in the context of limited data

Jan N. Fuhg, Craig M. Hamel+3 more|arXiv (Cornell University)|Oct 15, 2022

Model Reduction and Neural Networks6 citations

TL;DR

This paper proposes a modular, physics-informed machine learning framework for elastoplasticity that combines classical phenomenological models with data-driven components, enabling accurate extrapolation beyond training data even with limited datasets. By enforcing thermodynamic consistency and allowing selective use of data-driven modules based on data availability, the method achieves robust generalization in low-data regimes.

ABSTRACT

The development of accurate constitutive models for materials that undergo path-dependent processes continues to be a complex challenge in computational solid mechanics. Challenges arise both in considering the appropriate model assumptions and from the viewpoint of data availability, verification, and validation. Recently, data-driven modeling approaches have been proposed that aim to establish stress-evolution laws that avoid user-chosen functional forms by relying on machine learning representations and algorithms. However, these approaches not only require a significant amount of data but also need data that probes the full stress space with a variety of complex loading paths. Furthermore, they rarely enforce all necessary thermodynamic principles as hard constraints. Hence, they are in particular not suitable for low-data or limited-data regimes, where the first arises from the cost of obtaining the data and the latter from the experimental limitations of obtaining labeled data, which is commonly the case in engineering applications. In this work, we discuss a hybrid framework that can work on a variable amount of data by relying on the modularity of the elastoplasticity formulation where each component of the model can be chosen to be either a classical phenomenological or a data-driven model depending on the amount of available information and the complexity of the response. The method is tested on synthetic uniaxial data coming from simulations as well as cyclic experimental data for structural materials. The discovered material models are found to not only interpolate well but also allow for accurate extrapolation in a thermodynamically consistent manner far outside the domain of the training data. Training aspects and details of the implementation of these models into Finite Element simulations are discussed and analyzed.

Motivation & Objective

• Address the challenge of developing accurate, generalizable constitutive models for path-dependent materials under limited experimental data.
• Overcome the limitations of purely data-driven models that require large, diverse datasets and often violate thermodynamic principles.
• Enable reliable extrapolation in stress space—beyond training data—by integrating physics-based constraints with modular data-driven components.
• Develop a hybrid modeling framework where individual elastoplastic components (elastic law, yield function, hardening) can be selected as either classical or data-driven based on data availability.
• Ensure thermodynamic consistency in the resulting models through hard constraints on free energy, yield function, and evolution laws.

Proposed method

• Formulate elastoplasticity using a modular framework where each component—elastic response, yield surface, hardening, and internal variables—can be independently modeled as either phenomenological or data-driven.
• Use physics-informed machine learning to embed thermodynamic principles (e.g., polyconvexity of free energy, consistency of evolution laws) as hard constraints in the training process.
• Employ neural networks to represent data-driven components such as the yield function and hardening laws, trained on synthetic or experimental data with limited loading paths.
• Implement a consistent time-integration scheme using Newton-Raphson methods with Jacobian matrices derived from analytical derivatives of the residual system, ensuring stability and convergence.
• Construct the residual vector and Jacobian matrix for the system of equations governing the incremental stress update, incorporating derivatives of free energy, yield function, and hardening laws.
• Train models using uniaxial synthetic data from simulations and cyclic experimental data, validating performance on unseen loading paths and extrapolation tasks.

Experimental results

Research questions

• RQ1 Can a modular machine learning framework for elastoplasticity generalize effectively in low-data regimes where traditional data-driven models fail?
• RQ2 How well can data-driven components enforce thermodynamic consistency while maintaining predictive accuracy across diverse loading paths?
• RQ3 To what extent can the framework extrapolate beyond the training data domain, especially in non-probed stress states and complex loading paths?
• RQ4 How does the modularity of the framework allow for flexible integration of classical and data-driven models based on data availability?
• RQ5 What is the impact of enforcing hard constraints (e.g., polyconvexity, consistency of evolution laws) on model generalization and robustness?

Key findings

• The proposed framework achieves accurate extrapolation in stress space far beyond the training data domain, a key limitation of standard data-driven models.
• Even with limited data—such as uniaxial or cyclic experimental data—the model generalizes well to unseen loading paths, including complex multiaxial and reverse loading conditions.
• The integration of physics-informed constraints ensures thermodynamic consistency, preventing unphysical behavior and improving model reliability.
• The modular design allows for selective use of data-driven components where data is available, while retaining classical models where data is scarce, enabling robust performance across data regimes.
• Finite element simulations using the learned models show stable convergence and accurate stress predictions, demonstrating compatibility with standard computational mechanics workflows.
• The method outperforms purely data-driven models in low-data scenarios, where the latter often fail due to lack of generalization and thermodynamic violations.