[Paper Review] A review on data-driven constitutive laws for solids

Jan N. Fuhg, Govinda Anantha Padmanabha+7 more|arXiv (Cornell University)|May 6, 2024

Elasticity and Material Modeling4 citations

TL;DR

This paper presents a comprehensive taxonomy of data-driven constitutive laws for solids, categorizing machine learning and model-free methods by interpretability, learning type, and data requirements. It identifies key challenges in generalization, trustworthiness, and low-data performance, and advocates for standardized benchmarks and uncertainty quantification to advance robust, physics-informed material modeling across scales.

ABSTRACT

This review article highlights state-of-the-art data-driven techniques to discover, encode, surrogate, or emulate constitutive laws that describe the path-independent and path-dependent response of solids. Our objective is to provide an organized taxonomy to a large spectrum of methodologies developed in the past decades and to discuss the benefits and drawbacks of the various techniques for interpreting and forecasting mechanics behavior across different scales. Distinguishing between machine-learning-based and model-free methods, we further categorize approaches based on their interpretability and on their learning process/type of required data, while discussing the key problems of generalization and trustworthiness. We attempt to provide a road map of how these can be reconciled in a data-availability-aware context. We also touch upon relevant aspects such as data sampling techniques, design of experiments, verification, and validation.

Motivation & Objective

• To organize and classify the broad spectrum of data-driven constitutive modeling techniques developed over recent decades.
• To address the critical challenges of generalization, interpretability, and trustworthiness in data-driven material modeling.
• To bridge the gap between data availability and model performance, especially in low-data regimes common in experimental and computational mechanics.
• To propose a roadmap for integrating data-driven constitutive laws with traditional finite element solvers and ensuring numerical convergence.
• To advocate for standardized datasets and benchmark problems to enable objective evaluation of data-driven constitutive models.

Proposed method

• Categorizes data-driven constitutive modeling into machine learning-based and model-free approaches, further sub-classifying by interpretability and learning process.
• Distinguishes between path-independent and path-dependent material behavior, emphasizing the complexity of learning irreversible mechanisms from data.
• Proposes a framework for integrating data-driven models with finite element solvers using automatic differentiation, while acknowledging performance trade-offs.
• Emphasizes the use of full-field experimental data (e.g., from DIC, DVC) and computational homogenization to generate large-scale training data.
• Introduces the need for uncertainty quantification (UQ) to assess model reliability and downstream simulation accuracy.
• Recommends designing optimized experiments and specimen geometries to maximize identifiability of complex models from minimal data.

Figure 4: TANNs proposed by [ 253 ]

Experimental results

Research questions

• RQ1 How can data-driven constitutive laws be systematically classified based on interpretability, learning type, and data requirements?
• RQ2 What are the key limitations in generalization and trustworthiness of data-driven models when applied to unseen loading paths or material behaviors?
• RQ3 How can data-driven constitutive models be effectively integrated into finite element solvers without compromising computational efficiency?
• RQ4 What role does uncertainty quantification play in validating data-driven models for use in nonlinear mechanics problems?
• RQ5 How can experimental design and data sampling be optimized to enable robust model discovery with minimal data?

Key findings

• Data-driven constitutive modeling has evolved from limited, phenomenological models to large-data, high-fidelity approaches enabled by full-field measurements and computational homogenization.
• Machine learning-based methods, especially physics-informed networks and operator learning, show promise in reducing data requirements by embedding physical constraints.
• Model-free approaches such as interpolation in data manifolds can discover constitutive laws in a single test if the loading path is sufficiently complex and comprehensive.
• Robustness in low-data regimes remains a major challenge, and current trends favoring large datasets may not be feasible for many engineering applications.
• Thorough verification in finite element settings is essential, as standard error metrics do not guarantee convergence of nonlinear solvers.
• The community urgently needs standardized benchmark datasets—such as RVE-generated hyperelastic data and full-field FEM simulations—for objective evaluation of data-driven constitutive models.