[Paper Review] Physics-based Machine Learning for Computational Fracture Mechanics

Fadi Aldakheel, Elsayed S. Elsayed+2 more|ArXiv.org|Feb 13, 2025

Landslides and related hazards3 citations

TL;DR

This paper introduces a physics-based ML framework (phi ML) that embeds governing equations and thermodynamics into neural networks to model brittle and ductile fractures, trained on phase-field fracture data, and demonstrates superior generalization with limited data compared to naive data-driven models.

ABSTRACT

This study introduces a physics-based machine learning framework for modeling both brittle and ductile fractures. Unlike physics-informed neural networks, which solve partial differential equations by embedding physical laws as soft constraints in loss functions and enforcing boundary conditions via collocation points, our framework integrates physical principles, such as the governing equations and constraints, directly into the neural network architecture. This approach eliminates the dependency on problem-specific retraining for new boundary value problems, ensuring adaptability and consistency. By embedding constitutive behavior into the network's foundational design, our method represents a significant step toward unifying material modeling with machine learning for computational fracture mechanics. Specifically, a feedforward neural network is designed to embed physical laws within its architecture, ensuring thermodynamic consistency. Building on this foundation, synthetic datasets generated from finite element-based phase-field simulations are employed to train the proposed framework, focusing on capturing the homogeneous responses of brittle and ductile fractures. Detailed analyses are performed on the stored elastic energy and the dissipated work due to plasticity and fracture, demonstrating the capability of the framework to predict essential fracture features. The proposed physics-based machine learning framework overcomes the shortcomings of classical machine learning models, which rely heavily on large datasets and lack guarantees of physical principles. By leveraging its physics-integrated design, the physics-based machine learning framework demonstrates exceptional performance in predicting key properties of brittle and ductile fractures with limited training data.

Motivation & Objective

• Motivate the need for reliable fracture modeling under inelastic conditions with limited data.
• Develop a phi ML framework that guarantees thermodynamic consistency by architecture design.
• Train and validate the phi ML model on synthetic phase-field fracture data to capture brittle and ductile behavior.
• Evaluate the phi ML model against purely data-driven FFNN baselines in terms of accuracy and extrapolation.
• Demonstrate robustness and data efficiency of the physics-integrated approach.

Proposed method

• Embed governing equations and thermodynamic consistency directly into a feedforward neural network architecture.
• Use two dependent subnetworks: FFNN^{ε^p,d} to predict plastic strain and phase-field, and FFNN^{ψ} to predict free energy.
• Train jointly with a combined loss L_total = L_σ + L_ψ + L_{ε^p} + L_d + L_D, enforcing dissipation consistency via D = ReLU(D_d) + ReLU(D_p).
• Enforce phase-field bounds with a modified ReLU function to keep d within [0,1].
• Generate synthetic training data from finite element-based phase-field simulations for brittle and ductile fracture under homogeneous responses.

Experimental results

Research questions

• RQ1 Can a physics-based ML model predict brittle and ductile fracture variables (stress, energy, phase-field, dissipation) with limited data?
• RQ2 Do physics-informed architectures improve generalization and thermodynamic consistency over naive data-driven models in fracture mechanics?
• RQ3 How does phi ML perform under data scarcity and extrapolation (lower/upper bounds) compared to purely data-driven FFNNs?
• RQ4 What is the impact of embedding constitutive behavior and history-dependent variables on predictive accuracy for fracture phenomena?

Key findings

• Naive FFNNs struggle to generalize, especially in extrapolation and with reduced data, showing negative or near-zero R^2 and high MAPE.
• The phi ML model achieves near-perfect agreement (R^2 > 0.99) across interpolation and extrapolation, even with reduced data, for brittle fracture outputs (energy, stress, phase-field, dissipation).
• For brittle fracture, full dataset shows phi ML R^2 of 0.996–0.999 and MAPE ~3.18% for phi ML vs 0.894–0.998 and MAPE 14.93–58.53% for naive FFNN.
• For ductile fracture, tables and figures indicate phi ML maintains high accuracy and robustness under limited data, outperforming naive models.
• Across brittle cases, the phi ML framework demonstrates robustness, reliability, and data efficiency by incorporating physics into the learning process.