🔬 materials science

An extended deep energy method for thermo-mechanical crack propagation

This paper presents an extended deep energy method that utilizes two neural networks and a sharp polyline representation to solve coupled thermo-mechanical crack propagation problems, achieving high accuracy in stress intensity factor extraction and crack path prediction across various loading conditions without requiring a regularization length.

Original authors: Han Zhang, Mehrisadat Makki Alamdari, Babak Shahbodagh, Mohammad Vahab, Cosmin Anitescu, Timon Rabczuk, Elena Atroshchenko

Published 2026-10-08

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CC BY 4.0

Original authors: Han Zhang, Mehrisadat Makki Alamdari, Babak Shahbodagh, Mohammad Vahab, Cosmin Anitescu, Timon Rabczuk, Elena Atroshchenko

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of engineering, materials often fail not just because they are pulled too hard, but because they are heated or cooled unevenly. When a ceramic plate is plunged into cold water or a metal component in a jet engine faces extreme heat, temperature differences create internal stresses that can be strong enough to crack the material. These cracks rarely travel in straight lines; they curve, branch, and twist in patterns that are difficult to predict. For engineers designing everything from nuclear reactors to spacecraft, understanding exactly how a crack will grow under these combined thermal and mechanical forces is a matter of safety and survival. Traditional computer methods for predicting this behavior often struggle because they must constantly reshape their digital grid to follow the jagged path of a moving crack, a process that is computationally expensive and prone to error.

A team of researchers has developed a new way to simulate this complex behavior using artificial intelligence, specifically a technique that treats the problem as an energy minimization task. Instead of forcing a computer to solve a series of difficult equations step-by-step, the new method asks a neural network to find the state of lowest energy for the entire system at once. The researchers built a system where two separate neural networks work in tandem: one predicts how heat flows through the material, and the other predicts how the material deforms under stress. Crucially, both networks are given a special "map" of the crack's location. This map allows the networks to understand that the material is split in two, enabling them to represent the sudden jump in temperature and the separation of the material faces without needing to redraw the computer grid every time the crack moves.

The researchers tested this approach on several challenging scenarios, including a metal plate with a notch that was heated and then pulled apart, and a cross-shaped specimen subjected to both heat and force. In every case, the method successfully predicted the path the crack would take. The simulations showed that the crack would grow in the direction that released the most energy, a rule that holds true in the real world. The team also verified that their method could accurately calculate the intensity of the stress at the very tip of the crack, a critical value that determines whether a crack will stop or keep growing. By comparing their results against known solutions and other advanced computer models, they found their predictions matched with high precision, often within a fraction of a percent.

One of the most significant aspects of this work is how it handles the sharp edge of a crack. Many modern simulation methods smooth out the crack into a blurry zone to make the math easier, but this can hide important details about how the crack tip behaves. The new method keeps the crack as a sharp, distinct line throughout the entire simulation. This allows the researchers to read the stress values directly from the tip without having to guess or estimate them from a blurry zone. The method also accounts for the fact that the material's resistance to cracking can change depending on how hot the tip of the crack is. By solving the heat flow and the mechanical stress together, the system captures the feedback loop where heat causes stress, and the resulting crack changes how heat flows.

The study confirms that this energy-based neural network approach is a powerful tool for understanding thermo-mechanical fracture. It avoids the need for complex, constantly changing computer grids and provides a direct, accurate way to see how cracks will behave under the harsh conditions found in real-world engineering. The researchers demonstrated that the method works for both simple materials and those with varying properties, such as functionally graded materials designed to withstand extreme thermal environments. By successfully predicting crack paths in tension, shear, and combined loading scenarios, the work offers a reliable new way to simulate failure, potentially helping engineers design safer structures that can withstand the dual threats of heat and force.

Technical Summary: An Extended Deep Energy Method for Thermo-Mechanical Crack Propagation

Problem Statement
Thermo-mechanical fracture involves the coupling of transient heat conduction on a cracked domain with crack propagation driven by evolving thermal and mechanical stresses. While neural energy solvers have recently been extended to represent sharp cracks in isothermal settings, they have not yet addressed the simultaneous treatment of heat conduction on a cracked domain and crack propagation under resulting thermal stresses. Existing approaches often rely on phase-field methods, which regularize the crack into a diffuse damage band, or traditional mesh-based methods (XFEM, cohesive zones) that require explicit geometric tracking and remeshing. A significant gap exists in neural network-based solvers that can handle a sharp crack, solve the temperature field on the cracked domain, and propagate the crack based on trained fields without regularization length or explicit mesh updates.

Methodology
The authors propose an Extended Deep Energy Method (EDM) that maintains the crack as a sharp polyline throughout the simulation. The methodology integrates three core components:

Discontinuity Embedding: The crack geometry is encoded via a scalar embedding function, $\rho(x)$ , which is discontinuous across the crack faces and smooth elsewhere. This function is added as an input to two separate neural networks: one for temperature ( $T$ ) and one for displacement ( $u$ ). This allows a single global network to represent displacement jumps and distinct temperature values on opposite crack faces without splitting the domain.
Staggered Energy Minimization: The solution proceeds in a staggered sequence at every load step:
- Thermal Solve: The temperature field is obtained by minimizing an incremental functional of transient heat conduction (using a backward Euler scheme). The crack faces are treated as adiabatic boundaries, a condition satisfied naturally by the energy functional's weak form, supplemented by a consistency term.
- Mechanical Solve: The displacement field is obtained by minimizing the thermoelastic potential energy. The displacement network is enriched near the crack tip using the Williams expansion with trainable amplitudes ( $a_I, a_{II}$ ), analogous to XFEM, to capture the $r^{-1/2}$ strain singularity.
- Integration: Energies are estimated via Monte Carlo integration on points stratified over square background elements. Crucially, these points are redrawn periodically during training to prevent spurious solutions (e.g., transition layers hiding between fixed points) and densified near the crack tip.
Crack Propagation and Driving Force: Crack growth is governed by Linear Elastic Fracture Mechanics (LEFM) criteria evaluated on the trained fields:
- SIF Extraction: Stress Intensity Factors (SIFs) are extracted using the interaction integral. Under thermal loading, this includes the Wilson and Yu area term, which is essential for path independence in non-uniform temperature fields.
- Admissibility Checks: To ensure reliability, SIF extraction is validated by sweeping the contour radius. A reading is accepted only if the drift across radii is within a set tolerance, confirming the presence of the correct singularity.
- Growth Criterion: The crack advances when the energy release rate of a kink in the direction of maximum hoop stress reaches a temperature-dependent critical value ( $G_c(T)$ ). The direction is determined by the maximum hoop stress criterion, and the resistance is evaluated at the crack-tip temperature.

Key Contributions

Unified Thermo-Mechanical Solver: This work presents a neural network solver that solves the temperature field on a cracked domain and propagates a sharp crack based on fracture criteria evaluated on those fields, a combination not previously reported in the literature.
Sharp Crack Representation without Regularization: By combining discontinuity embedding with near-tip enrichment, the method represents a sharp crack and accesses SIFs directly, avoiding the regularization length and computational cost associated with phase-field diffuse bands.
Robust Thermal Fracture Extraction: The paper introduces a rigorous protocol for extracting SIFs under thermal load, specifically incorporating the Wilson and Yu area term and enforcing a radius-sweep admissibility check to filter out unreliable readings.
Handling of Complex Geometries: The embedding function is constructed to handle kinks and corners in the crack polyline, ensuring the discontinuity remains well-defined even as the crack geometry evolves.

Results
The method was assessed on four benchmarks:

Stationary Thermal Edge Crack: The extracted SIF agreed with a published value to within 0.11% on the finest grid, validating the extraction procedure including the area term.
Single-Edge Notched Tension (Homogeneous): Under three thermal loads ( $\Delta T = -50, 0, +50$ K), the predicted crack paths and peak loads matched phase-field references.
Single-Edge Notched Shear (Functionally Graded): The method reproduced initiation and propagation paths consistent with both phase-field references and an independent sharp-crack finite element solution.
Notched Cruciform Specimen: Under mechanical, thermal, and combined loading, the predicted crack paths followed published solutions from six independent studies.

In all cases, the method successfully reproduced crack paths and the influence of thermal loading on initiation and peak load, while maintaining the crack as a sharp geometric entity.

Significance
The paper claims that this approach bridges the gap between the efficiency of energy-based neural solvers and the physical fidelity required for sharp-crack thermo-mechanical fracture. By retaining the sharp crack and direct access to fracture mechanics quantities (SIFs, crack-tip temperature), the method avoids the indirect recovery of these quantities inherent in phase-field approaches. Furthermore, the proposed numerical procedures—specifically the resampling of integration points and the radius-sweep validation of SIFs—address stability and accuracy issues that have hindered the application of deep energy methods to complex, coupled fracture problems. The work demonstrates that neural networks can effectively solve coupled transient heat conduction and quasi-static elasticity on evolving discontinuous domains without the need for explicit mesh tracking or regularization.

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