# An extended deep energy method for thermo-mechanical crack propagation

arXiv: https://arxiv.org/abs/2610.09433 · HTML: https://mlsanity.com/paper/2610.09433/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 94 references that are in the map.
- First posted on arXiv 7 Oct 2026.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Physics-informed neural networks for solving thermo-mechanics problems of functionally graded material](https://mlsanity.com/paper/2111.10751/) (2021)
- [DEDEM: Discontinuity Embedded Deep Energy Method for solving fracture mechanics problems](https://mlsanity.com/paper/2407.11346/) (2024)
- [A mesh-free multiresolution deep energy method with phase-field modeling of brittle fracture](https://mlsanity.com/paper/2608.24126/) (2026)

## Abstract (arXiv)

Thermo-mechanical fracture couples transient heat conduction on a cracked domain with a crack that grows as the temperature and the displacement evolve. Neural energy solvers have been proposed for phase-field fracture and later extended to represent a sharp crack through the network input, but heat conduction on the cracked domain and crack propagation under the resulting thermal stresses have not yet been treated together in these solvers. We present an extended deep energy method for thermo-mechanical crack propagation in which the crack remains a sharp polyline. Two networks represent the temperature and the displacement and receive the crack through a scalar embedding function, discontinuous across the crack and smooth elsewhere, so that both fields can jump across it without a regularization length, and the displacement is enriched near the tip by the Williams expansion with trainable amplitudes. The two fields are obtained by minimizing an incremental conduction functional and the thermoelastic potential energy in a staggered sequence, with Monte Carlo integration on points stratified over background elements, densified near the tip and redrawn during training. The stress intensity factors are extracted by the interaction integral with the area term of Wilson and Yu and checked by a sweep of the contour radius, and the crack advances at the maximum hoop stress angle when the energy release rate of the kink reaches the critical value at the crack-tip temperature. On a stationary thermal edge crack the extracted stress intensity factor agrees with the published value to 0.11%, in a functionally graded shear test initiation agrees with an independent sharp-crack finite element solution to within one load step, and on a notched cruciform specimen the crack paths follow the published solutions under mechanical, thermal and combined loading.