PaperlayerView exact revision on arXiv

Source-linked AI summary

An extended deep energy method for thermo-mechanical crack propagation

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

arXiv:2610.09433v1cs.LGcond-mat.mtrl-scics.NAmath.NA

TL;DR

Neural energy solvers had not combined heat conduction on a cracked domain with propagation of a sharp crack under thermal stresses. This paper develops a staggered extended deep energy method and evaluates it on four benchmarks, reporting agreement with published and independent reference results.

  • Problem

    Neural solvers had not combined temperature solved on a cracked domain with crack propagation driven by the resulting thermal stresses.

    Source: §1 · p. 4

  • Method

    Two networks share a scalar embedding of a sharp polyline crack and obtain temperature and displacement by staggered minimization of thermal and thermoelastic energies.

    Source: §1 · p. 4

  • Results

    Across four benchmarks, the method reproduces crack paths and thermal-load effects; the stationary thermal edge-crack SIF agrees with the published value to 0.11%.

    Source: §1 · p. 4, §7 · p. 36

  • Takeaways & Limitations

    The benchmarks show that crack paths, SIFs, and crack-tip temperature can be obtained with the crack retained as a sharp geometric entity.

    Source: §1 · p. 4

    The separation rule leaves the last increment before physical separation of the two halves unresolved.

    Source: §5.5 · p. 23

Abstract

from arXiv · show

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.

1 Introduction

The introduction motivates sharp-crack thermo-mechanical propagation as a gap between neural solvers for thermal fields and neural methods for growing sharp cracks. It presents an energy-based approach and assesses it against four fracture benchmarks.

1 Introduction

  • Sharp-crack neural methods avoid a regularization length and provide direct crack-tip quantities, while phase-field methods avoid explicit crack tracking but recover SIFs indirectly.The introduction identifies explicit geometry tracking as a cost of sharp-crack approaches and fine resolution of the damage band as a cost of phase-field approaches.Source: §1 · p. 3
  • Thermal crack growth has not previously been treated in a neural solver that both solves temperature on the cracked domain and propagates the crack using trained fields.Earlier network studies either solved graded thermo-mechanics without a crack, embedded cracks without temperature, or held thermally loaded cracks fixed and stopped at SIFs.Source: §1 · p. 4
  • The method combines a sharp polyline crack, a shared scalar embedding for temperature and displacement networks, staggered energy minimization, and Williams near-tip displacement enrichment.The transient thermal field follows an incremental functional; the displacement includes trainable Williams amplitudes, and integration points are redrawn and densified near the tip.Source: §1 · p. 4
  • Reliable thermal SIF extraction requires the Wilson–Yu area term and a contour-radius sweep; growth uses the maximum hoop stress direction and tip-temperature-dependent critical energy release rate.The kink energy release is used to retain sensitivity to the sign of thermal loading.Source: §1 · p. 4
  • Across four benchmarks, the method reproduces crack paths and thermal-load effects on initiation and peak load, while retaining direct access to stress intensity factors and crack-tip temperature.The comparisons include published paths, a phase-field reference, and an independent sharp-crack finite element solution for graded shear.Source: §1 · p. 4

2 Governing equations and energy functional

The governing formulation treats transient heat conduction and quasi-static thermoelasticity on a fixed cracked domain at each step. Temperature affects mechanical stress through thermal strain, while the mechanical field changes the next thermal problem only through crack advance.

2 Governing equations and energy functional

  • For a fixed crack, the model uses one-way thermo-mechanical coupling: temperature induces thermal expansion, while deformation-related heat generation and mechanical feedback into the thermal balance are neglected.The treatment is stated to suit quasi-static brittle fracture with small, slow deformations.Source: §2 · p. 5

2.1 Cracked domain and primary fields

  • The temperature and displacement are solved on the cracked domain, with the crack held fixed during each field solve and advanced only by the growth rule.The crack faces have their own boundary conditions; in two dimensions the energy integrals can be evaluated over the background domain when fields admit distinct traces on the faces.Source: §2 · p. 5, §2.1 · p. 6

2.2 Transient heat conduction

  • Transient heat conduction uses Fourier flux and an implicit backward Euler step, whose discrete equation is obtained by minimizing an incremental thermal functional.The crack faces are adiabatic natural boundaries, and no internal heat source is used in the numerical examples.Source: §2.2 · pp. 6–7

2.3 Thermoelastic constitutive response

  • The mechanical stress follows thermoelasticity: elastic strain subtracts thermal strain from total strain, with thermal strain measured from the stress-free reference temperature.The formulation supports spatially varying material parameters, including functionally graded materials; plane strain uses an effective thermal expansion coefficient.Source: §2.3 · pp. 7–8

2.4 Quasi-static mechanical equilibrium

  • At each step, the mechanical field is a quasi-static equilibrium solution that minimizes thermoelastic potential energy for the current temperature and fixed crack.The crack faces are traction-free and assumed open; local crack closure would require a unilateral contact condition.Source: §2.4 · p. 8

3 Explicit crack representation

The crack is represented as an explicit polyline, while a scalar embedding lets a single global network distinguish points across its faces. The crack-tip frame and embedding are built from the current geometry.

3 Explicit crack representation

  • A single global network uses an added scalar crack embedding to distinguish points on opposite sides, allowing a displacement jump without splitting the physical domain.The crack itself remains an explicitly represented sharp geometric entity.Source: §3 · p. 8

3.1 Crack geometry

  • The crack geometry is an ordered control-point polyline whose final point is the active tip; its final segment supplies the tangent and normal for tip-local calculations.That local frame is used for both near-tip enrichment and crack-tip post-processing.Source: §3.1 · p. 9

3.2 Crack embedding function

  • The embedding is a weighted sign function: its sign distinguishes crack faces, while a smooth localization weight confines the discontinuity to the crack region.For the propagating polyline, segment contributions are blended to keep representative signed distance and projected arc length continuous across segment junctions.Source: §3.2 · p. 11
  • The implemented embedding closes its jump continuously beyond the active tip, but permits a small discontinuity over the endpoint-overlap distance δc ahead of the tip.The implementation also departs locally from ideal embedding conditions near polyline corners and endpoint-normal lines.Source: §3.2 · pp. 10–11

3.3 Local crack-tip frame

  • The crack-tip tangent defines a local frame, and its coordinates are used for the near-tip asymptotic enrichment.The frame is attached to the active tip.Source: §3.3 · p. 12

4 Enriched deep energy formulation

Separate crack-aware networks represent temperature and displacement, and sequential energy minimization couples the thermal and mechanical fields. Boundary liftings enforce prescribed values, while Williams modes enrich displacement near the crack tip.

4 Enriched deep energy formulation

  • Separate crack-aware temperature and displacement networks minimize incremental functionals, with automatic differentiation supplying derivatives and boundary liftings enforcing prescribed values exactly.The temperature network uses a sinusoidal first-layer map to help resolve steep thermal gradients.Source: §4 · pp. 12–13

4.1 Temperature approximation

  • The temperature network uses boundary liftings but no tip enrichment: its crack-face jump is embedded, while its integrable singular gradient suffices for the interaction integral’s area term.Zero normal flux on adiabatic crack faces is a natural condition of the thermal functional, supplemented by a small consistency term.Source: §4.1 · p. 13

4.2 Thermal energy loss

  • Latin hypercube samples stratified by background element use area-preserving weights, with denser sampling near the tip; periodic redrawing prevents fixed gaps from hiding spurious transition layers.Every element integrates to its own area, making the functional estimate unbiased; the near-tip sample multiple is 8 on notched tests and 48 on the cruciform.Source: §4.2 · p. 14, Figure 3 · p. 14

4.3 Enriched displacement approximation

  • The embedded crack permits a global displacement network, while localized trainable Williams modes supply singular near-tip strain; behind corners, their attenuated jump can lie off the crack.The extraction contour remains on the straight crack segment ending at the tip, where the enrichment discontinuity and crack coincide.Source: §4.3 · pp. 15–17

4.4 Mechanical energy loss

  • For fixed crack and temperature, the displacement network minimizes thermoelastic potential energy using elastic strain after subtracting thermal strain; built-in essential conditions need no Dirichlet penalty.Traction-free crack faces remain a natural condition of the mechanical energy statement.Source: §4.4 · pp. 17–18

4.5 Staggered solution procedure

  • At each time level, minimize the thermal functional then hold temperature fixed for the mechanical solve; after propagation, repeat on the rebuilt crack while retaining the previous temperature at that time level.Both networks warm-start from the preceding converged configuration, and periodic resampling is paired with Adam optimization.Source: §4.5 · p. 18

5 Crack driving force and propagation

The method extracts mixed-mode stress intensity factors (SIFs), checks their admissibility, and uses the crack-tip temperature and kink driving force to decide whether and where the crack advances. Each accepted extension updates the crack embedding and triggers retraining; unreliable readings and proximity to the far boundary constrain propagation.

5.1 Extraction of the stress intensity factors

  • The thermal interaction integral combines contour and area terms; the area term accounts for temperature-dependent elastic energy and vanishes for uniform temperature.The contour is evaluated directly and the area term over the enclosed disc.Source: §5.1 · pp. 19–20
  • A contour crossing a polyline corner uses a mismatched straight-crack auxiliary field and can place post-kink mode-I SIF about one-third below the kinked-crack solution.The nominal radius is 0.9 ∆a and stays within the straight stretch ending at the tip.Source: §5.1 · p. 20, Figure 7 · p. 21

5.2 Admissibility of an extraction

  • The convergence check rejects an extraction when χ exceeds a fixed threshold multiple or rises by more than a factor of two from the previous extension in that load step.Settled and unsettled extensions differ by an order of magnitude in χ in the reported tests.Source: §5.2 · p. 20
  • The radius sweep accepts SIFs only when p < 0.1172, using medians across retained radii; a failed clipped sweep may be released at most twice after a wider-span pass.The sweep tests singular structure, not whether the integral includes the thermal area term.Source: §5.2 · p. 21

5.3 Crack-tip temperature and fracture resistance

  • Fracture resistance is evaluated at the crack-tip temperature, estimated by averaging the thermal field over a small ring straddling both crack faces.The comparison with an independent conduction solution supports this ring estimate over a point value.Source: §5.3 · pp. 21–22

5.4 Growth direction and driving force

  • The maximum hoop stress criterion sets the kink angle, and the driving force is the energy release rate for that kink rather than straight-ahead growth.For mixed mode, the kink driving force distinguishes thermal stresses that open the crack from same-sized stresses that close it.Source: §5.4 · pp. 22–23

5.5 Propagation criterion and crack update

  • With χc = 1, the crack advances by one increment when the kink energy release rate reaches the temperature-dependent resistance, then the embedding is rebuilt and fields retrained.The extraction, checks, and criterion repeat within a load step until growth stops or reaches its extension cap.Source: §5.5 · p. 23
  • Propagation terminates near the far boundary when the remaining ligament is thinner than the contour radius; the final increment before physical separation is unresolved.After the separation margin is reached, no further SIFs are extracted and reaction force is set to zero.Source: §5.5 · p. 23

6 Numerical examples

The numerical examples test stress-intensity-factor extraction for a stationary thermal crack and crack propagation under tensile, graded-shear, and cruciform loading, with comparisons for the first three cases and an assessment of which published cruciform solutions are comparable. For the cruciform, computed crack paths track published paths under mechanical, thermal, and combined loading, while exploratory stress-intensity-factor comparisons show case-dependent differences qualified by setup differences.

6.1 Stress intensity factor of a stationary crack under thermal load

  • On the finest grid, the stationary thermal-crack SIF is KI/K0 = 0.4995, 0.11% from Wang et al.’s published 0.500 value [77].Across grids of 48, 72, and 96 elements across the strip width, the extracted values are 0.4953, 0.4955, and 0.4995; all radius sweeps pass the admissibility threshold p < 0.1172.Source: §6.1 · p. 24, Figure 9 · p. 25

6.2 Single-edge notched tension test

  • In tension, peak-load displacement rises 89%, from 0.35 × 10^-3 mm at −50 K to 0.66 × 10^-3 mm at +50 K, while peak load changes 1.3%.Peak loads agree with the phase-field reference by +4.1%, +1.3%, and −0.8% at −50, 0, and +50 K; cooling advances initiation and heating delays it. The crack advances at KI = 120–123 N/mm3/2, and the response is mode I to extraction accuracy.Source: §6.2 · pp. 26–28, Figure 11 · p. 27

6.3 Single-edge notched shear test with functionally graded material

  • In the heated graded-shear case, the trained temperature field has 0.08 K root-mean-square error against the sharp-crack finite element field on a 501 × 501 grid.The ring-sampled tip temperature agrees with the reference to 0.24 K, supporting the test’s choice to leave temperature without tip enrichment.Source: §6.3 · p. 29
  • The graded shear predictions match a sharp-crack finite element reference within 1% to 2% on the loading ramp, with initiation within one load step in all three thermal cases.The predicted downward kink is not prescribed: it follows from the extracted SIFs under the maximum hoop stress criterion, and the resulting path is consistent with the phase-field reference.Source: §6.3 · p. 29, Figure 14 · p. 30
  • The heated graded-shear initiation reading has KI = −40 and KII = 178 N/mm3/2 at load step 51, with χ = 1.03 and a −75° kink angle under traction-free faces.The negative KI corresponds to slight face interpenetration; the sharp-crack reference uses the same traction-free assumption, and contact would reduce the compressive KI and shrink the heated-case delay.Source: §6.3 · pp. 30–31

6.4 Notched cruciform test under combined thermo-mechanical loading

  • For the cruciform comparison, single-load published paths agree within reported spreads, while combined-load comparisons use only displacement-controlled solutions because the published problem families differ.The families vary in mechanical loading and material data, including a factor-of-36 difference in thermal expansion; combining their case-III paths would create a nearly 40° spread that no single problem exhibits.Source: §6.4 · pp. 31–32
  • SIF comparisons were exploratory because published values existed only for the traction-controlled statement and each family was read along its own crack path.The differing Poisson’s ratios, plane conditions, load application, and paths mean residual discrepancies may reflect the problem statement; path comparison remains the primary validation.Source: §6.4 · pp. 33–34
  • At 12 mm from the notch, path-angle offsets from published references were 1.5°–1.7° in case I, 2.4°–4.1° in case II, and 1.3°–3.2° in case III.The three repeats stayed within 1.6 mm of one another, less than half a crack increment; the comparison measures initial direction, not the full path.Source: §6.4 · p. 33, Table 6 · p. 33, Figure 18 · p. 34
  • In case I, mean KI ratios at matched crack length were 0.92 against Chen et al., 0.95 against Greco et al., and 0.93 across both; KII/KI had median magnitude 0.013.The KI ratio ranged from 0.89 to 1.04 along the history; away from the notch, KII/KI ranged from −0.038 to +0.018.Source: §6.4 · p. 34, Figure 19 · p. 35
  • In case II, KI/K0 fell below published curves, with its ratio declining from 0.88 to 0.48 as crack advance increased; KII/K0 ranged from 0.004 to 0.019.The residual grew with crack length, but differences in Poisson’s ratio, plane condition, and path prevent attributing it to any one factor.Source: §6.4 · pp. 34, 36, Figure 19 · p. 35
  • Case III had no published SIF reference for this load combination, and the three repeats agreed in KI within 9%.The only published SIFs under both loads concern a cruciform with cooled side arms.Source: Figure 19 · p. 35, §6.4 · p. 36

7 Conclusions

The paper presents an extended deep energy method for thermo-mechanical crack propagation with a sharp, explicitly represented crack. Four benchmarks assess its SIF extraction and crack paths, alongside stated scope limits and future extensions.

7 Conclusions

  • The method represents the crack as a polyline, shares its embedding across temperature and displacement networks, and advances it using fracture criteria evaluated from trained fields.It minimizes incremental conduction and thermoelastic potential energies in a staggered sequence, enriches displacement near the tip, and redraws integration points densified near the tip.Source: §7 · p. 36
  • Including Wilson and Yu’s area term makes stationary-benchmark SIF readings path independent to within 3%–7% across a fourfold contour-radius span.Without the area term, the SIF from the same trained field changes with contour radius; a sweep refuses crack extension when drift exceeds the chosen tolerance.Source: §7 · p. 36
  • On the stationary thermal edge crack, extracted SIF agrees with the published value to 0.11%; graded-shear initiation agrees with an independent sharp-crack finite element solution within one load step.The graded-shear agreement holds in all three thermal cases.Source: §7 · p. 36
  • For the notched cruciform specimen, predicted chord angles fall 1.3°–4.1° outside the nearest published curve, while published solutions themselves spread 5.4°–18.9° per case.Exploratory SIF comparison uses published curves solving the traction-controlled benchmark statement rather than the displacement-controlled one computed here. Along growth paths, mechanical ratios average 0.93, while thermal KI follows the published decline with an offset growing from 0.015 to 0.03 load-measure units.Source: §7 · p. 36
  • The study is limited to two-dimensional, quasi-static growth of a single open, traction-free crack; it does not treat branching, merging, or crack closure.The crack advances by a prescribed increment and direction, leaving the last increment before separation unresolved; the extraction contour also sets a lower bound on increment size for a given grid.Source: §7 · p. 36
  • Multiple or merging cracks, crack nucleation, three-dimensional problems, and two-way thermo-mechanical coupling are left for future work.A propagating case takes 3.5–12 hours on one GPU, and no propagating reference solution was timed on the same task.Source: §7 · p. 37

A Numerical parameters, cost and sensitivity

The appendix reports numerical settings, runtime, repeatability checks, and sensitivity to selected parameters. It also describes implementation details that affect the embedding, crack-face flux sampling, and SIF extraction.

A Numerical parameters, cost and sensitivity

  • Nine integration points per background element are used in all four examples, while network architecture, learning rate, and length scales vary by case.The appendix also reports Adam stopping criteria and random seeds; the seed does not govern integration-point draws in the notched tests.Source: A Numerical parameters, cost and… · p. 37, Table A.1 · p. 38
  • Near corners, the embedding’s sign change can depart from the polyline and leave two face samples on one side, so the flux term may act on only one face.Away from corners, face-offset sampling attenuates the jump by at most 2.5% in notched tests and 15% for the cruciform flux offset.Source: A Numerical parameters, cost and… · pp. 38–39
  • Propagating cases take 3.5–12 hours on one GPU, with both networks retrained at every load step and after every extension.A shear case comprises about 160 staggered solves of 2.5–4.5 minutes each; changing integration-point count or network width affects runtime less than the optimizer-iteration count.Source: A Numerical parameters, cost and… · p. 39
  • Every graded-shear run lies within one load step of the sharp-crack solution across the three thermal cases.The two runs per case initiate 2, 1, and 0 load steps apart at ΔT = 0, −30, and +30 K, respectively; their peak loads differ by 4.5%, 2.4%, and 0.2%.Source: A Numerical parameters, cost and… · p. 39, Table A.2 · p. 39
  • In the isothermal shear sensitivity study, every variant that grows a crack peaks 2.0%–4.1% above baseline, within the 4.5% repeat variation.Variants initiate at step 46 or 47 against step 45; doubling the increment reaches the separation margin after 11 or 12 extensions against 23.Source: A Numerical parameters, cost and… · p. 39, Table A.3 · p. 39
  • Extracted SIFs span 0.4921–0.4979, a 1.2% spread, despite trained enrichment amplitudes ending 7.8% apart across five initializations, including one wrong-sign start.Changing GPU model moves no recomputed grid value by more than 0.7%.Source: A Numerical parameters, cost and… · p. 40

B Reference solutions

The reference solutions benchmark stress-intensity extraction and crack-growth predictions against published and independent finite-element results. They also quantify mesh sensitivity and temperature-field agreement near the crack.

B Reference solutions

  • The phase-field reference uses AT2 with ℓ = 0.01 mm, elements about ℓ/5, and staggered temperature, displacement, and phase-field solves.Temperature is advanced first with backward Euler and crack-face conductivity degraded by (1 − ϕ)^2.Source: B Reference solutions · p. 40
  • Proposed-method peaks lie 7% to 10% below Tangella et al.'s published curves, while the reference peaks lie 8% to 13% below them.The coarser published mesh's toughness inflation accounts for 6.6% of the peak difference and raises peak load by 12%.Source: B Reference solutions · pp. 40–41
  • The independent sharp-crack reference estimates straight-ahead G from extended-notch energy differences with Richardson extrapolation, then fits crack-face openings to obtain the SIF ratio.Its SIFs enter the same kink-angle and temperature-dependent initiation criterion as the proposed method.Source: B Reference solutions · p. 41
  • The implementation extrapolates to KI/K0 = 0.4966 against the published 0.500, a 0.7% difference, and doubling the graded-run mesh leaves the first-growth step unchanged.The doubled mesh changes reaction force by 0.1% and G(θc) by 0.3%.Source: B Reference solutions · p. 41
  • Across crack faces, trained and reference temperature jumps agree within 1.4 K except over the final 0.025 mm; ahead of the tip, differences are 1.9 K within overlap and 0.5 K beyond.Within the endpoint overlap, the embedding prolongs the discontinuity past the tip by construction.Source: B Reference solutions · p. 41

C Stress intensity factor extraction under thermo-mechanical loading

The area term is essential to stable thermal SIF extraction: it produces a radius sweep that flattens, while contour-only readings drift. The sweep shape can distinguish the two even when an individual contour reading appears admissible.

C Stress intensity factor extraction under thermo-mechanical loading

  • Omitting the area term leaves finite SIFs but causes contour-radius drift, whereas the complete thermoelastic interaction integral is path independent.The area contribution is the interaction form of the thermoelastic J-integral term derived by Wilson and Yu.Source: C Stress intensity factor extraction… · p. 41
  • On the stationary benchmark, contour-only and area-term readings differ by 0.7% on the smallest contour and 5.9% on the largest.The contour-only reading still passes the admissibility check, making the flat-versus-monotonic sweep shape the distinguishing signal.Source: C Stress intensity factor extraction… · p. 42
  • In the cruciform thermal case at mid load, the contour-only reading has p = 0.54 versus 0.18 with the area term; the run recorded p = 0.176 and did not advance the crack.The area-term extraction therefore remains closer to the run's recorded value at this step.Source: C Stress intensity factor extraction… · p. 42
  • With the area term, the sweep flattens at KI/K0 = 0.495, 1.1% below the published value; contour-only readings fall 9.7% across the sweep.The contour-only value reduced from the sweep is 0.484, while the area-term reading has p = 0.039.Source: Figure C.1 · p. 42, C Stress intensity factor extraction… · p. 42
Loading 2610.09433v1…