cs.LGOct 7, 2026
SaveAn extended deep energy method for thermo-mechanical crack propagation
Organizations: University of New South Wales, Sydney, NSW, Australia · Central Queensland University, Melbourne, VIC, Australia · Bauhaus-Universität Weimar, Weimar, Germany
Abstract
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.
Figures & tables
Fig. 1: The thermo-mechanical problem on a cracked body. The crack has adiabatic, traction-free faces, whose outward normals point into the opening, and the active tip . Temperature and displacement are prescribed on and , heat flux and traction on and . For a fixed crack the temperature enters the mechanical problem through the thermal strain , and the mechanical field acts on the heat conduction problem only through the crack it advances.
| Symbol | Meaning | Symbol | Meaning |
|---|---|---|---|
| , | body, cracked domain | crack embedding function | |
| crack faces | crack-aware network input | ||
| , | displacement, temperature | , | network parameters |
| , , | total, elastic, thermal strain | , | trainable enrichment amplitudes |
| , | stress, elastic energy density | , | tip polar frame, tangent angle |
| , , | density, specific heat, conductivity | , , | extracted SIFs, -integral |
Table 1: Main symbols. The load or time step index appears as a superscript. Descriptive subscripts are upright ( , ). The mass density is written so that can denote the crack embedding function, and is the Euclidean norm.
Fig. 2: The crack embedding function ( 44 )–( 45 ) on the crack at load step 70 of a heated graded shear run of Section 6.3 ( , , ). (a) changes sign across the crack and decays away from it, the color scale clipped at so that the sign change stays visible toward the tip. (b) Transverse profiles at three stations , with the arc length. The jump is largest where the crack meets the boundary and nearly closed at the tip. (c) The embedded jump sampled at the offset , as markers, against , as the dashed line, where is the arc-length window of ( 45 ) before squaring. The zero just past the notch corner is the corner effect of Section 3.1 .
Fig. 3: The integration scheme. The bounding box of the specimen is partitioned into square background elements of side , each with Latin hypercube samples of weight . An element whose center, marked by a cross, lies within a set radius of the crack tip, the dashed circle, receives a multiple of samples, each weight divided by the same multiple, so that every element still integrates to its own area. The multiple is in the sketch and on the notched tests and on the cruciform. The set is redrawn at a fixed interval during training.
Fig. 4: Structure of the enriched approximation. The crack polyline defines the embedding, which enters both networks through the crack-aware input. Both network outputs are lifted onto the essential conditions, and the displacement is enriched near the tip by the Williams modes with trainable amplitudes. The thermal and mechanical losses are minimized in sequence, as set out in Section 4.5 , the trained temperature entering the mechanical loss through the thermal strain.
Fig. 5: Magnitude of the mode I enrichment with its decay, computed from ( 68 ) and ( 72 ) with the tension test’s and , on the kinked crack of Fig. 2 .
Fig. 6: Flow chart of the staggered solution and crack growth cycle. The ratio is formed before the admissibility checks of Section 5.2 , since the first check acts on it. A reading that fails the checks authorizes no extension, except under the bounded release for a clipped sweep. While ( 94 ) holds and the extensions in the step are below their cap, the crack is extended and the fields are retrained on the new embedding.
Fig. 7: The two geometric rules of the extraction. (a) The nominal contour stays inside the straight stretch that ends at the tip. A larger contour, such as , dashed, would cross the kink, and the auxiliary field, a straight-crack solution, would no longer match the crack it encloses. (b) Schematic of the radius sweep, flat for a field with the singularity and drifting as for a field without it, with the admissible drift , across a fourfold span, shaded.
Fig. 8: The thermal edge crack of Wang et al. [ 77 ] , with the imposed temperature difference drawn schematically. The rollers prevent the cooled face from contracting, which puts into tension across the crack.
Fig. 9: SIF of the stationary thermal edge crack, normalized by with K -1 and K. (a) The three grids against the published value of Wang et al. [ 77 ] , with a band of 2%. (b) The radius sweep of each grid with its exponent , all below the admissibility threshold . The innermost contour of the finest grid lies above the rest of its sweep.
Fig. 10: Geometry and boundary conditions of the single-edge notched specimen. (a) Tension, with a vertical displacement on the upper edge. (b) Shear, with a horizontal displacement on the upper edge. In both tests the lower edge is fixed and held at , the upper edge is held at and the vertical edges are adiabatic. The temperature shading is schematic.
| Parameter | Symbol | Value |
|---|---|---|
| Young’s modulus | ||
| Poisson’s ratio | ||
| Critical energy release rate | ||
| Density | ||
| Thermal conductivity | ||
| Specific heat capacity |
Table 2: Material parameters for the homogeneous single-edge notched tension test.
Fig. 11: Force–displacement response of the single-edge notched tension test at , and , cooled in blue, isothermal in black and heated in red, with the phase-field reference dashed.
Fig. 12: Temperature and vertical displacement in the single-edge notched tension test at , at load steps 0, 66 and 70 of 106. Step 66 is the last before the crack advances and step 70 follows separation. Each row shares one color scale. The region below the notch stays near the initial temperature since the crack faces are adiabatic.
Fig. 13: SIFs of the single-edge notched tension test against the applied displacement: (a) , (b) , with the legend of (a). The test is mode I to the accuracy of the extraction. Readings after the tip has reached the far edge, where no contour fits inside the material, are not plotted.
| Parameter | Symbol | ||
|---|---|---|---|
| Young’s modulus | |||
| Poisson’s ratio | |||
| Critical energy release rate | |||
| Density | |||
| Thermal conductivity | |||
| Specific heat capacity |
Table 3: Material parameters for the functionally graded single-edge notched shear test, adopted from the implementation of Tangella et al. [ 85 ] . The values at are those of the homogeneous tension test in Table 2 .
Fig. 14: Force–displacement response of the graded single-edge notched shear test at , and , with the colors and line styles of Fig. 11 . The stars mark initiation in the sharp-crack finite element solution, which holds the crack stationary.
Fig. 15: Temperature and horizontal displacement in the graded single-edge notched shear test at , at load steps 0, 70 and 140 of 141. Each row shares one color scale. The region below the crack stays near the initial temperature since the crack faces are adiabatic.
Fig. 16: SIFs of the graded shear test against the applied displacement: (a) , (b) , with the legend of (a). grows while the notch is stationary and collapses when the first extension turns out of its plane, where takes over. Readings after the tip has reached the separation margin of Section 5.5 are not plotted.
Fig. 17: Geometry of the notched cruciform specimen and the three load cases. (a) Case I, the top edge displacement alone, isothermal. (b) Case II, the temperatures alone, with the top edge traction free. (c) Case III, both. In all three the side arms are held at the reference temperature with and the bottom edge is fixed. Dimensions are marked on (a). The temperature shading is schematic.
| Parameter | Symbol | Value |
|---|---|---|
| Young’s modulus | ||
| Poisson’s ratio | ||
| Fracture energy | ||
| Tensile strength | ||
| Thermal expansion coefficient | ||
| Density |
Table 4: Material parameters for the notched cruciform test.
| Solution | Crack | Method | Load | ( ) | Plane | Used in | |
|---|---|---|---|---|---|---|---|
| Wang [ 89 ] | sharp | meshfree | traction | – | I, II | ||
| Nguyen et al. [ 88 ] | sharp | enriched elements | traction | – | I, II | ||
| Chen et al. [ 90 ] | sharp | smoothed elements | traction | strain | I, II, F | ||
| Greco et al. [ 91 ] | sharp | moving mesh | traction | strain | I, II, F | ||
| Mandal et al. [ 37 ] | phase field | finite elements | displacement | stress | I, II, III | ||
| Chen et al. [ 92 ] | phase field | scaled boundary elements | displacement | – | I, II, III |
Table 5: Published solutions of the notched cruciform specimen and the statement of the problem that each solves. Every source that states Young’s modulus gives . The traction on the top edge is Pa and the displacement mm. An asterisk marks data that the source does not state, taken from Prasad et al. [ 87 ] , to whose statement of the problem the source refers, and a dash marks what neither states. The last column gives the load cases in which the path is compared, and F marks the two solutions whose SIFs are compared.
| Case | Proposed method | Published | Outside the published span |
|---|---|---|---|
| I, mechanical | ( to ) | to | to above |
| II, thermal | ( to ) | to | to below |
| III, both | ( to ) | and | to below |
Table 6: Chord angles of the crack paths at mm from the notch tip, measured from the global axis. The proposed method is given as the first of three repeats with the range over the three, the published solutions as the span of those drawn in Fig. 18 , and the last column gives the distance of the repeats from the nearest published value.
Fig. 18: Crack paths of the three load cases against the published solutions, labeled by crack description. (a) Case I, mechanical. (b) Case II, thermal. (c) Case III, both, with the two displacement-controlled references only, since the traction-controlled family solves a different combined problem. One of the three repeats is drawn per case, the other two staying within of it.
Fig. 19: Both SIFs against the crack advance , read along the path the proposed method grows for itself, as the published curves are read along theirs. Case I is read with the crack held at each position of that path under the final load, cases II and III from the propagating runs. Markers give the median over the three repeats, which lie within of each other in case I, in case II and in case III, except at in (c), where one repeat passed the radius sweep at a single load step, and at and in case II, where two repeats and one passed the radius sweep. Case III has no reference, since the only published SIFs under both loads are for a cruciform with cooled side arms.
Appendix figures & tables5 assets
Supplementary material from the paper’s appendix.
Appendix
| Stationary crack | Tension | Graded shear | Cruciform | |
|---|---|---|---|---|
| Section 6.1 | Section 6.2 | Section 6.3 | Section 6.4 | |
| Background elements across the width | 48, 72, 96 | 100 | 100 | 150 |
| Resampling interval | none | 500 iterations | 500 iterations | 500 iterations |
| Displacement network | 3–50–50–50–2 | 3–50–50–50–2 | 3–50–50–50–2 | 3–50–50–50–2 † |
| Adam iterations per load step, thermal / mech. | 8000 | 3000 / 3000 | 3000 / 3000, 5000 / 5000 at step 1 | 2000 / 3000 |
| Unloaded step 0, Adam then L-BFGS, thermal / mech. | – | 5000 / 7000, then 2500 / 3000 | 3000 / 3000, then 3000 / 1500 | 5000 / 7000, then 2500 / 3000 |
Table A.1: Numerical parameters of the four examples. Layer widths run from the input to the output. The thermal network has a sinusoidal first layer with , the displacement network hyperbolic tangents throughout. All four examples share nine integration points per background element, the thermal network – – – – and the learning rate . The stationary example has no crack increment and no redraw. Lengths are in millimeters, the notched tests being computed on the unit square of side mm and the cruciform in its physical dimensions, and of ( 45 ) is given in millimeters or in units of the background element size . The cruciform entries marked are those of cases I and III, and case II uses the displacement network – – – – – with .
| Proposed method | 40 | 45 | 51 |
| Proposed method, repeat | 41 | 47 | 51 |
| Sharp-crack finite elements | 41 | 46 | 51 |
| Phase field | 46 | 51 | 56 |
Table A.2: Load step at which the crack first advances in the graded shear test, two runs per case, against the two references of Section 6.3 . One load step is of applied displacement. The phase-field entry is the step of its peak load, since it has no discrete initiation. The two runs differ only in the draw of the integration points.
| Variation | Peak | First growth | Extensions |
| Baseline | N | step | |
| Repeat, integration point draw resampled | |||
| Increment halved to mm, contour at elements | no growth | – | |
| Increment doubled to mm, radii | |||
| Increment doubled, contour pinned at mm | |||
| Both networks narrowed to width |
Table A.3: Sensitivity of the isothermal shear test to its numerical parameters, one change per row. Peak forces are read inside the growth window and given relative to the baseline. The repeat differs only in the draw of the integration points and sets the run-to-run scatter against which the other rows are read.
| Case | Run | Extensions | Sweep refusals | Convergence | Released, extended | |
|---|---|---|---|---|---|---|
| drift | no contour | refusals | ||||
| I | reported | 14 | 6 | 37 | 0 | 0, 0 |
| I | repeat 1 | 14 | 16 | 37 | 0 | 0, 0 |
| I | repeat 2 | 15 | 7 | 36 | 0 | 1, 1 ∗ |
| II | reported | 7 | 37 | 0 | 0 | 2, 0 |
| II | repeat 1 | 7 | 33 | 0 | 0 | 3, 0 |
Table A.4: Readings refused and released in the nine cruciform runs over load steps each, the reported run of each case first. The sweep refusals are split into a drift beyond the threshold and no fitting contour. The last column gives the readings released on a clipped sweep and, after the comma, the extensions taken on them, and an asterisk marks an entry that includes one extension taken on a single radius near the boundary.
Fig. C.1: The area term isolated on the stationary benchmark. A field trained separately on the finest grid is read with the area term ( 85 ) and from the contour alone. With the area term the sweep flattens inside the band, and from the contour alone the reading falls monotonically. The dashed line is the published value of Wang et al. [ 77 ] , the shaded band 2%.