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On phase-field regularization in dynamic fracture with brittle and cohesive formulations

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Phase-field regularization itself, not numerics, causes oscillations and widening in dynamic brittle fracture; a cohesive variant recovers sharp-crack behavior.

desk verdict Solid, useful paper: it gives a credible mechanistic explanation for the oscillations and widening in dynamic phase-field fracture, and a new dynamic cohesive model with an analytical opening law; the TMM/FEM split-model gap and the inertia-free eigenstrain assumption are the main soft spots. read the letter →

arxiv 2607.13599 v2 pith:IODFGUB7 submitted 2026-07-15 cs.CE

Jonas Heinzmann , Francesco Vicentini , Pietro Carrara , Laura De Lorenzis This is my paper · ORCID
classification cs.CE
keywords phase-fieldfracturedynamiccohesiveelastodynamicswave-crackinteractionacousticimpedancecrackbranchingeigenstrain
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper examines what happens when an elastic wave strikes a pre-existing crack in phase-field models of dynamic fracture. It argues that for the standard brittle model with stiffness degradation, the high-frequency oscillations and widening of the damaged band are inherent consequences of the regularization, not numerical artifacts, and that they are controlled by the single ratio ℓ/λ of regularization length to elastic wavelength, acting through the acoustic-impedance profile. It also shows that degrading the mass density along with stiffness does not fix the problem and violates mass conservation. The paper then extends a strength-degrading cohesive phase-field model to elastodynamics and shows it preserves the sharp-crack wave response under a condition on ℓ/λ and the stress amplitude ratio σ̃/σc, under which it further derives an analytical dynamic cohesive opening law governed by c0/ℓch. The 2D benchmarks indicate the cohesive model branches like the brittle stiffness-degraded model, while the density-degraded model fails to branch.

What carries the argument

The key object is the acoustic impedance profile Z(α) = sqrt(g(α)E0ρ0) and the WKB adiabaticity parameter δ(x) = |Z'(x)|/Z(x) · c(x)/ω. For the optimal AT1 phase-field profile, δ tends to λ/(2πℓ) near the crack center, making ℓ/λ the single governing parameter that separates reflecting (sharp-crack-like) from transmitting (homogeneous-like) regimes. For the cohesive model, the key machinery is the eigenstrain field η and the strength-degradation function a(α)=1−α, which leave the bulk wave equation unchanged; the condition (59) ensures the strength criterion is met only at the crack, and the complementary relations yield the linear ODE (60) whose solution is the analytical dynamic cohesive l

What would settle it

Compute the reflection and transmission power coefficients for a fixed phase-field crack as a function of ℓ/λ, e.g. by transfer-matrix or spectral simulations; if the transition does not fall near ℓ/λ ≈ 0.08–0.221, the impedance-based explanation is wrong.

Watch

Extended reading notes

Core claim

On the paper's own terms: for a stiffness-degrading brittle phase-field model, a crack is not a free surface but a smooth impedance well; a harmonic wave is reflected only when the adiabatic parameter δ ≈ λ/(2πℓ) is large, and transmitted when ℓ/λ is large, with transfer-matrix results giving the sharp-crack-like reflection regime for ℓ/λ ⪅ 0.08 and transmission for ℓ/λ ⪆ 0.221. The same mechanism, combined with the local strength profile of the AT1 model, explains the widening of the damaged band. Degrading the density in addition to stiffness removes the wave-speed dip but deepens the impedance well, yielding total reflection for all ℓ/λ in tension and also altering compression. The newly

Load-bearing premise

The cohesive model's dynamic law rests on the assumption that the eigenstrain carries no inertia and its pointwise evolution is identical to the quasi-static stationarity condition at every instant.

Editorial extensions

If this is right

  • The reported 'numerical' artifacts in dynamic brittle phase-field simulations will persist under mesh refinement and even with numerical dissipation, because they stem from the regularization itself as long as ℓ/λ is not small.
  • For brittle stiffness-degrading models, the sharp-crack response is recovered only as ℓ/λ→0, which conflicts with using ℓ as a material parameter to set the nucleation stress.
  • The stiffness+density degradation variant, despite constant wave speed in tension, is not a remedy: it produces full reflection for all ℓ/λ, fails on compressive waves, and breaks mass conservation.
  • For the new cohesive model, the regularization length ℓ is decoupled from the material strength and can be chosen purely from numerical considerations, with the dynamic response governed by c0/ℓch, which is testable.
  • In the 2D benchmark, crack branching is captured by both the stiffness-degraded brittle model and the cohesive model, while the density-degraded model produces straight cracks; conclusions about branching from the latter should be reconsidered.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, the finding that the bulk wave equation is untouched in the cohesive model implies that any spurious wave–crack interactions in existing simulations using brittle models can be quantified a priori by computing ℓ/λ; a practical quality metric for dynamic phase-field simulations would be to report it.
  • Beyond the paper, the condition (59) is derived for the incident-wave stress envelope of a half-sine pulse; for broadband or multiply reflected waves, the boundary in Fig. 13b may shift, so a more general criterion would be needed before using it as a universal safeguard.
  • Beyond the paper, the analytical opening law suggests that the dimensionless number c0/ℓch is the natural time-scale for dynamic cohesive crack opening, which could be used to design experiments that distinguish cohesive from brittle dynamic response in a material.
  • Beyond the paper, the similarity of the branching patterns between the brittle and cohesive models—despite very different mechanisms—suggests that branching itself may be insensitive to the regularization details provided the elastic domain and dissipation are matched; testing this hypothesis would require a systematic parameter sweep.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper analyzes three phase-field formulations for dynamic fracture in 1D and 2D. For brittle stiffness-degradation models, it argues—through WKB/scattering analysis, transfer-matrix computations, and FEM experiments with two time integrators and two mesh resolutions—that high-frequency oscillations and phase-field widening in wave-crack interaction are intrinsic consequences of the regularization, governed by the ratio ℓ/λ of regularization length to wavelength. It shows that adding density degradation restores constant wave speed but violates mass balance and still fails compressive-wave transmission. The paper then extends a recently proposed eigenstrain-based cohesive phase-field model (strength degradation only) to elastodynamics, derives a condition (ℓ/λ, σ̃/σc) for sharp-crack-like response, and derives a closed-form dynamic cohesive opening law governed by c0/ℓch, verified against FEM. A 2D pre-notched plate benchmark compares branching behavior.

Significance. If the claims hold, the paper clarifies a long-debated issue: the oscillations reported in dynamic phase-field fracture are not merely numerical artifacts but an intrinsic feature of stiffness-degrading regularization, and the proposed cohesive formulation offers a principled alternative. The manuscript has notable strengths: the scattering analysis is self-contained, the transfer-matrix predictions are falsifiable, the analytical opening law is checked against direct FEM solutions of the same PDE system (Fig. 27), the numerical studies include systematic mesh and time-integration checks, and the implementations are publicly available. These features make the paper a potentially valuable contribution to computational fracture mechanics. However, the quantitative threshold claims for the brittle model rest on an unsplit surrogate, and the cohesive model's predictions rest on a rate-independent elimination of the eigenstrain that is stated but not discussed in depth; these caveats need attention.

major comments (3)
  1. [§2.1.3 / §2.1.5, Eq. (16), (20), Fig. 2 & 5] The TMM thresholds in Fig. 2 (ℓ/λ≈0.08 and 0.221) are computed for the unsplit heterogeneous wave equation (16) with the fixed optimal AT1 profile, whereas the FEM simulations of §2.1.5 use the tension–compression split (20), as the paper itself states ('only qualitative'). This gap is load-bearing for the quantitative claim that the wave-crack interaction is governed by the impedance profile with these specific thresholds. Since incident and reflected waves superpose during the interaction, both signs of u′ coexist in the phase-field support; in the split model the effective stiffness profile becomes solution-dependent, so the impedance profile from the TMM is not the one actually experienced by the FEM solution. Moreover, the pulse (21) is a truncated sinusoid, so its spectral content is not captured by the monochromatic TMM parameter ℓ/λ̃. I therefore do not consider the quantitative
  2. [§3.1 footnote / Appendix D, Eq. (46)] The dynamic extension of the cohesive model eliminates the eigenstrain η by invoking, in the §3.1 footnote, that 'the eigenstrain carries no inertia' and that pointwise stationarity coincides with quasi-static optimality. This is a strong modeling assumption, and every subsequent cohesive result—the condensed energy (46), the diffuse-jump criterion (59), and the opening law (60)-(62)—inherits it. The manuscript does not discuss the physical scope of this assumption (e.g., what a finite relaxation time for η would change, or why zero inertia is the appropriate limit for a fracture process). Because this is the first dynamic extension of this cohesive model, the authors should state this limitation explicitly in the conclusions and, if possible, support the rate-independent elimination by an asymptotic argument. As written, the 'dynamic cohesive law' is dynamic only in the displacement fie
  3. [§3.4, Eqs. (57)–(59), Fig. 13] The no-diffuse-jump condition (56) is converted into the regime diagram of Fig. 13b using the stress envelope (57) and the strength profile (58). Equation (57) is stated without derivation; it is not the incident-wave maximum, since for a free-end reflection of a half-sine pulse the total (incident+reflected) stress at a fixed point has this form. The derivation should be given, including the assumptions (half-sine pulse, no secondary reflections, α frozen at its initial profile before the criterion is violated). The figure caption calls (b) 'numerically obtained limits', but it is obtained by solving the inequality analytically; this is misleading. Since Fig. 13b defines the regime in which the cohesive model is claimed to recover sharp-crack behavior, this is a central element and needs a rigorous derivation or a direct FEM verification along the boundary.
minor comments (5)
  1. [§2.1.5, Eq. (21), Fig. 5] The relationship between the nominal wavelength λ̃ of the truncated sinusoid and the monochromatic λ of the TMM deserves a word; the pulse's spectral width is probably why FEM at ℓ/λ̃=0.075 still shows visible transmission although the monochromatic TMM would place it in the near-full-reflection regime.
  2. [§3.1 footnote] The footnote could be more precise: the action (6) contains no kinetic term for η by construction; this is a modeling choice, not a consequence of the quasi-static model. Please phrase it that way.
  3. [Fig. 13 caption] Replace 'numerically obtained limits' with 'analytically obtained from (59)' to avoid implying a parameter sweep over FEM solutions.
  4. [§4.2.1, Figs. 20–23] The 2D comparison uses ℓ/∆x≈5 only. Given the 1D sensitivity to discretization, a brief mesh-convergence statement for the branching patterns would increase confidence.
  5. [Eq. (57) / §2.1.3] The notation λ̃ vs λ is used inconsistently between the pulse section and the TMM section; please unify and clarify which quantity is being used in each threshold.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: the TMM analysis and the dynamic cohesive opening law are derived from stated model equations and checked against FEM solutions of the same system; self-citation to the prior cohesive model is legitimate background, with only acknowledged limitations (qualitative FEM/TMM bridge, inertia-free eigenstrain) but no reduction-by-construction.

full rationale

The claimed derivations are self-contained. The ℓ/λ control parameter and the reflection/transmission thresholds come from a WKB criterion and transfer-matrix solution of the 1D heterogeneous wave equation (Eq. 16, Appendix A/B), i.e. from stated material profiles, not from the FEM results. The paper explicitly flags that the subsequent FEM comparison with the strain-split model is only qualitative: 'The comparison between FEM and TMM results is therefore only qualitative' (§2.1.5, repeated in §2.2.4). That is a limitation on quantitative transfer, not a circular fit. The dynamic cohesive opening law (ODE (60), solution (61)-(62), Appendix E) is derived from d'Alembert solutions, stress continuity, and the model's own KKT/cohesive relations (50)-(55); it is not an assumption imported to match data. Its validation in Fig. 27 is against direct FEM solutions of the same PDE system ('Clearly, analytical and FEM results coincide'), which is consistency testing rather than circular prediction. The main self-reference is the adoption of the cohesive phase-field model from [39,40]; that is prior published work and supplies the model ingredients, not the paper's conclusions about dynamic behavior. Two assumptions are flagged as such: the inertia-free elimination of the eigenstrain ('the eigenstrain carries no inertia, so the point-wise stationarity condition...', §3.1 footnote), and the incident-wave-only stress envelope used in criterion (59) ('For an incoming wave (21), the temporal maximum at each spatial point reads...', Eq. (57)). Both are limitations/assumptions, not self-definitional reductions of the claimed results. No fitted parameter is renamed as a prediction, and no uniqueness claim from the authors' prior work is used to force the model choice. Overall score 1 reflects minor self-citation and these acknowledged assumptions, but no circular derivation.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The central claims rest on standard continuum-mechanics axioms plus three model-specific premises: the rate-independent, inertia-free elimination of the eigenstrain; the special-case strength surface; and the envelope criterion for diffuse jumps. Hand-set numerical floors (g0, h0, ε) and strength inputs (σc, pc, τc) are declared. No quantity is fitted to experimental data, and no new physical entity is postulated — the eigenstrain η, the jump set J(z), and the strength-degradation mechanism are inherited from the authors' quasi-static cohesive model [39,40].

free parameters (5)
  • g0 (residual stiffness) = 1e-6
    Floor in degradation function g(α)=(1−α)²+g0 for the brittle models, set for convexity and numerical stability (§2.1.1). It sets the impedance and wave-speed floor at the crack center and therefore participates in the fine structure of the reflection coefficient in Fig. 2a; the leading-order WKB analysis in §2.1.3 ignores its effect for |x| < g0·ℓ.
  • h0 (residual density) = 1e-2
    Residual density in h(α)=(1−α)²+h0 for the stiffness+density model (§2.2.4), deliberately raised from 1e-6 to suppress numerical oscillations and instabilities induced by the degraded mass. FEM results for this model depend on this choice; the paper notes the TMM scattering result does not.
  • epsilon (residual energy density) = 1e-7
    Added to all branches of the condensed cohesive potential (46) to avoid rank-deficiency from the linear branches; by the authors' statement it does not affect the stationarity conditions for η (Appendix D).
  • sigma_c (tensile strength, 1D cohesive simulations) = 5 MPa
    Chosen to match one of the brittle model's calibrated strengths σ̂c(0) (8.7 MPa and 5 MPa, §2.1.5) for a fair comparison. It is an input material parameter, not fitted to experimental data.
  • p_c, tau_c (2D cohesive strength surface) = 8.94 MPa, 10.95 MPa
    Set equal to the brittle model's initial critical pressure and shear (§4.2) and consistent with the compact-reformulation special case p²c/τ²c = κ0/(2μ0) (≈0.6665 vs 0.6667); this makes the initial elastic domains identical for all three models at α=0.
assumptions (7)
  • standard math Linear isotropic elasticity, infinitesimal strains, volumetric–deviatoric split (3)
    Constitutive framework of all three models, introduced in §2.1.1.
  • domain assumption Rate-independent damage evolution with irreversibility α̇≥0 and KKT conditions (12)
    Standard phase-field/damage modeling assumption adopted throughout; it drives the damage-widening argument in §2.1.5.
  • domain assumption Sharp-crack benchmark: a fully open crack reflects a tensile wave completely and transmits a compressive wave undistorted
    The ground truth against which regularized models are judged (Fig. 4, §2.1.5, §3.4). If this target response were not physically correct, the entire 'anomaly' framing would change.
  • standard math WKB/scattering criterion (18) for smoothly graded media
    Classical criterion (Bremmer/Brekhovskikh) applied to the phase-field profile in §2.1.3; quantitative thresholds are then obtained from exact TMM (Appendix B), so the criterion only identifies regimes.
  • ad hoc to paper Eigenstrain carries no inertia; pointwise stationarity with respect to η in the action functional coincides with quasi-static optimality
    Footnote in §3.1 — the key premise of the dynamic extension. All cohesive results (reflection regimes, opening law (60)-(62), phase-IV shock waves) inherit this static, rate-independent elimination of η.
  • ad hoc to paper Special case p²c/τ²c = κ0/(2μ0) enabling the condensed formulation (46)
    §3.2; restricts the admissible strength surfaces so that the compact reformulation is exact. The 1D model is unaffected; the 2D inputs are chosen to satisfy it approximately.
  • ad hoc to paper Stress envelope (57) for the diffuse-jump criterion (59): temporal max of stress equals the incident wave's amplitude profile
    §3.4; ignores superposition of reflected waves near the crack and the strain-split physics. The paper checks it only against the two FEM cases it classifies (Fig. 13b).

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Cite this review

Pith. "Pith review of On phase-field regularization in dynamic fracture with brittle and cohesive formulations." pith.science (2026). https://pith.science/paper/IODFGUB7

@misc{pith2026260713599,
  author       = {Pith},
  title        = {Pith review of: On phase-field regularization in dynamic fracture with brittle and cohesive formulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IODFGUB7}},
  note         = {Machine review of arXiv:2607.13599}
}
read the original abstract

Phase-field models of fracture are widely used for simulating crack nucleation and propagation, yet the role of the phase-field regularization in the dynamic regime is not fully understood and depends critically on how the damage variable is coupled to the displacement field. In this paper, we analyze three alternative formulations: the brittle model with stiffness degradation, its variant with stiffness+density degradation, and our recently proposed phase-field regularization of cohesive fracture, which we extend to elastodynamics. By studying the interaction of a tensile and a compressive elastic wave with a phase-field crack in a one-dimensional bar, we determine for which models and under which conditions the phase-field regularization preserves the features of the wave-crack interaction expected for a sharp crack, and we theoretically explain which variables control the behavior. For the new cohesive model extended to dynamics, we further derive an analytical dynamic cohesive opening law. Finally, we study the dynamic behavior including branching of a two-dimensional notched plate at two loading intensities.

Figures

Figures reproduced from arXiv: 2607.13599 by the authors.

Figure 1
Figure 1. Optimal phase-field profile (a), resulting local wave speed (b) and local acoustic impedance (c) for the [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Reflected (r) and transmitted (t) fractions of the incident power depending on the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. 1D setup for the study of an elastic wave interacting with a pre-existing crack. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (32 more)
Figure 3
Figure 3. Figure 3: 1D setup for the study of an elastic wave interacting with a pre-existing crack. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png]
Figure 4
Figure 4. Figure 4: Interaction of a sinusoidal stress wave with a sharp crack in terms of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Interaction of a sinusoidal stress wave with a [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Interaction of a sinusoidal, compressive stress wave with a [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Interaction of a sinusoidal stress wave with an [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Reflected (r) and transmitted (t) fractions of the incident power depending on the [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: Interaction of a sinusoidal stress wave with a [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: Interaction of a sinusoidal stress wave with an [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: Visualization of the model behavior in the 1D setting, in terms of reduced elastic energy density (a), cohesive law [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 11
Figure 11. Figure 11: Visualization of the model behavior in the 1D setting, in terms of reduced elastic energy density (a), cohesive law [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: Interaction of a sinusoidal stress wave with an [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: Comparison between maximum stress of an elastic wave and locally degraded strength (a), and numerically obtained [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: Interaction of a sinusoidal stress wave with an [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]
Figure 14
Figure 14. Figure 14: Interaction of a sinusoidal stress wave with an [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]
Figure 15
Figure 15. Figure 15: Time evolution of the jump (a), the phase field (b) and the stress (c) at the crack for the test in Fig. 14 for various [PITH_FULL_IMAGE:figures/full_fig_p022_15.png]
Figure 16
Figure 16. Figure 16: Visualization of the dynamic cohesive law for different [PITH_FULL_IMAGE:figures/full_fig_p023_16.png]
Figure 17
Figure 17. Figure 17: Phase-field profile and its effect on local wave speed, acoustic impedance, and material strength in the brittle [PITH_FULL_IMAGE:figures/full_fig_p023_17.png]
Figure 17
Figure 17. Figure 17: Phase-field profile and its effect on local wave speed, acoustic impedance, and material strength in the brittle [PITH_FULL_IMAGE:figures/full_fig_p024_17.png]
Figure 18
Figure 18. Figure 18: Strength surfaces S(α) in the volumetric-deviatoric stress space for the cohesive (a) and the brittle model with stiffness degradation (b). The ’closeness’ of a stress state to the damaged strength surface s(x) is depicted with green arrows for an example with positiv…
Figure 19
Figure 19. Figure 19: Setup for the pre-notched plate (a), as well as the loading ramp (b) for the traction on the upper and lower edges. [PITH_FULL_IMAGE:figures/full_fig_p026_19.png]
Figure 19
Figure 19. Figure 19: Setup for the pre-notched plate (a), as well as the loading ramp (b) for the traction on the upper and lower edges. [PITH_FULL_IMAGE:figures/full_fig_p027_19.png]
Figure 20
Figure 20. Figure 20: Phase field and s(x) at various time instants for the pre-notched plate with fˆy = 1 N/mm. While the brittle model with stiffness degradation and the cohesive model both branch, the brittle model with stiffness and density degradation produces straight crack propagati…
Figure 20
Figure 20. Figure 20: Phase field and s(x) at various time instants for the pre-notched plate with fˆy = 1 N/mm. small values of g(α) meet very large strains. As the crack faces continue to separate under the sustained loading, the strains in the band — and with them this stored energy — k…
Figure 21
Figure 21. Figure 21: Slices of s(y) (a-c) and α(y) (d-f) at various x-coordinates for the pre-notched plate with fˆy = 1 N/mm at the final time step (t = 80 µs). the cohesive model the crack tip emits ripples into the domain alongside the elastic energy release. For the brittle model with…
Figure 21
Figure 21. Figure 21: Slices of s(y) (a-c) and α(y) (d-f) at various x-coordinates for the pre-notched plate with fˆy = 1 N/mm at the final time step (t = 80 µs). 0.0 0.2 0.4 P e in N mm mm (a) 0.0 0.2 0.4 0.6 P f in N mm mm (b) 0 20 40 60 80 t in µs 0.98 0.99 1.00 m/m0 (c) 0 20 40 60 80 t…
Figure 22
Figure 22. Figure 22: Energy contributions (a,b), mass (c) and crack tip speed (d) monitoring for the pre-notched plate with [PITH_FULL_IMAGE:figures/full_fig_p029_22.png]
Figure 23
Figure 23. Figure 23: The region around the branching point x ∈ [60, 70] mm × [17.5, 22.5] mm at the final time step t = 80 µs for the brittle model with stiffness degradation (a), and the cohesive model (b). For reference, the outline of a single element is drawn in green, while the green…
Figure 24
Figure 24. Figure 24: Phase field and s(x) at various time instants for the pre-notched plate with fˆy = 2 N/mm. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_24.png]
Figure 25
Figure 25. Figure 25: Phase field, s(x), and eigenstrain obtained with the cohesive model at various time instants for the pre-notched plate with fˆy = 2 N/mm, zoomed in the region x ∈ [50, 85] mm × [12.5, 27.5] mm around the branching points. equation unchanged with respect to the one of …
Figure 26
Figure 26. Figure 26: Setup for the study of the dynamic cohesive response (a), phase field and stress field during the four phases of the [PITH_FULL_IMAGE:figures/full_fig_p041_26.png]
Figure 26
Figure 26. Figure 26: Setup for the study of the dynamic cohesive response (a), phase field and stress field during the four phases of the [PITH_FULL_IMAGE:figures/full_fig_p042_26.png]
Figure 27
Figure 27. Figure 27: Comparison of numerical and theoretical results on the cohesive crack evolution, in terms of the jump (a), the [PITH_FULL_IMAGE:figures/full_fig_p044_27.png]

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