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Phase-field modelling of cohesive fracture. Part III: From mathematical results to engineering application

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Pith's one-line read Phase-field models built from a prescribed cohesive traction-separation law can look different internally yet deliver the same macroscopic fracture response, independent of the regularization length.

desk verdict Useful closed-form phase-field cohesive models, but the core construction lives in unpublished companions and the exponential example carries an unquantified simplification. read the letter →

arxiv 2507.22072 v1 pith:KLNUW2FO submitted 2025-07-17 math.AP cond-mat.mtrl-sci

Roberto Alessi , Francesco Colasanto , Matteo Focardi This is my paper · ORCID
classification math.APcond-mat.mtrl-sci MSC 74R1074G6549J45
keywords phase-fieldfracturecohesivezonemodeltraction-separationlawΓ-convergencegradientdamageinternallengthsofteninglawsone-dimensionalbar
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

This paper is the third part of a programme that turns prescribed cohesive traction-separation laws into phase-field fracture models with proven variational underpinnings. It works out, in a one-dimensional tensile bar, the mechanical response of models built for five standard softening laws: linear, bilinear, exponential, hyperbolic, and Dugdale. The main claim is that the global stress–displacement response reproduces the target cohesive law and does not depend on the internal length, even though the phase-field and displacement profiles differ between models and vary with that length. For the linear law, three different phase-field models are shown to give one and the same cohesive response. The paper's purpose is to show that the abstract construction of Part II yields directly usable engineering models.

What carries the argument

The central objects are the pair of material functions $(l(\alpha), w(\alpha))$ — the local phase-field dissipation potential and the auxiliary function defining the degradation through $g_\ell(\alpha) = 1/(1+(2G_c E/(\ell \bar{\sigma}^2)) w(\alpha)/l(\alpha))$ — together with the closed-form parameterization of the response by the maximum phase-field value $\bar{\alpha}$. The limit stress $\sigma(\bar{\alpha})=\sqrt{K w(\bar{\alpha})/h_\ell(\bar{\alpha})}$ and the crack opening $\delta(\bar{\alpha})$ obtained by integrating the complementary degradation over the localization profile are independent of $\ell$ whenever the degradation has the specific smooth form used throughout the paper. This is the mechanism that makes models with very different phase-field evolutions deliver the same traction-separation law: the construction theorems of Part II choose $(l,w)$ so that the parameterized curve $(\delta(\bar{\alpha}), \sigma(\bar{\alpha}))$ traces the target law, and the length cancels out.

What would settle it

Take the model (E) for the exponential law, compute the traction-separation curve up to very large crack openings in the one-dimensional bar, and compare the area under $\sigma(\delta)$ with $G_c$ and the tail with the target exponential law: if the deviation grows beyond the small-parameter regime, the claim that the modification is negligible fails. Equivalently, run the same test for (L1) with the internal length set to a sizable fraction of the bar length; the claimed independence from $\ell$ predicts no change in the global response, so a visible shift would falsify it.

Watch

Extended reading notes

Core claim

For each of the five considered cohesive laws, the reconstruction procedure of Part II produces phase-field energy functionals whose one-dimensional uniaxial response coincides with the prescribed traction-separation law. The paper exhibits explicit material functions for each law: for the linear law three distinct triples (L1), (L2), and (L12), and for Dugdale two models (D1) and (D2). Through the parameterization by the maximum phase-field value, it shows that the limit stress and the crack opening are independent of the internal length $\ell$; only the width of the phase-field localization is controlled by $\ell$. The surface fracture energy recovered from the model has area $G_c$ matching the fracture toughness, with the expected exception that Dugdale's law produces a snap-back response in which the dissipated energy exceeds $G_c$. The paper also notes that a linear term in the local dissipation potential is sufficient but not necessary for an explicit elastic stage.

Load-bearing premise

The load-bearing premise is that the construction theorems and hypotheses of the two companion papers are correct and apply to these five laws, and that for the exponential law the small modification made to satisfy the hypotheses can be neglected without affecting the response.

Editorial extensions

If this is right

  • The global response of each constructed model reproduces the target cohesive response, so the internal length can be chosen for mesh resolution without altering the macroscopic softening curve.
  • Because many phase-field models realize the same cohesive law, a model can be selected for auxiliary properties: for example, (L12) has a constant finite localization support and thus automatically satisfies the irreversibility condition, while (L2) has unbounded support.
  • Closed-form material functions are available for the exponential law, avoiding the fitted polynomials used previously; cohesive forces never vanish because $w(\alpha)\to\infty$ as $\alpha\to1$.
  • A phase-field model for a bilinear softening law is constructed, and its response is again independent of the internal length.
  • For Dugdale's law the construction yields a snap-back response: at a critical opening the phase-field jumps to its fully damaged profile, and the dissipated energy exceeds $G_c$ although the area under the traction-separation curve is still $G_c$.

Reading between the lines

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

  • The multiplicity result suggests a design principle: within the family of models realizing a given cohesive law, one can search for a member that satisfies side constraints such as monotone growth of the localization support, which is exactly the route the paper proposes for enforcing irreversibility.
  • The length independence shown in one dimension plausibly explains why the smooth degradation form avoids the numerical regularization and $\ell$-dependent critical stress reported for truncated-degradation models; a direct finite-element comparison would test this transfer.
  • For the exponential law, dropping the small-parameter modification is stated without a quantitative error bound; a targeted numerical check of the tail of $\sigma(\delta)$ would reveal how large an opening is needed before the deviation matters.
  • The Dugdale construction shows the method can handle traction-separation laws whose crack opening is non-monotone in the phase-field variable, which may extend to other plateau-type or snap-back laws.
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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 presents one-dimensional mechanical responses of phase-field cohesive models constructed in Part II [33] for linear, bilinear, exponential, hyperbolic, and Dugdale traction-separation laws. It derives closed-form expressions for the material functions l and w, computes phase-field profiles, crack openings, and global responses through analytical formulas, and claims that the global response is independent of the internal length and that several distinct models produce identical cohesive responses. The central mathematical construction is delegated to the companion papers [32,33]; for the exponential law, an admissible modified version from [33] is replaced by the unmodified law (3.8) without a quantitative error estimate.

Significance. If the companion results and the exponential simplification are valid, the paper provides a practically useful catalog: exact (or near-exact) phase-field realizations of common cohesive laws with length-insensitive macroscopic response, and explicit evidence that phase-field profile evolution is not unique. The analytical formulas (2.16), (2.23), and (2.25) are internally consistent, and the claimed independence from the internal length follows from the scaling of the material functions. The paper also correctly identifies the non-monotone displacement jump in the Dugdale models and the associated snap-back response, which is a useful observation for engineering applications. However, the main claims are conditional on theorems stated only in the submitted companion paper [33], and the exponential example is not yet certified.

major comments (3)
  1. [Sec. 3.3 and Eq. (E)] The exponential model (E) is the one example for which the construction in [33] is not used directly: the text states that [33, Sec. 3.3.6] treats a 'slightly modified version' of (3.9) needed to satisfy the hypotheses of [32, Theorem 1.1], and that neglecting this modification the response 'is well captured'. No error bound, no small parameter, and no explicit modified law are given, and hypotheses (Hp1)-(Hp4) and (Hp6)-(Hp8) are not checked for (E). Because the paper's headline claim is exact reproduction of a prescribed cohesive law, this uncontrolled approximation must be either removed (by using the modified law from [33]) or quantified (by an explicit estimate on the difference between the responses of (E) and law (3.8)).
  2. [Sec. 2.3 and Sec. 3] The reconstruction of the functions {l,w} for every example is delegated to Theorems 3.1, 3.2, and 2.18 of the unpublished companion paper [33]. These theorems are not stated, and their hypotheses are only partially quoted: (Hp6)-(Hp8) are listed in Sec. 2.3, while (Hp1)-(Hp4) are cited by reference. As a result, the reader cannot verify that the displayed functions (L1), (L2), (L12), (B), (H1), (H2), (D1), and (D2) indeed correspond to the prescribed laws. The authors should state the relevant theorems and either verify the hypotheses for each example or include the verification in an appendix.
  3. [Secs. 3.1 and 3.5] For models (L1), (L2), (D1), and (D2), the function l(alpha) is obtained by numerical interpolation of an inverse that has no explicit analytic form, but the interpolation method and its error are not reported. Since the plotted traction-separation and global responses are computed through (2.16) and (2.23) using these interpolated functions, the claimed identity of the responses is only as accurate as the interpolation. The authors should specify the interpolation scheme and provide an error estimate or a convergence check with respect to the interpolation parameter.
minor comments (5)
  1. [Sec. 3.1, first paragraph] The sentence 'the tree models (L1)-(L12) are describing the same cohesive fracture response' contains a typo: 'tree' should be 'three'.
  2. [Sec. 3.3, text before Eq. (3.9)] The phrase 'the linear traction-separation law within the mathematical framework (3.8)' refers to an exponential law; 'linear' appears to be a typo and should be corrected.
  3. [Sec. 3.4, Eq. (H1)] The coefficient 0.170 in the expression for w(alpha) is introduced without derivation; it should be traced to condition (2.12) and the stated value k0 approx 0.386/k, and the computation should be shown explicitly.
  4. [Fig. 9 caption] The caption states 'for (L12) and ell = 10 mm' but also highlights displacement profiles for smaller internal lengths; please clarify which curves correspond to which value of ell.
  5. [Sec. 2.2, Eqs. (2.16) and (2.17)] The notation sigma-bar is used for both the critical stress in (2.17) and the argument of the limit stress function sigma(alpha-bar) in (2.16); the distinction should be made explicit to avoid confusion.

Circularity Check

3 steps flagged · score 4.0 of 10

Sec. 3.3 drops the admissible modification from [33] and asserts, with no error control, that the simplified exponential model (E) 'well captures' law (3.8); combined with unquantified l(alpha)-interpolations for (L1),(L2),(D1),(D2) and hypotheses (Hp1-8) cited only from the submitted companion papers, the exact-reproduction headline is only partially certified.

  1. other [Sec. 3.3, exponential softening law, construction of (E)]
    "In fact, a slightly modified version of (3.9) was considered in [33, Sec. 3.3.6] to ensure that all the assumptions of [32, Theorem 1.1] were satisfied, specifically the continuity of the pair {l, omega}. ... Neglecting this modification, which affects only the asymptotic behavior, the corresponding engineering material functions l and w for the exponential model (3.8) simplify to: [E]."

    The only proven construction (Theorem 3.1 in [33], subject to the hypotheses of [32, Theorem 1.1]) is stated to apply to a 'slightly modified version' of (3.9), introduced precisely to satisfy those hypotheses. The paper then drops the modification, never exhibits it or its small parameter, and asserts without any quantitative error bound that (E) 'well captures' the target response. The confirming sigma-delta curves are computed from the same parametrization (2.25) that defines the model's own response, so the claim that exponential model (E) reproduces law (3.8) does not follow from the cited theorem chain; for one of the five headline laws the exact reproduction statement is therefore unverified, an omitted proof rather than a derived result.

  2. other [Sec. 3.1 (linear models L1, L2) and Sec. 3.5 (Dugdale models D1, D2)]
    "The functions l(alpha) for models (L1) and (L2) do not admit an explicit analytical inverse. As a consequence, interpolation functions to reconstruct l(alpha) are hereafter adopted. ... The function l(alpha) for models (D1) and (D2) does not admit an explicit analytical inverse. As a consequence, an interpolation function to reconstruct l(alpha) is hereafter adopted, similar to what have been done for (L1) and (L2)."

    For (L1), (L2), (D1) and (D2), the construction theorems of [33] determine an exact function l(alpha) (or its inverse) that is supposed to deliver the target traction-separation law. The paper replaces this exact function by an unspecified interpolation whose error is never estimated. All displayed response curves and the claimed identity of the cohesive responses for these models are consequently certified only up to this unquantified numerical step; the paper states that the exact agreement with the target law follows from the cited construction, while the quantities actually evaluated use a surrogate l(alpha) rather than the constructed one.

1 more flagged steps
  1. self citation load bearing [Sec. 2.3, identification of the phase-field model for a prescribed cohesive law]
    "In Part II of this work [33], we have shown, within a mathematical framework, how to assign a given cohesive law, either linear or superlinear for small amplitude of the displacement jump, by appropriately selecting the parameters of the phase-field model in (2.1). ... we define the corresponding functionals F_epsilon and demonstrate that g0 equals the surface energy density of their Gamma-limit g, as established in [32, Theorem 1.1]. The hypotheses (Hp 1-4) are reported in [32, Sec. 1.2]."

    The paper's central premise - that each displayed phase-field model reproduces the prescribed cohesive response exactly - is imported from the authors' own submitted papers [32] (Theorem 1.1) and [33] (Theorems 3.1, 3.2, 2.18). The examples in Sec. 3 evaluate the forward response of the constructed models, but they do not prove the construction theorems, and the hypotheses (Hp1-4, Hp6-8) are only cited, not verified for the five chosen laws and their displayed material functions. If the companion theorems fail or the examples violate the hypotheses, the claimed equivalence collapses within this paper; this is load-bearing self-citation, partially mitigated by the closed-form response evaluation and by the external match of (L12) with the independent model of [29].

full rationale

The core one-dimensional derivation in Sec. 2.2 is self-contained: the limit stress (2.16), the phase-field profile relation (2.19), the crack-opening formula (2.23) and the traction-separation parametrization (2.25) all follow from the equilibrium and Kuhn-Tucker conditions (2.13)-(2.15) of the energy functional (2.4), independently of the companion papers. The 'identical cohesive fracture response' claim, however, is the content of the inverse construction developed in the authors' submitted papers [32, 33]: the material functions {l, omega} are designed there so that the model's Gamma-limit surface energy equals a prescribed g0, so the response curves of Sec. 3 mostly re-evaluate that construction rather than independently predicting it. That is legitimate verification rather than fitting-renamed-as-prediction: no parameter is tuned to data, the target laws are not defined in terms of the model outputs, and model (L12) matches the independent published model of [29]. Nevertheless, three load-bearing steps are only informally certified: (i) in Sec. 3.3 the admissible 'slightly modified' exponential law of [33] is dropped and (E) is asserted to 'well capture' (3.8) with no error estimate and no check of hypotheses (Hp1-8) for (E), an omitted proof affecting one of the five laws; (ii) in Secs. 3.1 and 3.5 the inverse l(alpha) for (L1), (L2), (D1), (D2) is replaced by an unquantified interpolation; (iii) the construction theorems themselves are load-bearing self-citations, cited as 'submitted' with hypotheses not verified for the displayed examples. Since the central derivation does not reduce to its inputs by definition and the main examples are internally and partly externally grounded, this is not a score-6-or-higher case; the uncertified exponential simplification and the unquantified interpolations make the strongest claim only partially supported, so the score is 4.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

No new physical entities (particles, forces, dimensions) are introduced. The novel objects are new material functions (l, w, g_l) for phase-field functionals, which fall under model construction rather than invented entities.

free parameters (6)
  • k0 scaling factors for linear models = ks = 1/(2k)
    Determined by normalization condition (2.12); connects physical cohesive law to non-dimensional mathematical law. Not fitted to experimental data.
  • k0 scaling factor for bilinear model (B) = 0.91/k
    From condition (2.12); quoted to two decimals, indicating numerical solution of the normalization integral.
  • k0 scaling factor for exponential model (E) = 1/k
    From condition (2.12).
  • k0 scaling factor for hyperbolic model (H1) = 0.386/k
    From condition (2.12); approximate value.
  • k0 scaling factor for hyperbolic model (H2) = 1/k
    From condition (2.12).
  • coefficient 0.170 in w(α) for (H1) = 0.170
    Appears without derivation; likely a normalization constant inherited from the construction in [33].
assumptions (7)
  • domain assumption Rate-independent energetic formulation (stability condition and energy balance) governs the evolution.
    Adopted from Mielke's framework [42,43] as the basis of the 1D response formulas in Sec. 2.2.
  • domain assumption Single symmetric localization at x = L/2, no boundary interaction, and phase-field vanishing at boundaries (or explicitly allowed to be nonzero).
    Stated in Sec. 2.2 and used for all profile and response computations; several models (L2, E, H2, D2) allow boundary values different from zero.
  • ad hoc to paper Assumptions (Hp1-4) and (Hp2) on the energy functional, including existence of the critical stress limit σ̄.
    Taken from Part I [32, Sec. 1.2]; not restated in full, yet needed for the Γ-convergence and reconstruction results.
  • ad hoc to paper Reconstruction Theorems 3.1 and 3.2 (and Theorem 2.18 for Dugdale) in [33] produce the material functions {l,ω} for a prescribed cohesive law.
    The core of the paper: all examples in Sec. 3 inherit their material functions from these theorems, which are not proved here.
  • standard math Normalization condition (2.12) fixes the scaling factor k0 for each example.
    An integral identity derived from the sharp-interface fracture energy; used to set the k0 values.
  • ad hoc to paper Assumption Q = ω/ς without loss of generality (from [33]).
    Used to obtain (2.26) and the engineering degradation functions.
  • ad hoc to paper For the exponential law, the modification of the cohesive law required in [33] can be neglected because it only affects asymptotically large crack openings.
    Explicitly stated in Sec. 3.3; no error bound is provided.

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Pith. "Pith review of Phase-field modelling of cohesive fracture. Part III: From mathematical results to engineering application." pith.science (2026). https://pith.science/paper/KLNUW2FO

@misc{pith2026250722072,
  author       = {Pith},
  title        = {Pith review of: Phase-field modelling of cohesive fracture. Part III: From mathematical results to engineering application},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KLNUW2FO}},
  note         = {Machine review of arXiv:2507.22072}
}
abstract

This paper concludes a three-part effort aimed at developing a consistent and unified framework for the phase-field modeling of cohesive fracture. Building on the theoretical foundations established in the first two parts, which included a $\Gamma$-convergence result for a broad class of phase-field energy functionals and the presentation of a rigorous analytical methodology for constructing models tailored to specific cohesive laws, this third paper explores the mechanical response of phase-field models, most of which are novel, associated with different cohesive fracture behaviors within a one-dimensional framework. Particular emphasis is placed on the possibility of formulating distinct phase-field models that, despite exhibiting different evolutions of their phase-field and displacement profiles, yield identical cohesive fracture responses. Thus, this work aims at providing a practical interpretation of the mathematical framework connecting the theoretical insights established in the previous parts for physical relevant applications.

Figures

Figures reproduced from arXiv: 2507.22072 by the authors.

Figure 2
Figure 2. The one-dimensional tensile problem (a): An example of evolution with a sharp-crack (b) and the corresponding phase-field regularization (c). 2.1. The mathematical and engineering frameworks In this section we would like to establish a link between the mathematical setting of the first two papers of this work, [32, 33] and the present more engineering-oriented contribution. Mathematical framework. From a mathematica… view at source ↗
Figure 3
Figure 3. Qualitative trends of the considered softening cohesive laws: ( [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. (a) Degradation and (b) local dissipation functions for the linear cohesive fracture models (L1), (L2) and (L12). To highlight the role of ℓ, three different values for the internal length have been considered, namely: ℓ = {10, 5, 1} (mm). (3.3) It turns out, as predicted, that the global response is independent of the value of the internal-length, provided it is sufficiently small compared to the domain size, [PIT… view at source ↗
Figures from the paper (30 more)
Figure 5
Figure 5. Figure 5: (a) Surface fracture energy and (b) traction-separation law as a function of the displacement jump, obtained using the parametrisation (2.25) of the displacement jump (c) and the critical stress (d) with respect to the maximum phase-field value for all considered linea…
Figure 6
Figure 6. Figure 6: Global response of the bar, exhibiting an independent response both with respect to the linear models (( [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Phase-field profiles at given maximum phase-field states for the linear models: ( [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Half-width of the support of the phase-field profiles for the three linear models ( [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Displacement profiles at given maximum phase-field states for ( [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: (a) Degradation and (b) local dissipation functions for the bilinear cohesive fracture model (B). sufficiently small compared to the bar length, only affects the phase-field localization size, as evident from the phase-field and displacement field profiles evolutions …
Figure 11
Figure 11. Figure 11: (a) Surface fracture energy and (b) traction-separation law of the bilinear model (B) as a function of the displacement jump, obtained using the parametrization (2.25) of the displacement jump (c) and the critical stress (d) with respect to the maximum phase-field val…
Figure 12
Figure 12. Figure 12: Global response for the bilinear model ( [PITH_FULL_IMAGE:figures/full_fig_p021_12.png]
Figure 13
Figure 13. Figure 13: Phase-field profiles at given maximum phase-field states for the bilinear model ( [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]
Figure 14
Figure 14. Figure 14: Half-width of the support of the phase-field profile for the bilinear model ( [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]
Figure 15
Figure 15. Figure 15: Displacement profiles at given maximum phase-field states for the bilinear model ( [PITH_FULL_IMAGE:figures/full_fig_p022_15.png]
Figure 16
Figure 16. Figure 16: (a) Degradation and (b) local dissipation functions for the exponential cohesive fracture model (E). 23 ACF-part-III.tex [September 5, 2025] [PITH_FULL_IMAGE:figures/full_fig_p023_16.png]
Figure 17
Figure 17. Figure 17: (a) Surface fracture energy and (b) traction-separation law of the exponential model (E) as a function of the displacement jump, obtained using the parametrization (2.25) of the displacement jump (c) and the critical stress (d) with respect to the maximum phase-field …
Figure 18
Figure 18. Figure 18: Global response for the exponential model ( [PITH_FULL_IMAGE:figures/full_fig_p025_18.png]
Figure 19
Figure 19. Figure 19: Phase-field profiles at given maximum phase-field states for the exponential model ( [PITH_FULL_IMAGE:figures/full_fig_p025_19.png]
Figure 20
Figure 20. Figure 20: Half-width of the support of the phase-field profile for the exponential model ( [PITH_FULL_IMAGE:figures/full_fig_p025_20.png]
Figure 21
Figure 21. Figure 21: Displacement profiles at given maximum phase-field states for the exponential model ( [PITH_FULL_IMAGE:figures/full_fig_p026_21.png]
Figure 22
Figure 22. Figure 22: (a) Degradation and (b) local dissipation functions for the hyperbolic cohesive fracture models. and the traction-separation law as a function of the displacement jump for all the considered hyperbolic models. Their global responses, which again are independent from t…
Figure 23
Figure 23. Figure 23: (a) Surface fracture energy and (b) traction-separation law as a function of the displacement jump, obtained using the parametrisation (2.25) of the displacement jump (c) and the critical stress (d) with respect to the maximum phase-field value for the considered hype…
Figure 24
Figure 24. Figure 24: Global response of the bar for the hyperbolic models (( [PITH_FULL_IMAGE:figures/full_fig_p029_24.png]
Figure 25
Figure 25. Figure 25: Half-width of the support of the phase-field profiles for the hyperbolic models with respect to the fracture [PITH_FULL_IMAGE:figures/full_fig_p029_25.png]
Figure 26
Figure 26. Figure 26: Phase-field profiles at given maximum phase-field states for the hyperbolic models: ( [PITH_FULL_IMAGE:figures/full_fig_p030_26.png]
Figure 27
Figure 27. Figure 27: Displacement profiles at given maximum phase-field states for the hyperbolic models: ( [PITH_FULL_IMAGE:figures/full_fig_p031_27.png]
Figure 28
Figure 28. Figure 28: (a) Degradation and (b) local dissipation functions for the Dugdale cohesive fracture models (D1) and (D2), for different internal lengths. associated critical phase-field value as δ ∗ := max α δ(α) and α ∗ := arg max α δ(α). (3.17) As a consequence, the energy dissip…
Figure 29
Figure 29. Figure 29: Traction-separation laws (a) and (b) for, respectively (D1) and (D2) as a function of the displacement jump, obtained using the parametrisation (2.25) of the displacement jump (c) and the critical stress (d) with respect to the maximum phase-field value. In (a) and (b…
Figure 30
Figure 30. Figure 30: Global responses (a) and (b) for respectively (D1) and (D2), independent of the internal length. The initial elastic response is shown in black. The dependence on the maximum phase-field state is also highlighted. At the critical displacement U∗, a snap back response …
Figure 31
Figure 31. Figure 31: (a) crack density function decomposition between the local and gradient contributions for (D1) and ℓ = 1 mm. (b) zoom on the gradient term. The arrows denote the evolution direction. 0 50 100 150 200 0.0 0.2 0.4 0.6 0.8 1.0 x HmmL Α α¯ 0 1 ( ¯α = 1) ℓ = 1 mm ℓ = 0.5 m…
Figure 32
Figure 32. Figure 32: Phase-field profiles at given maximum phase-field states for the Dugdale models: ( [PITH_FULL_IMAGE:figures/full_fig_p036_32.png]
Figure 33
Figure 33. Figure 33: Half-width of the support of the phase-field profiles for the two Dugdale models with respect to the fracture [PITH_FULL_IMAGE:figures/full_fig_p037_33.png]
Figure 34
Figure 34. Figure 34: Displacement profiles (a) and (b) at given maximum phase-field states for, respectively, (D1) and (D2) with ℓ = 1 mm. The critical displacement profiles at α¯ = α ∗ = 0.186 for (D1) and α¯ = α ∗ = 0.120 for (D2) for smaller internal lengths are also highlighted corres…

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Strength-degradation phase-field regularization of cohesive fracture: the antiplane case

    cond-mat.mtrl-sci 2026-07 conditional novelty 6.0 of 10

    In antiplane shear, a strength-degrading phase-field model yields an ℓ-independent equivalent cohesive law with independent strength and toughness, verified by closed-form solutions and simulations.

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