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Unified analysis of phase-field models for cohesive fracture

T0 review · 3 major / 9 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Phase-field models of cohesive fracture deliver their intended traction-separation law only when the regularization length enters the dissipation and degradation and the crack band never shrinks during failure.

desk verdict A competent unified review of phase-field cohesive fracture that makes a real conceptual point, but whose optimality claims are scoped more narrowly than the abstract suggests. read the letter →

arxiv 2412.03836 v1 pith:7Z5Y4KL7 submitted 2024-12-05 cond-mat.mtrl-sci

Jian-Ying Wu This is my paper · ORCID
classification cond-mat.mtrl-sci MSC 74R1074A4574G6549J45
keywords cohesivefracturephase-fieldmodeltraction-separationlawcrackirreversibilitygeometricfunctiondegradationdissipationGamma-convergence
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 proposes a unified way to read every phase-field model of cohesive fracture, treating them all as regularizations of the Barenblatt cohesive zone model. Three characteristic functions tell the models apart: the geometric function sets the crack profile, the degradation function sets the stress-strain relation, and the dissipation function sets the crack driving force; in the associated formulation the last two coincide, and in the non-associated formulation they are chosen independently. The central claim is that such a model reproduces a cohesive traction-separation law only under two conditions: the regularization length must enter the dissipation and degradation functions so the failure strength and softening curve are well defined, and the crack bandwidth must never shrink during failure so that crack irreversibility does not distort the law. Under those conditions the polynomial geometric function $\alpha(d)=2d-d^2$ is shown to be optimal, and the non-associated $\mu$PF-CZM extends validity to almost arbitrary softening laws, including concave ones. If correct, this gives practicing engineers a direct recipe for choosing phase-field parameters from the material's strength, fracture energy, and softening slope, rather than tuning the length scale to fit experiments.

What carries the argument

The engine of the analysis is the triplet of characteristic functions $(\alpha(d),\omega(d),\varpi(d))$, together with the derived combination $\eta(d)=\alpha(d)/(b\mu(d))$ in which $\mu$ is the dissipation kernel. The geometric function $\alpha(d)$ fixes the crack profile and the ultimate bandwidth $D_u = b\int_0^1 \alpha^{-1/2}\,d\vartheta$; the degradation/cracking function $\omega(d)=1/(1+\varphi(d))$ controls the constitutive relation; the dissipation function $\varpi(d)$ (or $\mu(d)$) controls the crack driving force; and $\eta(d)$ converts these choices into the traction-separation law $\sigma(d_*)$ and $w(d_*)$. The two load-bearing conditions are the scaling requirement $\mu(d)\propto 1/b$ and $\varphi(d)\propto 1/b$ for a well-defined strength, and the non-shrinking-band condition $D_0\le D_u$, with full monotonicity of $D(d_*)$ quoted from earlier work. The optimality argument runs through the initial bandwidth formula (5.9) and shows that only $\xi=2$ in $\alpha(d)=\xi d+(1-\xi)d^2$ keeps $D_0\le D_u$ for the full range of softening slopes and traction orders.

What would settle it

Take the optimal geometric function $\alpha(d)=2d-d^2$ with the non-associated $\mu$PF-CZM and any admissible softening law, for instance exponential softening with traction exponent $p=1.25$ or Park et al. concave softening with $m=1.15$, and evaluate the half bandwidth $D(d_*)$ from Eq. (5.31) at many intermediate $d_*$ values. If $D(d_*)$ is ever smaller than $D_0$ for some $d_*$ in $(0,1)$, then the irreversibility-modified 1D response will differ from the target softening curve, and the central claim that $\alpha(d)=2d-d^2$ always guarantees a non-shrinking band for arbitrary softening is false.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is a set of working conditions that separate phase-field models which genuinely behave as cohesive zone models from those that only approximate them. The paper derives, in one dimension, the traction $\sigma(d_*)$ and separation $w(d_*)$ generated by any choice of geometric function, degradation function, and dissipation function, and shows that the failure strength is length-scale independent exactly when $\eta_0 = c_\alpha/(2l_{\rm ch})$, which forces the dissipation and cracking functions to scale as $1/b$. It then shows that the analytically derived softening law is the law the model actually delivers only if the half bandwidth $D(d_*)$ is non-decreasing, equivalently at least the endpoint check $D_0\le D_u$; otherwise material points inside the crack band unload and the irreversibility condition changes the effective traction-separation curve. The quadratic geometric function of the earlier Conti et al. model gives a truncated, length-scale-convergent model that recovers the strength only in the vanishing limit and only one special softening curve, whereas the PF-CZM with a rational-fraction degradation function is length-scale insensitive. Among polynomial geometric functions, $\alpha(d)=2d-d^2$ is optimal: for the associated PF-CZM it guarantees a non-shrinking band for linear and convex softening, and for the non-associated $\mu$PF-CZM it does so for arbitrary softening curves and any traction exponent $p\ge1$ while decoupling the softening law from the crack bandwidth.

Load-bearing premise

The load-bearing premise is that the endpoint comparison $D_0\le D_u$ is enough to certify that the crack bandwidth never shrinks at any intermediate stage; the full monotonicity of $D(d_*)$ is cited from earlier work rather than proved here. If for some admissible softening law the bandwidth dipped below $D_0$ partway through failure, crack irreversibility would bend the very traction-separation curve the model claims to reproduce.

Editorial extensions

If this is right

  • A user can calibrate a phase-field cohesive model directly from the material's tensile strength, fracture energy, and initial softening slope; the length scale then needs no experimental tuning.
  • The associated PF2-CZM exactly reproduces linear softening and approximates exponential and Cornelissen-type softening, while the non-associated $\mu$PF-CZM also handles concave laws such as Park et al. softening.
  • Models built on the quadratic geometric function and truncated degradation (Conti type) should be understood as length-scale-convergent rather than length-scale-insensitive: they require $b\to0$ and fine meshes to reach the intended strength.
  • Including the second surface-energy term in Eq. (2.12) removes the previously reported discrepancy between phase-field and cohesive-zone energy dissipation, so the dissipated energy of the PF-CZM matches the Barenblatt CZM.
  • Because the traction order parameter $p\ge1$ does not change the traction-separation curve or the numerical responses in the examples, simulations can choose $p$ for numerical convenience.

Reading between the lines

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

  • Inference beyond the paper: the endpoint criterion $D_0\le D_u$ could be promoted to a full-path monotonicity diagnostic; a simple numerical monitor of $D(d_*)$ during loading would let any phase-field code audit whether its softening law is being honored, which the paper does not propose.
  • Inference beyond the paper: because the non-associated formulation decouples the softening law from the crack bandwidth, the same strategy may simplify other smeared-crack schemes (plasticity-damage or gradient-damage formulations) that currently couple constitutive softening to localization width.
  • Inference beyond the paper: the paper restricts to elastic solids and quasi-static loading; whether $\alpha(d)=2d-d^2$ remains optimal when inelastic deformation precedes crack nucleation, or under fatigue where the band widens cyclically, is an open question that a natural extension would test.
  • Inference beyond the paper: the optimality result is stated within the polynomial family $\alpha(d)=\xi d+(1-\xi)d^2$; non-polynomial geometric functions might also satisfy the non-shrinking condition, so 'optimal' should be read as optimal within this parameterized class, not globally.
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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 / 9 minor

Summary. The manuscript presents a unified analysis of phase-field models for cohesive fracture, organized around three characteristic functions: the geometric function α(d), the degradation function ω(d) (or φ(d)), and the dissipation function ϖ(d) (or μ(d)). The central thesis is that a phase-field model can represent a Barenblatt-type cohesive zone model only if the regularization length b is incorporated into the degradation/dissipation functions and if the crack bandwidth is non-decreasing during the failure process, so that crack irreversibility does not alter the intended traction–separation law (TSL). The author analyzes the Conti et al. model and its variants, the associated PF-CZM family with parameterized or analytically solved degradation functions, and the non-associated μPF-CZM. It is claimed that, within the considered polynomial family of geometric functions, only α(d)=2d−d² guarantees a non-shrinking crack band for general non-concave softening in the associated case and for almost arbitrary softening in the non-associated case. Representative numerical examples for concrete dams, beams, and adhesive joints support the qualitative conclusions and show insensitivity to the traction order parameter p.

Significance. If the central optimality and non-shrinking-band claims are established, the paper provides a useful organizing framework for selecting and designing phase-field cohesive fracture models, and it clarifies the role of the length scale and the energy-dissipation identity against earlier misinterpretations. The manuscript contains several closed-form analytical results: the 1D traction–separation formulas, explicit parameter calibration from the initial slope and ultimate opening, the PF-CZM counterpart of the Conti model, and the associated/non-associated μPF-CZM constructions. Numerical validation on multiple benchmarks is a strength. However, the key monotonicity/optimality theorems are not proved herein but cited from the author's own prior work, and the manuscript uses an endpoint inequality as though it were a sufficient design rule. This gap weakens the universal claims in the abstract and conclusions and prevents the review from being fully self-contained.

major comments (3)
  1. [Section 5.3.2, Eq. (5.40)] The paper replaces the full-path monotonicity requirement ∂D/∂d* ≥ 0 with the endpoint inequality D0 ≤ Du, explicitly noting that closed-form monotonicity is generally unavailable. As written, D0 ≤ Du is only a necessary condition: a bandwidth that decreases on a subinterval and then increases above D0 would satisfy Eq. (3.10) while violating Eq. (3.7), causing points in the crack band to unload and the identity (3.1) to fail. This endpoint condition is then used as the design criterion in Eqs. (5.9), (5.14), (5.23), and (5.32) to select α(d)=2d−d² as 'optimal' and to assert that this geometric function 'automatically guarantees a non-shrinking crack band' (Section 5.2.3) and is 'optimal ... in the sense that the resulting crack band is non-shrinking' (Section 5.3.2). The sufficiency proof is deferred to the author's own Wu (2017, 2024); no theorem, statement, or sketch is provided in this manuscript. If monotonicity fails for some admissible softening law, the reproduced TSL is altered by irreversibility, invalidating the abstract's 'only' and 'almost any arbitrary' claims. Please either provide a proof (or a precise theorem with hypotheses) that α(d)=2d−d² yields ∂D/∂d*≥0 for all d*∈[0,1] and for the stated classes of softening laws, or qualify the claims to 'satisfies the necessary endpoint condition and is verified numerically for the shown softening laws.'
  2. [Section 5.3.2, Eq. (5.40)] The statement that 'only the geometric function ξ=2 is optimal for arbitrary softening curves and any traction order p≥1' is a universal claim over an infinite-dimensional family of softening laws and over all p≥1. The manuscript provides numerical evidence for p=1, 1.5, 2 and for the specific softening laws shown in Figures 6(d) and 10, but no general proof or even a monotonicity argument over a continuous range of p is included; the proof is again cited to Wu (2024). Consequently, the abstract's phrase 'to (almost) any arbitrary one' is not established in this manuscript. Please either supply the missing proof or explicitly restrict the conclusion to 'satisfies the necessary endpoint condition for the considered class and is demonstrated for the shown examples.'
  3. [Section 5.2, Eqs. (5.12)–(5.13); Section 5.3, Eq. (5.6)] The 'reproduction' of a target traction–separation law is by construction: the degradation-function parameters a1, a2, ... are calibrated to the initial slope and ultimate opening of the given law (Eq. 5.13), and in Section 5.3 the degradation function is solved analytically from that law (Eq. 5.6). The manuscript should state explicitly that the TSL is an input to the model, not a prediction, and that the contribution is an exact (or asymptotically close) matching construction. As written, statements such as 'the linear and commonly adopted convex softening curves can be reproduced or approximated with sufficient precision' could mislead readers about the direction of the mapping and about what is being predicted versus fitted.
minor comments (9)
  1. [Abstract] The abstract uses 'uPF-CZM' while the body uses 'µPF-CZM'; please unify the notation.
  2. [Section 3.2, text before Eq. (3.8)] The sentence 'usually neither the closed-form nor the monotonicity of the crack bandwidth (3.8) is unavailable' contains a double negative that obscures the meaning; it should read 'is available' or be rephrased.
  3. [Section 4.1, Eq. (4.8)] The limit expression for η0 is written with the McAuley bracket but the limiting process is ambiguous; please clarify that the truncation is active for d²<b̄, so that η0=+∞ for any fixed b>0, and that the finite value is recovered only in the limit b→0.
  4. [Section 5.1, Eq. (5.6)] The integral formula for the solved cracking function appears garbled (the integrand contains '−1/2 ¯w(ϑ)h′(ϑ)/√h(ϑ)' with unclear placement of the denominator); please check against the cited source (Polyanin and Manzhirov, 2008) and correct the typesetting.
  5. [Section 5.2.3, Eq. (5.23)] The step '2πξ²/cα ≥ (πb/Du)³ =⇒ ξ=2' relies on the preceding statement that both D0 and Du decrease monotonically with ξ and that D0 decreases more rapidly; please provide the explicit expressions or a reference for this monotonicity.
  6. [Section 6.4] There is a typo: 'Ths simulation' should be 'The simulation'.
  7. [Conclusions, first paragraph] In the list of PF-CZM variants, 'Wang, 2000' should be 'Wang et al. (2020)' to match the reference list and the text in Section 5.2.
  8. [Section 4.1.2, Eq. (4.14)] The expression '0 < (0, +∞)' should be '0 ∉ (0, +∞)' to state that the limit is not in the required set.
  9. [Section 5.2.3 and Section 5.3.2] Please state explicitly that the optimality of α(d)=2d−d² is relative to the parameterized polynomial family (2.20); outside this family, other geometric functions might also satisfy the endpoint condition or even the full monotonicity condition.

Circularity Check

3 steps flagged · score 6.0 of 10

TSL reproduction is by construction (fitted a1/a2 and inverted φ), and the 'only optimal' geometric function is imported from the author's own prior work; partial circularity, though numerical benchmarks are independent.

  1. fitted input called prediction [Section 5.2, Eqs. (5.13) and Figure 6]
    "Accordingly, for a given TSLσ(w) the parameters a1, a2 ....., are determined as ... in terms of the ratios ¯k0 := k0/k0L and ¯wc := wc/wcL, respectively. ... Figure 6 compares the softening curves predicted by Eqs. (5.5a) and (5.11a) against the target ones. Among them, the linear softening curve is exactly reproduced."

    The two parameters a1 and a2 are solved from the initial slope k0 and ultimate crack opening wc of the target TSL via Eq. (5.13). Any curve reproduced by the PF2-CZM in Figure 6 therefore matches those two endpoints by construction; the 'predicted' softening curve is a calibrated interpolation within the chosen P(d) ansatz, not an independent output. The linear law is exact because the ansatz was designed to contain it. The interior shape is not fully forced, so the circularity is partial, but the endpoints are fitted inputs renamed as predictions.

  2. self definitional [Section 5.3, Eq. (5.6) and Figure 7]
    "Vice versa, for a given TSLσ(w) is given, the cracking functionϕ(d) can be solved as (Polyanin and Manzhirov, 2008; Feng et al., 2021; Wu, 2024) ... such that the crack opening (5.5b) becomes w(d∗) = wcL ¯w(d∗). ... The resulting softening curve is depicted in Figure 7. As expected, all the linear, convex and concave softening curves can be reproduced independently of the exponent p≥ 1 (Wu, 2024)."

    Here φ(d) is not assumed; it is the solution of the integral equation that enforces w(d∗)=wcL ar w(d∗), so the target traction–separation law is inserted as the input. The subsequent statement that linear, convex and concave laws 'can be reproduced' is a consistency check of the inversion, not a prediction from the phase-field theory. The reproduction is by construction, making the claimed capability self-definitional, modulo the well-posedness of the integral inversion.

1 more flagged steps
  1. uniqueness imported from authors [Section 5.2.3 and Section 5.3.2, Eqs. (5.23) and (5.40)]
    "That is, the geometric functionα(d) = 2d− d2 is optimal for the non-concave (linear and convex) softening curves with ¯k0≥ 1 since the resultingPF 2-CZM automatically guarantees a non-shrinking crack band (Wu, 2017). ... As proved in Wu (2024), regarding the parameterized polynomial function (2.20) with the parameter ξ∈ [0, 2], only the geometric function ξ = 2 =⇒ α(d) = 2d− d2, cα =π is optimal for arbitrary softening curves and any traction orderp≥ 1 in the sense that the resulting crack band is non-shrinking as expected; see Figure 10."

    The central optimality claim — 'only with the optimal geometric function' — is stated as proved in the author's own Wu (2017, 2024) papers, and no proof is reproduced here. The uniqueness part ('only') is therefore imported from a self-citation rather than established in this article, and it is load-bearing because it selects α=2d−d2 and justifies the 'almost any arbitrary' applicability of the non-associated model. Unless the cited proof is independently verified, the conclusion rests on the author's own authority.

full rationale

The derivation of the 1D traction–separation law from the phase-field balance equations is self-contained, and the numerical examples against experimental benchmarks (Koyna dam, SENB, DENB, DCB) are independent external checks, so this is not wholesale circularity. However, two load-bearing claims are circular in the specific sense defined here: (i) the PF-CZM softening curves in Section 5.2 are presented as predictions although a1 and a2 are calibrated to the initial slope and ultimate opening of the very target TSL; (ii) the µPF-CZM's claimed ability to reproduce arbitrary softening laws follows from solving φ(d) from the target TSL, so the reproduction is built into the construction. In addition, the uniqueness/optimality of α=2d−d2 is imported from the author's own prior Wu (2017, 2024) papers and is therefore self-referential authority rather than an in-paper proof. These features make the central 'can reproduce / only optimal' assertions partially circular; the independent experimental benchmarks keep the overall contribution from being wholly reducible to its inputs. The D0≤Du sufficiency gap is a correctness concern and is not scored here as circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central framework is the author's own; the paper depends on prior proofs of convergence and optimality that are cited but not re-derived. The free parameters are calibration constants for the target softening laws, making the TSL reproduction a designed fit rather than a prediction.

free parameters (4)
  • Traction order exponent p = 1.0, 1.25, 1.35, 1.5, 2.0
    Free exponent p>=1 in muPF-CZM controlling the parameterized softening shape; chosen for each target softening law, not derived from data.
  • Polynomial coefficients a1, a2, ... = Linear: a1=0, a2=0; Exponential: a1=0.4748; Cornelissen: a1=1.8868, a2=3.7081
    Calibrated to match the target TSL's initial slope and ultimate crack opening via Eq. (5.13).
  • Regularization length scale b = 0.3 m, 1.0 mm, 2.0 mm, 0.1 mm
    User-selected phase-field length parameter; affects mesh size and crack bandwidth, not fitted to fracture experiments.
  • Coefficients c1...c6 for 6th-order softening fits = listed in Appendix A for Cornelissen and Park softening
    Fitted to approximate the analytical softening curves in the muPF-CZM representation.
assumptions (5)
  • domain assumption The Barenblatt cohesive zone model with surface energy G(w) is the target functional for regularization.
    Stated in Section 1; all compared models aim to reproduce this CZM.
  • domain assumption Quasi-static, infinitesimal strain, elastic solids; crack nucleation coincides with peak stress.
    Section 1 and Appendix B; inelastic deformation before nucleation excluded.
  • domain assumption The crack band remains fully in a loading state so the phase-field evolution identity Eq. (3.1) holds.
    Needed for the analytical traction-separation derivations in Section 3.
  • ad hoc to paper The endpoint condition D0 <= Du is used to select the optimal geometric function.
    Eq. (3.10); a necessary condition for non-shrinking crack bandwidth, treated as a design rule; sufficiency rests on a proof in Wu 2024.
  • standard math Gamma-convergence proofs for the Conti et al. and Lammen et al. models are accepted from prior literature.
    Invoked in Sections 4.1 and 5 without reproduction.

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Cite this review

Pith. "Pith review of Unified analysis of phase-field models for cohesive fracture." pith.science (2026). https://pith.science/paper/7Z5Y4KL7

@misc{pith2026241203836,
  author       = {Pith},
  title        = {Pith review of: Unified analysis of phase-field models for cohesive fracture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7Z5Y4KL7}},
  note         = {Machine review of arXiv:2412.03836}
}
read the original abstract

We address in this review unified analysis of phase-field models for cohesive fracture. Aiming to regularize the Barenblatt (1959) cohesive zone model, all the discussed models are distinguished by three characteristic functions, i.e., the geometric function dictating the crack profile, the degradation function for the constitutive relation and the dissipation function defining the crack driving force. The latter two functions coincide in the associated formulation, while in the non-associated one they are designed to be different. Distinct from the counterpart for brittle fracture, in the phase-field model for cohesive fracture the regularization length parameter has to be properly incorporated into the dissipation and/or degradation functions such that the failure strength and traction-separation softening curve are both well-defined. Moreover, the resulting crack bandwidth needs to be non-decreasing during failure in order that imposition of the crack irreversibility condition does not affect the anticipated traction-separation law (TSL). With a truncated degradation function that is proportional to the length parameter, the Conti et al.(2016) model and the latter improved versions can deal with crack nucleation only in the vanishing limit and capture cohesive fracture only with a particular TSL. Owing to a length scale dependent degradation function of rational fraction, these deficiencies are largely overcome in the phase-field cohesive zone model (PF-CZM). Among many variants in the literature, only with the optimal geometric function, can the associated PF-CZM apply to general non-concave softening laws and the non-associated uPF-CZM to (almost) any arbitrary one. Some mis-interpretations are clarified and representative numerical examples are presented.

Figures

Figures reproduced from arXiv: 2412.03836 by the authors.

Figure 1
Figure 1. The softening curve and surface energy density function predicted by the [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. The degradation function ω(d) adopted in the Conti et al. (2016, 2024) model with the parameters d0 = 2 √ b¯ 1 + √ b¯ , bˆ 1 = [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. The degradation function ω(d) of rational fraction adopted in the PF-CZM Accordingly, the cracking and dissipation functions are given by ϕ(d) = a0 d 2 [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (23 more)
Figure 4
Figure 4. Figure 4: The softening curve and surface energy density function predicted by the [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 5
Figure 5. Figure 5: The softening curve and surface energy density function given by the [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]
Figure 6
Figure 6. Figure 6: The softening curves given by the PF 2-CZM with the optimal geometric function α(d) = 2d − d 2 . the exponential one are invisible. For the Cornelissen et al. (1986) softening curve, the discrepancy is also acceptable and can be reduced by increasing the order of P(d) …
Figure 7
Figure 7. Figure 7: Softening curves predicted by the µPF-CZM (Wu, 2024). The resulting softening curve is depicted in [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]
Figure 8
Figure 8. Figure 8: The surface energy density function and crack band width predicted by the associated [PITH_FULL_IMAGE:figures/full_fig_p031_8.png]
Figure 9
Figure 9. Figure 9: The surface energy density function and crack band width predicted by the associated [PITH_FULL_IMAGE:figures/full_fig_p032_9.png]
Figure 10
Figure 10. Figure 10: Evolution of the crack bandwidth given by the non-associated [PITH_FULL_IMAGE:figures/full_fig_p033_10.png]
Figure 11
Figure 11. Figure 11: Koyna dam under overflow pressure. Left: Geometry (Unit of length: m), boundary and loading conditions; [PITH_FULL_IMAGE:figures/full_fig_p035_11.png]
Figure 12
Figure 12. Figure 12: Koyna dam under overflow pressure: Numerical crack patterns predicted by the [PITH_FULL_IMAGE:figures/full_fig_p035_12.png]
Figure 13
Figure 13. Figure 13: Koyna dam under overflow pressure: Numerical curves of overflow height [PITH_FULL_IMAGE:figures/full_fig_p036_13.png]
Figure 14
Figure 14. Figure 14: Single edge-notched beam (Schlangen, 1993). Left: Geometry (Unit of length: mm), loading and boundary conditions; Right: Experimentally observed crack paths. In the numerical simulation, the following mechanical parameters for concrete were adopted: Young’s modulus E0…
Figure 15
Figure 15. Figure 15: Single edge-notched beam: Numerical crack patterns predicted by the [PITH_FULL_IMAGE:figures/full_fig_p038_15.png]
Figure 16
Figure 16. Figure 16: Single edge-notched beam test: Applied load–CMSD curves predicted by the [PITH_FULL_IMAGE:figures/full_fig_p039_16.png]
Figure 17
Figure 17. Figure 17: Double edge-notched beam (Bocca et al., 1990). Left: Geometry (Unit of length: mm), loading and boundary conditions; Right: Experimentally observed crack paths. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_17.png]
Figure 18
Figure 18. Figure 18: Double edge-notched beam: Numerical crack patterns predicted by the [PITH_FULL_IMAGE:figures/full_fig_p040_18.png]
Figure 19
Figure 19. Figure 19: Double edge-notched beam: Applied load–CMSD curves predicted by the [PITH_FULL_IMAGE:figures/full_fig_p041_19.png]
Figure 20
Figure 20. Figure 20: Double edge notched specimens (Nooru-Mohamed, 1992). Left: Geometry (Unit of length: mm), loading and boundary conditions; Right: Experimentally observed crack paths. The following mechanical parameters for concrete were adopted in the numerical simulation: Young’s mo…
Figure 21
Figure 21. Figure 21: Double edge notched specimen (DENS-4a): Numerically predicted damage profiles at [PITH_FULL_IMAGE:figures/full_fig_p043_21.png]
Figure 22
Figure 22. Figure 22: Double edge notched specimen (DENS-4b): Numerically predicted damage profiles at [PITH_FULL_IMAGE:figures/full_fig_p043_22.png]
Figure 23
Figure 23. Figure 23: Double edge notched specimens: Applied load – displacement curves predicted by the [PITH_FULL_IMAGE:figures/full_fig_p043_23.png]
Figure 24
Figure 24. Figure 24: Double cantilever beam (DCB) test: Geometry (unit of length: mm), loading and boundary conditions. [PITH_FULL_IMAGE:figures/full_fig_p044_24.png]
Figure 25
Figure 25. Figure 25: Double cantilever beam (DCB) test: Predicted crack profiles at CMOD = 2.5 mm. [PITH_FULL_IMAGE:figures/full_fig_p045_25.png]
Figure 26
Figure 26. Figure 26: Double cantilever beam (DCB) test: Force–CMOD curves predicted by the non-associated [PITH_FULL_IMAGE:figures/full_fig_p045_26.png]

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

  1. Phase-field modelling of cohesive fracture. Part II: Reconstruction of the cohesive law

    math.AP 2025-07 accept novelty 7.0 of 10

    A rigorous inverse-construction procedure maps any admissible cohesive traction law to a phase-field model whose Gamma-limit reproduces it exactly, via Abel integral inversion.

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Works this paper leans on

59 extracted references · 56 canonical work pages · cited by 1 Pith paper

  1. [1]

    , author Furgiuele, F

    author Alfano, M. , author Furgiuele, F. , author Lenonardi, A. , author Maletta, C. , author Paulino, G. , year 2009 . title Mode- I fracture of adhesive joints using tailed cohesive zone models . journal International Journal of Fracture volume 157 , pages 193--204

    work page 2009

  2. [2]

    , author Furgiuele, F

    author Alfano, M. , author Furgiuele, F. , author Pagnotta, L. , author Paulino, G.H. , year 2010 . title Analysis of fracture in aluminum joints bonded with a bi-componnet epoxy adhesive . journal Journal of Testing and Evaluation volume 39 , pages JTE102753

    work page 2010

  3. [3]

    , author Gerasimov, T

    author Ambati, M. , author Gerasimov, T. , author de Lorenzis, L. , year 2015 . title A review on phase-field models for brittle fracture and a new fast hybrid formulation . journal Comput. Mech. volume 55 , pages 383--405

    work page 2015

  4. [4]

    , author Tortorelli, V.M

    author Ambrosio, L. , author Tortorelli, V.M. , year 1990 . title Approximation of functional depending on jumps by elliptic functional via Gamma -convergence . journal Communications on Pure and Applied Mathematics volume 43 , pages 999--1036

    work page 1990

  5. [5]

    , year 2015

    author Ayachit, U. , year 2015 . title The ParaView guide: A parallel visualization application . publisher Kitware, ISBN 978-1930934306

    work page 2015

  6. [6]

    , year 1959

    author Barenblatt, G.I. , year 1959 . title The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially -symmetric cracks . journal Journal of Applied Mathematics and Mechanics volume 23 , pages 622--636

    work page 1959

  7. [7]

    , author Planas, J

    author Ba z ant, Z.P. , author Planas, J. , year 1997 . title Fracture and size effects in concrete and other quasi-brittle materials . publisher CRC Press, New York

    work page 1997

  8. [8]

    , author Carpinteri, A

    author Bocca, P. , author Carpinteri, A. , author Valente, S. , year 1990 . title Size effects in the mixed mode crack propagation: Softening and snap-back analysis . journal Engineering Fracture Mechanics volume 35 , pages 159--170

    work page 1990

Show all 59 references
  1. [9]

    , author Francfort, G

    author Bourdin, B. , author Francfort, G. , author Marigo, J.J. , year 2000 . title Numerical experiments in revisited brittle fracture . journal J. Mech. Phys. Solids volume 48 , pages 797--826

    2000

  2. [10]

    , author Francfort, G

    author Bourdin, B. , author Francfort, G. , author Marigo, J.J. , year 2008 . title The variational approach to fracture . publisher Springer, Berlin

    2008

  3. [11]

    , year 1998

    author Braides, A. , year 1998 . title Approximation of free-discontinuity problems . publisher Springer science & Business Media, Berlin

    1998

  4. [12]

    , author de Borst, R

    author Chen, L. , author de Borst, R. , year 2021 . title Phase-field modeling of cohesive fracture . journal European Journal of Mechanics / A Solids volume 90 , pages 104343

    2021

  5. [13]

    , author de Borst, R

    author Chen, L. , author de Borst, R. , year 2022 . title Phase-field regularised cohesive zone model for interface modelling . journal Theoretical and Applied Fracture Mechanics volume 122 , pages 103630

    2022

  6. [14]

    , author Focardi, M

    author Conti, S. , author Focardi, M. , author Iurlano, F. , year 2016 . title Phase field approximation of cohesive fracture models . journal Annales de l'Institut Henri Poincare (C) Non Linear Analysis volume 33 , pages 1033--1067

    2016

  7. [15]

    , author Focardi, M

    author Conti, S. , author Focardi, M. , author Iurlano, F. , year 2024 . title Phase-field approximation of a vectorial, geometrically nonlinear cohesive fracture energy . journal Arch. Rational Mech. Anal. volume 248 , pages 21

    2024

  8. [16]

    , author Hordijk, D

    author Cornelissen, H. , author Hordijk, D. , author Reinhardt, H. , year 1986 . title Experimental determination of crack softening characteristics of normalweight and lightweight concrete . journal Heron volume 31 , pages 45--56

    1986

  9. [17]

    , author Fan, J

    author Feng, F. , author Fan, J. , author Li, J. , year 2021 . title Endowing explicit cohesive laws to the phase-field fracture theory . journal Journal of the Mechanics and Physics of Solids volume 152 , pages 104464

    2021

  10. [18]

    , author Marigo, J

    author Francfort, G. , author Marigo, J. , year 1998 . title Revisting brittle fracture as an energy minimization problem . journal J. Mech. Phys. Solids volume 46 , pages 1319--1342

    1998

  11. [19]

    , author Iurlano, F

    author Freddi, F. , author Iurlano, F. , year 2017 . title Numerical insight of a variational smeared approach to cohesive fracture . journal J. Mech. Phys. Solids volume 98 , pages 156--171

    2017

  12. [20]

    , author Liu, Y

    author Geelen, R.J.M. , author Liu, Y. , author Hu, T. , author Tupek, M.R. , author Dolbow, J.E. , year 2019 . title A phase-field formulation for dynamic cohesive fracture . journal Comput. Methods Appl. Mech. Eng. volume 34 , pages 680--711

    2019

  13. [21]

    , author Remacle, J.F

    author Geuzaine, C. , author Remacle, J.F. , year 2009 . title Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities . journal Int. J. Numer. Engng volume 79(11) , pages 1309--1331

    2009

  14. [22]

    , year 1921

    author Griffith, A.A. , year 1921 . title The phenomena of rupture and flow in solids . journal Philosophical Transactions of the Royal Society of Londres volume 221 , pages 163--198

    1921

  15. [23]

    , author Kumar, A

    author Kamarei, F. , author Kumar, A. , author Lopez-Pamies, O. , year 2024 . title The poker-chip experiments of synthetic elastomers explained . journal J. Mech. Phys. Solids volume 188 , pages 105683

    2024

  16. [24]

    , author Bourdin, B

    author Kumar, A. , author Bourdin, B. , author Francfort, G.A. , author Lopez-Pamies, O. , year 2020 . title Revisiting nucleation in the phase-field approach to brittle fracture . journal Journal of the Mechanics and Physics of Solids volume 142 , pages 104027

    2020

  17. [25]

    , author Hilsdorf, H.K

    author Kupfer, H. , author Hilsdorf, H.K. , author Rusch, H. , year 1969 . title Behaviour of concrete under biaxial stress . journal Proc. ACI volume 66 , pages 656--666

    1969

  18. [26]

    , author Conti, S

    author Lammen, H. , author Conti, S. , author Mosler, J. , year 2023 . title A finite deformation phase field model suitable for cohesive fracture . journal Journal of the Mechanics and Physics of Solids volume 178 , pages 105349

    2023

  19. [27]

    , year 2024

    author Larsen, C. , year 2024 . title A local variational principle for fracture . journal Journal of the Mechanics and Physics of Solids volume 187 , pages 105625

    2024

  20. [28]

    , author Dolbow, J

    author Larsen, C. , author Dolbow, J. , author Lopez-Pamies, O. , year 2024 . title A variational formulation of griffith phase-field fracture with material strength . journal International Journal of Fracture volume doi: https://doi.org/10.1017/s10704-024-00786-3

    2024 doi

  21. [29]

    , author Song, Y.C

    author Lee, S.K. , author Song, Y.C. , author Han, S.H. , year 2003 . title Biaxial behavior of plain concrete of nuclear containment building . journal Nucl. Eng. Des. volume 227 , pages 143--153

    2003

  22. [30]

    , author Guo, Z.H

    author Li, W.Z. , author Guo, Z.H. , year 1991 . title Experimental research for strength and deformation of concrete under biaxial tension–compression loading . journal J. Hydraul. Eng. (in Chinese) volume 8 , pages 51--56

    1991

  23. [31]

    , year 2017

    author Lorentz, E. , year 2017 . title A nonlocal damage model for plain concrete consistent with cohesive fracture . journal Int. J. Fract. volume 207 , pages 123--159

    2017

  24. [32]

    , author Cuvilliez, S

    author Lorentz, E. , author Cuvilliez, S. , author Kazymyrenko, K. , year 2012 . title Modelling large crack propagation: from gradient-damage to cohesive zone models . journal Int. J. Fract. volume 178 , pages 85--95

    2012

  25. [33]

    , author Godard, V

    author Lorentz, E. , author Godard, V. , year 2011 . title Gradient damage models: Towards full-scale computations . journal Comput. Methods Appl. Mech. Engrg. volume 200 , pages 1927--1944

    2011

  26. [34]

    , author Bennani, L

    author Marboeuf, A. , author Bennani, L. , author Budinger, M. , author Pommier-Budinger, V. , year 2020 . title Electromechanical resonant ice protection systems: numerical investigation through a phase-field mixed adhesive/brittle fracture model . journal Engineering Fractur...

    2020

  27. [35]

    , author Bourdin, B

    author Mesgarnejad, A. , author Bourdin, B. , author Khonsari, M. , year 2015 . title Validation simulations for the variational approach to fracture . journal Comput. Methods Appl. Mech. Engrg. volume 290 , pages 420--437

    2015

  28. [36]

    , author Welschinger, F

    author Miehe, C. , author Welschinger, F. , author Hofacker, M. , year 2010a . title Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field fe implementations . journal Int. J. Numer. Meth. Engng. volume 83 , pages 1273--1311

  29. [37]

    , author Shah, J

    author Mumford, D. , author Shah, J. , year 1989 . title Optimal approximations by piecewise smooth functions and associated variational problems . journal Communications on Pure and Applied Mathematics volume 42 , pages 577--685

    1989

  30. [38]

    , author Giraldo-Londono, O

    author Muneton-Lopez, R.A. , author Giraldo-Londono, O. , year 2024 . title A phase-field formulation for cohesive fracture based on the park-paulino-roesler ( PPR ) cohesive fracture model . journal Journal of the Mechanics and Physics of Solids volume 182 , pages 105460

    2024

  31. [39]

    , author Yvonnet, J

    author Nguyen, T.T. , author Yvonnet, J. , author Zhu, Q.Z. , author Bornert, M. , author Chateau, C. , year 2016 . title A phase-field method for computational modeling of interfacial damage interacting with crack propagation in realistic microstructures obtained by microtomo...

    2016

  32. [40]

    , author Nguyen, V.P

    author Nguyen-Thanh, C. , author Nguyen, V.P. , author de Vaucorbeilc, A. , author Mandal, T.K. , author Wu, J.Y. , year 2020 . title Jive: an open source, research-oriented c++ library for solving partial differential equations . journal Advances in Engineering Software , pag...

    2020

  33. [41]

    , year 1992

    author Nooru-Mohamed, M. , year 1992 . title Mixed-mode fracture of concrete: an experimental approach . Ph.D. thesis. Delft University of Technology

    1992

  34. [42]

    , author Paulino, G

    author Park, K. , author Paulino, G. , author Roesler, J. , year 2009 . title A unified potential-based cohesive model for mixed-mode fracture . journal Journal of the Mechanics and Physics of Solids volume 57 , pages 891--908

    2009

  35. [43]

    , author Amor, H

    author Pham, K. , author Amor, H. , author Marigo, J.J. , author Maurini, C. , year 2011 . title Gradient damage models and their use to approximate brittle fracture . journal International Journal of Damage Mechanics volume 20 , pages 618--652

    2011

  36. [44]

    , author Manzhirov, A

    author Polyanin, A. , author Manzhirov, A. , year 2008 . title Handbook of integral equations . publisher CRC Press

    2008

  37. [45]

    , year 1993

    author Schlangen, E. , year 1993 . title Experimental and numerical analysis of fracture processes in concrete . Ph.D. thesis. Delft University of Technology

    1993

  38. [46]

    , author de Borst, R

    author Verhoosel, C.V. , author de Borst, R. , year 2013 . title A phase-field model for cohesive fracture . journal Int. J. Numer. Meth. Engng. volume 96 , pages 43--62

    2013

  39. [47]

    , author Zhou, W

    author Wang, Q. , author Zhou, W. , author Feng, Y.T. , year 2020 . title The phase-field model with an auto-calibrated degradation function based on general softening laws for cohesive fracture . journal Applied Mathematical Modelling volume 86 , pages 185--206

    2020

  40. [48]

    , year 2000

    author Wang, T. , year 2000 . title Analysis of strip electric saturation model of crack problem in piezoelectric materials . journal International Journal of Solids and Structures volume 37 , pages 6031–6049

    2000

  41. [49]

    , year 2017

    author Wu, J.Y. , year 2017 . title A unified phase-field theory for the mechanics of damage and quasi-brittle failure in solids . journal Journal of the Mechanics and Physics of Solids volume 103 , pages 72--99

    2017

  42. [50]

    , year 2018a

    author Wu, J.Y. , year 2018a . title A geometrically regularized gradient-damage model with energetic equivalence . journal Comput. Methods Appl. Mech. Engrg. volume 328 , pages 612--637

  43. [51]

    , year 2018b

    author Wu, J.Y. , year 2018b . title Robust numerical implementation of non-standard phase-field damage models for failure in solids . journal Computer Methods in Applied Mechanics and Engineering volume 340 , pages 767--797

  44. [52]

    , year 2024

    author Wu, J.Y. , year 2024 . title A generalized phase-field cohesive zone model ( PF-CZM ) for fracture . journal Journal of the Mechanics and Physics of Solids volume 192 , pages 105841

    2024

  45. [53]

    , author Cervera, M

    author Wu, J.Y. , author Cervera, M. , year 2015 . title On the equivalence between traction- and stress-based approaches for the modeling of localized failure in solids . journal Journal of the Mechanics and Physics of Solids volume 82 , pages 137--163

    2015

  46. [54]

    , author Cervera, M

    author Wu, J.Y. , author Cervera, M. , year 2016 . title A thermodynamically consistent plastic-damage framework for localized failure in quasi-brittle solids: Material model and strain localization analysis . journal International Journal of Solids and Structures volume 88-89...

    2016

  47. [55]

    , author Huang, Y

    author Wu, J.Y. , author Huang, Y. , author Nguyen, V.P. , year 2020 a. title On the BFGS monolithic algorithm for the unified phase field damage theory . journal Comput. Methods Appl. Mech. Engrg. volume 360 , pages 112704

    2020

  48. [56]

    , author Nguyen, V.P

    author Wu, J.Y. , author Nguyen, V.P. , year 2018 . title A length scale insensitive phase-field damage model for brittle fracture . journal Journal of the Mechanics and Physics of Solids volume 119 , pages 20--42

    2018

  49. [57]

    , author Nguyen, V.P

    author Wu, J.Y. , author Nguyen, V.P. , author Nguyen, C.T. , author Sutula, D. , author Sinaie, S. , author Bordas, S. , year 2020 b. title Phase field modeling of fracture . journal Advances in Applied Mechancis volume 53 , pages 1--183; https://doi.org/10.1016/bs.aams.2019.08.001

    2020 doi

  50. [58]

    , author Yao, J.R

    author Wu, J.Y. , author Yao, J.R. , author Le, J.L. , year 2023 . title Phase-field modeling of stochastic fracture in heterogeneous quasi-brittle solids . journal Comput. Methods Appl. Mech. Engrg. volume 416 , pages 116332

    2023

  51. [59]

    , author Maurini, C

    author Zolesi, C. , author Maurini, C. , year 2024 . title Stability and crack nucleation in variational phase-field models of fracture: Effects of length-scales and stress multi-axiality . journal J. Mech. Phys. Solids volume 192 , pages 105802

    2024

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.