{"id":"4b8ea657-e682-4d4a-8897-7ec4b22c99d8","arxiv_id":"2412.03836","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A unified framework for phase-field cohesive fracture models identifies conditions for length-scale-insensitive traction-separation laws and shows a non-associated formulation can reproduce almost arbitrary softening curves.","lead":"This review paper unifies several phase-field models for cohesive fracture under one mathematical framework based on three characteristic functions. It explains why the crack bandwidth must not shrink during failure and how a newer non-associated model can reproduce nearly any traction-separation softening law.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The optimality claim rests on D0 ≤ Du (Eq. 3.10), but that endpoint inequality is only necessary for the required monotonicity of D(d*); the sufficiency proof is deferred to self-cited work, so irreversibility may still alter the TSL for admissible softening laws.","rationale":"The paper is a coherent review that develops a useful unified framework and demonstrates its predictions on several benchmarks; the analytical machinery around the parameterized TSL is a genuine contribution. However, the strongest claim depends on a non-shrinking crack band throughout the entire failure path, not just at the endpoints. The manuscript itself acknowledges that monotonicity of D(d*) is generally not available in closed form, and then uses D0 ≤ Du as the selection criterion. Since D0 ≤ Du is necessary but not sufficient, the optimality of α(d)=2d−d^2 is not fully established within this paper for all admissible softening laws. The reader's conditional verdict already flags exactly this gap, and my stress-test confirms it and locates it precisely in Eqs. (3.7)-(3.10) and in the transition from Section 3.2 to Sections 5.2.3 and 5.3.2. I do not see a separate, more severe flaw; the numerical examples provide supporting evidence, but not a proof covering the abstract's universal quantifier. Thus the appropriate disposition remains conditional, with the requested condition being a rigorous monotonicity proof or a counterexample.","tokens_in":31519,"tokens_out":7848,"duration_ms":95782,"concrete_test":"For the PF2-CZM with α(d)=2d−d^2, select a non-concave target softening that is not among the three shown in Figure 6(d), e.g., a piecewise-linear curve with steep initial drop followed by a long tail; calibrate P(d) via Eq. (5.13), then evaluate D(d*) from Eq. (5.8) on a fine grid (at least 10^4 points) over d*∈[0,1]. If min D(d*) < D0, or if D(d*) decreases on any subinterval, the endpoint condition Eq. (3.10) is insufficient and the optimality claim needs an explicit monotonicity proof before it can stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim — that the geometric function α(d)=2d−d^2 guarantees a non-shrinking crack band for any non-concave softening law in the associated PF-CZM and for almost arbitrary softening in the non-associated µPF-CZM — is load-bearing. Section 3.2 correctly states that the analytical TSL holds only if ∂D/∂d* ≥ 0, and then immediately replaces this full-path condition with the endpoint comparison D0 ≤ Du in Eq. (3.10), explicitly noting that closed-form monotonicity is generally unavailable. The step from D0 ≤ Du to global monotonicity is not derived in this paper; it is attributed to the author's own Wu (2017, 2024). Figure 6(d) shows monotonicity for linear, exponential, and Cornelissen softening, but the abstract's 'only' and 'almost any arbitrary' claims require it for every admissible softening curve. If for some admissible law D(d*) dips below D0, or decreases on a subinterval, then some material points in the crack band unload while d* increases, so the identity (3.1) fails and the reproduced traction-separation law is altered by the irreversibility constraint. The endpoint condition cannot detect such a dip. This is a correctness gap in the central optimality assertion, not merely a disagreement with the field's consensus.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":31832,"tokens_out":14542,"duration_ms":132583,"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":[{"comment":"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.'","section":"Section 5.3.2, Eq. (5.40)"},{"comment":"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.'","section":"Section 5.3.2, Eq. (5.40)"},{"comment":"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.","section":"Section 5.2, Eqs. (5.12)–(5.13); Section 5.3, Eq. (5.6)"}],"minor_comments":[{"comment":"The abstract uses 'uPF-CZM' while the body uses 'µPF-CZM'; please unify the notation.","section":"Abstract"},{"comment":"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.","section":"Section 3.2, text before Eq. (3.8)"},{"comment":"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.","section":"Section 4.1, Eq. (4.8)"},{"comment":"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.","section":"Section 5.1, Eq. (5.6)"},{"comment":"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.","section":"Section 5.2.3, Eq. (5.23)"},{"comment":"There is a typo: 'Ths simulation' should be 'The simulation'.","section":"Section 6.4"},{"comment":"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.","section":"Conclusions, first paragraph"},{"comment":"The expression '0 < (0, +∞)' should be '0 ∉ (0, +∞)' to state that the limit is not in the required set.","section":"Section 4.1.2, Eq. (4.14)"},{"comment":"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.","section":"Section 5.2.3 and Section 5.3.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is essentially a review and synthesis of the author's own prior publications (Wu 2017, 2018a,b, 2024). The key theorems on monotonicity and optimality of the geometric function are cited from those works rather than proved or even stated precisely in this paper. Given that the abstract makes universal claims ('only', 'almost any arbitrary'), the editor may wish to require that the author either include the missing proofs (perhaps as an appendix) or explicitly reduce the claims to what is demonstrated. The paper is likely of interest to the phase-field fracture community, but its impact as a review article depends on whether it is self-contained in its central assertions. The numerical examples are valuable, but they do not substitute for the missing analytical proofs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, this is a review masquerading as a unified theory, and mostly a very competent one. The genuinely useful parts are the clean three-function decomposition (geometric, degradation, dissipation), the correction to Chen and de Borst on the missing second term in the dissipation identity, and the explicit inter-relation between the Conti et al. model and Wu's PF-CZM. It also makes a sharp point that for cohesive fracture the length scale must enter the degradation/dissipation functions, and that the crack band must not shrink during failure. Those are real conceptual contributions for a community that often borrows brittle-fracture phase-field machinery without checking these conditions.\n\nThe numerical examples are multiple and mostly convincing: Koyna dam, SENB, DENB, DCB, covering mode-I and mixed-mode, with p-independence demonstrated. The DCB with concave Park softening is the right acid test and it works. That is good evidence the non-associated muPF-CZM does something the associated PF-CZM cannot.\n\nThe soft spots are real but not disqualifying. First, the \"only\" and \"almost any arbitrary\" claims in the abstract are stronger than what is actually shown. The paper derives the endpoint condition D0<=Du as a design rule, but the sufficiency for monotonicity of D(d*) is deferred to a self-cited proof in Wu (2024) and illustrated on selected softening laws. The endpoint condition is necessary, not sufficient, and Eq. (3.10) makes it look like an equivalence when it is only a one-way check. The stress-test note is right: if monotonicity fails on a subinterval for some admissible law, irreversibility will alter the TSL. The author should either prove monotonicity for the polynomial family or state the claim only for the class where it is verified. Second, the reproduction of target TSLs is by calibration: the degradation functions are solved or fitted from the softening law itself, so the agreement with the TSL is construction, not prediction. The paper says this openly, but the abstract presents the capability as a result. Third, no code or data are shipped, so the numerical examples are not independently reproducible as-is.\n\nWho gets value: anyone working on phase-field or gradient-damage models for quasi-brittle materials, especially people choosing among variants. It is a review, but the kind of review that sets the taxonomy for the field. It deserves serious peer review; the claims just need to be scoped precisely.","headline":"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.","tokens_in":32357,"tokens_out":3448,"would_cite":true,"duration_ms":43893,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["74R10","74A45","74G65","49J45"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["cohesive fracture","phase-field model","traction-separation law","crack irreversibility","geometric function","degradation function","dissipation function","Gamma-convergence"],"falsifier":"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.","tokens_in":31291,"feed_emoji":"💥","tokens_out":10343,"duration_ms":92506,"temperature":0.7,"pith_summary":"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.","feed_headline":"A shrinking crack band breaks phase-field fracture laws","feed_subtitle":"Unified analysis picks alpha = 2d - d^2 so the softening curve survives crack irreversibility and matches laboratory tests.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the cohesive zone model whose traction-separation law every phase-field model in the paper is meant to regularize.","marker":"(Barenblatt, 1959)"},{"why":"Supplies the truncated, length-scale-proportional degradation model whose Gamma-convergence and limitations are analyzed and whose failure strength exists only in the vanishing limit.","marker":"(Conti et al., 2016, 2024)"},{"why":"Provides the unified phase-field theory with parameterized polynomial geometric function and rational-fraction degradation function from which the PF-CZM and the optimal-geometry analysis emerge.","marker":"(Wu, 2017)"},{"why":"Introduces the phase-field cohesive zone model (PF-CZM) with length-scale-insensitive traction-separation response; the PF2-CZM follows from this line.","marker":"(Wu, 2018a; Wu and Nguyen, 2018)"},{"why":"Contributes the gradient-damage model with length-scale-dependent rational degradation function that the paper classifies as PF1-CZM and compares for exponential and linear softening.","marker":"(Lorentz, 2017)"},{"why":"Shows how an explicit cohesive law can be endowed by solving the degradation function from the softening curve; the paper treats it as the p=1 limit of the associated solved-degradation muPF-CZM.","marker":"(Feng et al., 2021)"},{"why":"Gives the non-associated muPF-CZM, the proof that alpha(d)=2d-d^2 is optimal for arbitrary softening curves, and the closed-form cracking functions used for the benchmark simulations.","marker":"(Wu, 2024)"},{"why":"States the earlier claim that phase-field models only partially reproduce cohesive response; the paper clarifies that a missing surface-energy term caused the discrepancy.","marker":"(Chen and de Borst, 2021)"}],"fun_headline_variants":["Non-shrinking crack band preserves traction-separation law","Unified analysis crowns alpha=2d-d^2 for cohesive fracture","Phase-field cohesive models need length-scale-aware degradation","One formula ensures softening law survives crack irreversibility"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Non-shrinking crack band preserves traction-separation law","Unified analysis crowns alpha=2d-d^2 for cohesive fracture","Phase-field cohesive models need length-scale-aware degradation","One formula ensures softening law survives crack irreversibility"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000658,"raw_usage":{"total_tokens":3114,"prompt_tokens":1150,"completion_tokens":1964,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":766,"completion_tokens_details":{"reasoning_tokens":1897}},"tokens_in":766,"tokens_out":1964,"duration_ms":15428,"temperature":1.0,"reasoning_tokens":1897,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:01:29.105709+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":", year 1959","cited_arxiv_id":null,"evidence_quote":"Defines the cohesive zone model whose traction-separation law every phase-field model in the paper is meant to regularize."},{"cited_title":", author Focardi, M","cited_arxiv_id":null,"evidence_quote":"Supplies the truncated, length-scale-proportional degradation model whose Gamma-convergence and limitations are analyzed and whose failure strength exists only in the vanishing limit."},{"cited_title":", year 2017","cited_arxiv_id":null,"evidence_quote":"Contributes the gradient-damage model with length-scale-dependent rational degradation function that the paper classifies as PF1-CZM and compares for exponential and linear softening."},{"cited_title":", author Fan, J","cited_arxiv_id":null,"evidence_quote":"Shows how an explicit cohesive law can be endowed by solving the degradation function from the softening curve; the paper treats it as the p=1 limit of the associated solved-degradation muPF-CZM."},{"cited_title":", author de Borst, R","cited_arxiv_id":null,"evidence_quote":"States the earlier claim that phase-field models only partially reproduce cohesive response; the paper clarifies that a missing surface-energy term caused the discrepancy."}],"review_version":1}