{"id":"84c136c2-8ce8-40a8-b6cd-bea961ff4712","arxiv_id":"2507.12172","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"This paper gives a recipe for choosing the free functions in a phase-field crack model so that the effective crack law matches a user-specified cohesive law. The construction is rigorous and covers both linear and superlinear small-opening behaviors, with explicit formulas for seven standard softening curves.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main reconstruction theorems are coherent, but the exponential-law approximation in §3.3.6 uses an Rδ inconsistent with the stated gδ, so the claimed handling of the Hp8 exception is unsupported.","rationale":"The reader's weakest_assumption identifies (Hp8) and the strict decrease of Φ as load-bearing, and notes that the exponential law is handled only by a δ-family approximation. My stress-test agrees that the main reconstruction theorems are sound: the constructions in Theorems 3.1, 3.2, 3.4, and 3.5 do verify the needed hypotheses, including strict decrease of Φ through convexity of (ϑ∘Ψ^{-1})^{1/2}. However, the paper's treatment of the exponential exception contains a concrete internal inconsistency: the Rδ displayed in §3.3.6 is not the Abel datum of the linearized gδ defined in the same section. The correct Rδ for that gδ is linear on the tail, not constant, and the printed constant-tail Rδ corresponds to a non-C¹ cutoff law whose R is not convex, so (Hp8) fails. Consequently, the construction of ωδ in §3.3.6 does not establish the claimed reconstruction of the exponential law, and the double Γ-limit step is not justified as written. This is a significant defect in the examples that the paper uses to illustrate the method, even though the central theorems are unaffected. A conditional acceptance requiring correction of §3.3.6 is therefore appropriate; if the authors replace the incorrect Rδ with the correct one (or explicitly work with the cutoff law and adjust the regularity discussion), the verdict should become ACCEPT.","tokens_in":37859,"tokens_out":54803,"duration_ms":530423,"concrete_test":"Recompute Rδ for the linearized gδ in §3.3.6. With a=e^{−ksδ}, for t∈(1−a²,1] set y=(1−t)^{1/2}=a(1−k(s−sδ)); integrate gδ′ to obtain Rδ(t)=gδ(s)=(1/k)[1−a/2−(1−t)/(2a)] and compare with the printed constant 1/k(1−(1−ˆsδ)^{1/2}). Then check convexity on [0,1]: for the printed Rδ, Rδ′ jumps downward from a positive value to 0 at ˆsδ, so Rδ is not convex and does not satisfy (Hp8); for the linearized law, Rδ′ is continuous at the junction with value 1/(2ka) and the convexity holds. If the printed Rδ is instead used to define ωδ, reconstructing via Theorem 3.1 and computing the Γ-limit surface density will yield the non-C¹ cutoff law, not gδ.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The main Theorems 3.1–3.5 hang together and I have no objection to the reconstruction claim for laws satisfying (Hp8) in the linear case or (Hp6′)–(Hp7′) in the superlinear case. The load-bearing weakness is the treatment of the exponential law in §3.3.6, presented as the way to overcome the failure of (Hp8). The displayed Rδ is not the Abel datum of the stated linearized law gδ. For gδ′(s)=e^{-ks} on [0,sδ] and e^{-ksδ}(1−k(s−sδ)) on (sδ,sδ+1/k), one computes with y=(1−t)^{1/2}=gδ′(s) that on the tail t>1−e^{−2ksδ} one has Rδ(t)=(1/k)[1−e^{−ksδ}/2−(1−t)/(2e^{−ksδ})], a linear function of t. The printed Rδ is constant there, which is instead the Abel datum of the discontinuous cutoff law g′(s)=e^{−ks} for s≤sδ and 0 for s>sδ. That cutoff law is not C¹ (violating (Hp6)), and its R has a downward jump in R′ at the junction, so it is not convex and fails (Hp8). Thus the construction of ωδ in §3.3.6 does not reconstruct the stated gδ, and the double Γ-limit conclusion is unsupported as written; the displayed monotonicity gδ2≥gδ1 for δ1<δ2 also appears reversed. This does not undermine Theorems 3.1–3.5 themselves, but it invalidates the paper's claim to have handled the exponential exception.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a systematic inverse-design procedure for phase-field cohesive fracture models. Starting from the Gamma-convergence framework established in Part I, it derives a scalar characterization of the surface energy density g(s), analyzes it through an auxiliary function Phi, and then solves the reconstruction problem: given a target cohesive law g0, choose the phase-field ingredients (l, Q, omega, phi) so that the Gamma-limit of the energies F_epsilon has surface energy density exactly g0. The main theorems are Theorem 3.1 and 3.2 for laws that are linear at small jumps, and Theorem 3.4 and 3.5 for laws that are superlinear at small jumps; the proofs use Abel inversion with a convexity assumption (Hp8) on the auxiliary profile R. Several closed-form examples are worked out, including Dugdale, linear, bilinear, hyperbolic, quadratic hyperbolic, exponential, and logarithmic softening laws. The exponential law is treated by a delta-family approximation in Section 3.3.6 because it fails (Hp8).","tokens_in":38183,"tokens_out":5204,"duration_ms":61327,"significance":"If the main theorems are correct, this is a valuable and substantial contribution: it turns the heuristic calibration of phase-field cohesive models into a rigorous inverse problem, giving explicit formulas for the degradation function or the damage potential for a broad class of target laws. The approach is genuinely constructive rather than a parameter fit, and it places earlier results such as FFL21 and BI24 in a unified framework. The analysis of the relaxed scalar functional in Section 2, in particular the role of the function Phi and the Abel inversion, is mathematically substantial and carefully developed. The main caveats are that the reconstruction relies on the Gamma-convergence theorem imported from the unpublished Part I, and that the treatment of the exponential example in Section 3.3.6 contains an internal inconsistency that needs to be corrected. These issues do not by themselves undermine Theorems 3.1-3.5, but they affect the scope of the claims made for the exponential law.","major_comments":[{"comment":"The displayed R_delta is not the Abel datum of the stated law g_delta. For the stated linearized tail, with a=e^{-k s_delta} and y=(1-t)^{1/2}=g_delta'(s), one obtains on the tail t>1-a^2 the affine increasing expression R_delta(t)=(1/k)[1-a/2-(1-t)/(2a)]. The printed R_delta is instead constant on (s_hat_delta,1], which is the Abel datum of the discontinuous cutoff law g'(s)=e^{-ks} for s<=s_delta and g'(s)=0 for s>s_delta. That cutoff law is not C^1, so it violates (Hp6), and its R is not convex, so it does not satisfy (Hp8). Consequently the construction of omega_delta in Section 3.3.6 does not reconstruct the g_delta defined in the text, and the double Gamma-limit conclusion for the exponential law is unsupported as written. The section should either recompute R_delta and phi_delta for the genuine linearized tail, or state explicitly that the approximated law is the discontinuous cutoff and justify the Gamma-limit claim by another argument.","section":"Section 3.3.6"},{"comment":"The monotonicity assertion before the double Gamma-limit is reversed: for 0<delta_1<delta_2<1, the cutoff point s_delta is larger for the smaller delta, so one expects g_{delta_1}(s)>=g_{delta_2}(s) pointwise, not g_{delta_2}>=g_{delta_1}. The appeal to [DM93, Proposition 5.4] therefore uses the wrong monotonicity and should be corrected together with the R_delta computation.","section":"Section 3.3.6"},{"comment":"The main reconstruction theorems rely on Theorem 1.1 of [ACF25a], which is cited as 'submitted' and is not reproduced in this manuscript. Since the Gamma-convergence statement is load-bearing for the whole reconstruction procedure, the paper should either include a complete statement (and proof or precise reference) of the needed theorem, or the editor should confirm that Part I is available for the refereeing process. As it stands, the correctness of the reconstruction chain cannot be independently verified from the present text.","section":"Section 1.2 and Theorems 3.1-3.5"}],"minor_comments":[{"comment":"The word 'analitically' should be 'analytically'.","section":"Section 1.1"},{"comment":"There is a typo: 'there exists e sequence' should read 'there exists a sequence'.","section":"Section 2.2, proof of Proposition 2.6"},{"comment":"In the displayed estimate for I_j, the factor involving sup(ϑϑ')^{-1/2} should be written with parentheses for clarity; the intended bound is clear but the notation is compressed.","section":"Section 2.3, proof of Proposition 2.10"},{"comment":"It would be helpful to specify in the caption of part (f) that the two curves correspond to g and an approximating g_delta, and to indicate the ordering of delta used in the figure.","section":"Figure 2 and Section 3.3"},{"comment":"The closing remark that omega_delta coincides with the damage potential of [FFL21, Section 4.2] on [delta^{1/2},1] should be revisited after the R_delta correction, since the matching region depends on the correct Abel datum.","section":"Section 3.3.6"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is part of a three-part work, and its central reconstruction theorems depend on the Gamma-convergence result of Part I, which is only cited as submitted. If Part I is not available to the referees, this is a significant verification gap. The exponential example in Section 3.3.6 is a genuine internal inconsistency, but it is localized and fixable; the main theorems for laws satisfying (Hp8) or (Hp6')-(Hp7') appear sound. I recommend major revision rather than rejection, with the expectation that the authors correct the R_delta/phi_delta computation and the reversed monotonicity in Section 3.3.6."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorems here are legitimate and represent a real step beyond the existing literature. The Abel-inversion construction in Section 3, with Theorems 3.1–3.5, gives explicit recovery formulas for prescribed cohesive laws in both the linear and superlinear regimes, and the supporting analysis in Section 2 (scalar relaxation of Gm,s, monotonicity of Gs, and the role of Φ) is coherent and carefully argued. The comparison with FFL21 and BI24 is fair and specific. Six of the seven worked examples are consistent with the stated hypotheses.\n\nThe soft spot is Section 3.3.6, the exponential softening example, which is the one place the paper claims to handle failure of (Hp8). The stress-test note is right: the displayed Rδ is the Abel datum of a discontinuous cutoff law, not the stated linearized gδ. For the actual linearized law, Rδ(t) on the tail is linear in t, not constant. So the construction of ωδ does not reconstruct the stated gδ, and the double-Γ-limit conclusion is unsupported. The monotonicity statement \"gδ2 ≥ gδ1 for δ1 < δ2\" is also reversed. This is a genuine error, but it is contained in the example; the main theorems do not depend on it.\n\nTwo smaller issues: the paper leans on Part I's Theorem 1.1, cited as submitted, which makes independent checking harder; and (Hp8) is a real restriction, though clearly stated and reasonable for the intended class. Neither is a red flag by itself.\n\nWho should read this? Anyone working on phase-field fracture or variational models of cohesive fracture; the inverse-design procedure is the main contribution. It deserves a serious referee. My recommendation: send it to peer review, and make sure the referee scrutinizes §3.3.6. If the authors correct or retract the exponential-exception claim, the rest of the paper stands.","headline":"Strong inverse-design theorems for cohesive phase-field laws, but the exponential example in §3.3.6 contains a real Abel-datum error and should not be trusted as written.","tokens_in":38715,"tokens_out":6798,"would_cite":true,"duration_ms":73660,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49J45","74R10","74G65","35A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that any admissible cohesive traction–separation law can be realized exactly as the surface energy density of a phase-field model in the Γ-convergence limit.","keywords":["phase-field fracture","cohesive traction-separation law","Gamma-convergence","Abel integral equation","degradation function","damage potential","surface energy density","fracture mechanics"],"falsifier":"Fix a target law g0 satisfying the hypotheses, build ω and Q by the formulas of Theorem 3.1 for l(t)=$t^{2}$, and evaluate the infimum in (2.13) numerically on a fine grid of jump amplitudes s; any deviation from g0(s) beyond quadrature error would disprove the reconstruction. For the exponential law, compute φ(t)= (1/kπ)$\\cosh$^{-1}((1-t)^{-1/2}) and observe that it is unbounded at t=1; if a single phase-field model with continuous ω and l exactly reproduced the exponential law, that would falsify the claim that (Hp8) is needed.","tokens_in":37624,"feed_emoji":"🧩","tokens_out":6011,"duration_ms":66069,"temperature":0.7,"pith_summary":"This paper proves a constructive converse to Γ-convergence for cohesive fracture: given a target traction–separation law g0 of the type used in engineering practice, one can choose the phase-field ingredients so that the Γ-limit surface energy density is exactly g0, not merely close to it. The construction works for laws with either a finite initial slope (linear regime) or an infinite initial slope (superlinear regime), and it runs in two directions: fix the damage potential and solve for the degradation function, or fix the degradation and solve for the damage potential. The paper also maps out the boundary of what is achievable. If the auxiliary function controlling the minimization is non-decreasing, the only possible law is the Dugdale plateau; the purely exponential law falls outside the main theorem and is recovered only as a limit of approximating laws.","feed_headline":"Every admissible softening law becomes a phase-field model","feed_subtitle":"The paper's Abel-equation construction turns the target traction–separation curve into explicit model ingredients.","key_machinery":"The load-bearing object is the auxiliary function $\\Phi(x)=B(x,\\vartheta^{1/2}(x))$, which separates the two regimes of the relaxed scalar problem: when $\\Phi(m)\\ge s$ the minimizer is $W^{1,1}$, and when $\\Phi(m)<s$ it has a jump at the minimum point of the phase field. When $\\Phi$ is strictly decreasing, $g$ is $C^1$ and concave with $g'(s)=\\vartheta^{1/2}(\\Phi^{-1}(s))$, and the reconstruction reduces to solving the Abel integral equation $R(t)=\\int_0^t \\phi(\\tau)/(t-\\tau)^{1/2}\\,d\\tau$ for a strictly increasing, continuous $\\phi$, which then defines $\\omega$ through the relation $\\omega(1-t)=((\\vartheta^{1/2}(t))'\\phi(1-\\vartheta(t)))^2$ or equivalently defines $l$ through its inverse. Abel inversion is the technical core that turns the target law into explicit model ingredients.","core_discovery":"The central claim, stated as Theorems 3.1, 3.2, 3.4 and 3.5, is that for every admissible target law g0 satisfying (Hp6)–(Hp8) in the linear case or (Hp6')–(Hp7') in the superlinear case, there exist model ingredients l, Q, ω, φ such that the phase-field energies Fε in (1.3) Γ-converge to a functional whose surface energy density equals g0. The proof route is to characterize the surface energy density g(s) as the infimum of scalar one-dimensional energies, to express its derivative through the auxiliary function Φ, and then to invert the resulting Abel integral equation so that the target law fixes the damage potential or the degradation function. A companion theorem shows that if Φ is non-decreasing, every phase-field model of the class has a Dugdale-type surface energy, so non-monotone or strictly concave target laws require the strictly decreasing regime in which Φ is invertible.","pith_inferences":["If the construction is correct, experimental traction–separation curves that satisfy the hypotheses can be fed directly into the model-design process, and the same macroscopic law can be realized by many phase-field models that differ in the localized damage profile; which profile is physically selected would then be an additional modeling choice.","The obstruction examples suggest a general principle: laws whose auxiliary function $R$ is not convex or not $W^{1,p}$ for some $p>2$ force a trade-off between exact reproduction and regularity of the phase-field ingredients; such laws may still be approximated in a two-step $\\Gamma$-limit.","The one-dimensional reconstruction likely lifts to vectorial and higher-dimensional settings through the companion vector-valued theory, in which case the same Abel-equation recipe would give interfacial cohesive laws for curved cracks."],"forward_implications":["For linear-at-zero laws, prescribing $\\vartheta$ (equivalently the crack geometric function $l$) yields the damage potential $\\omega$ by an explicit formula, and prescribing $\\omega$ yields $l$ by an explicit inverse formula, both depending only on the Abel-inverted function $\\phi$.","In the superlinear case the same inversion produces models with $\\varsigma=\\infty$, reproducing laws with infinite initial traction, which are used to describe ductile fracture via strain-gradient plasticity.","Standard engineering laws—Dugdale, linear, bilinear, hyperbolic, quadratic hyperbolic, and logarithmic softening—admit closed-form reconstructions; the quadratic hyperbolic law is the boundary case where $R'$ lies in $L^p$ only for $p<4$.","The exponential law, whose Abel-inverted function is unbounded, cannot be reproduced with continuous ingredients by the main theorems; it is obtained as the $\\Gamma$-limit as $\\delta\\to0$ of linearized approximating laws, so the exact exponential law is a limit rather than a direct reconstruction."],"supporting_citations":[{"why":"Supplies the Γ-convergence theorem for the broad class of phase-field energies that the reconstruction procedure targets.","marker":"[ACF25a]"},{"why":"Introduced the integral relation and heuristic Abel-equation calibration for linear cohesive laws that this paper rigorously generalizes.","marker":"[FFL21]"},{"why":"Provides the Abel integral equation inversion theorems used to recover the damage potential and degradation functions.","marker":"[GV91]"},{"why":"Proposed the next-generation phase-field cohesive models with polynomial crack geometric functions that the reconstruction fits into.","marker":"[Wu17]"},{"why":"Established the variational phase-field model for cohesive fracture that is a special case of the framework analyzed here.","marker":"[CFI16]"},{"why":"Gives an alternative characterization of surface energy densities in one-dimensional cohesive fracture, compared with the paper's new scalar minimization.","marker":"[BCI21]"},{"why":"Prior results on regularity and differentiability of the surface energy density that the paper extends to the reconstruction setting.","marker":"[BI24]"},{"why":"Recent numerical and mathematical work on approximating arbitrary traction–separation laws, whose concavity condition is discussed in Remark 2.11.","marker":"[LCM25]"}],"fun_headline_variants":["Every softening law now yields its phase-field model","Abel inversion maps cohesive laws to phase-field ingredients","Reconstruct phase-field models from arbitrary softening curves","Prescribed cohesive law fixes phase-field via Abel equation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction depends on the target law's auxiliary function R being convex on [0,$ς^{2}$] and $W^{{1,p}}$ with p>2, because that is what makes the Abel-inverted function φ continuous, strictly increasing, and admissible; the purely exponential law fails this and is recovered only as a limit.","fun_headline_variants_meta":{"raw":{"variants":["Every softening law now yields its phase-field model","Abel inversion maps cohesive laws to phase-field ingredients","Reconstruct phase-field models from arbitrary softening curves","Prescribed cohesive law fixes phase-field via Abel equation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000139,"raw_usage":{"total_tokens":1131,"prompt_tokens":896,"completion_tokens":235,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":512,"completion_tokens_details":{"reasoning_tokens":174}},"tokens_in":512,"tokens_out":235,"duration_ms":3283,"temperature":1.0,"reasoning_tokens":174,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:53:13.341804+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix a target law g0 satisfying the hypotheses, build ω and Q by the formulas of Theorem 3.1 for l(t)=$t^{2}$, and evaluate the infimum in (2.13) numerically on a fine grid of jump amplitudes s; any deviation from g0(s) beyond quadrature error would disprove the reconstruction. For the exponential law, compute φ(t)= (1/kπ)$\\cosh$^{-1}((1-t)^{-1/2}) and observe that it is unbounded at t=1; if a single phase-field model with continuous ω and l exactly reproduced the exponential law, that would falsify the claim that (Hp8) is needed.","supporting_citations":[],"review_version":1}