{"id":"2c58db48-9d0f-4d53-8e54-1569ba92efa9","arxiv_id":"2507.22072","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Closed-form phase-field cohesive fracture models are derived for linear, bilinear, exponential, hyperbolic, and Dugdale softening laws, with several distinct models sharing the same overall response.","lead":"This paper constructs phase-field damage models for five standard cohesive fracture laws and shows that different models can produce identical global force-displacement responses while having different damage profiles. It demonstrates how the authors' earlier mathematical framework translates into concrete engineering tools for simulating quasi-brittle fracture.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sec. 3.3 drops the admissible modification from [33] without error control; until the simplified exponential model (E) is checked against the exact modified law, the claimed reproduction of law (3.8) is unverified.","rationale":"The paper's engineering claim relies on two transfers: from Part II's construction theorem to the specific material functions, and from those functions to the mechanical response. The first transfer is not independently checkable because [32] and [33] are cited as submitted rather than available. The second transfer is analytical and self-consistent for most models, and the length-independence is plausible from the scaling of (2.16)-(2.23). The exponential law is the single case where the paper explicitly changes the data of the problem, namely the cohesive law itself, without a quantitative justification. Since the central claim is about exact reproduction of prescribed cohesive laws, an uncontrolled modification of one of the five laws is the most load-bearing weak point. This matches the reader's identified weakest assumption, including the exponential simplification. A conditional disposition is therefore appropriate: acceptance should require access to the companion papers and a quantitative check, or replacement, of the exponential simplification. I see no basis for rejection: the analytical derivations presented here are internally coherent, and the issue is localized rather than structural.","tokens_in":27497,"tokens_out":13075,"duration_ms":165716,"concrete_test":"Obtain Part II [33, Sec. 3.3.6], and with its exact modified exponential cohesive law and the associated admissible pair {l,omega} from Theorem 3.1, compute sigma(delta) from (2.16)-(2.25) using the parameters of Table 1. Repeat the same computation with the simplified functions (E), and compare the two traction-separation curves, reporting the sup-norm and pointwise relative error on the plotted range delta in [0, 0.15] mm. If the deviation exceeds 2% of the critical stress (or any pre-set tolerance), the assertion that the simplified exponential model 'is well captured' is not justified, and the example in Sec. 3.3 should use the exact modified law or be accompanied by a derived error bound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the displayed phase-field models reproduce prescribed cohesive laws exactly. In Section 3.3 the target is the exponential law (3.8), but the only admissible construction derived in [33] is for a 'slightly modified version' introduced to satisfy the hypotheses of [32, Thm. 1.1]. The paper then drops that modification and asserts, without a quantitative error estimate, that the simplified functions (E) 'well capture' the target response. This is the one place where the authors deliberately sever the connection to the proven construction. Neither the actual modified law nor the small parameter is exhibited, and hypotheses (Hp1)-(Hp8) are not checked for (E). Consequently, the exponential example is valid only up to an unquantified approximation, so the headline 'identical cohesive fracture responses' is not established for one of the five displayed laws. A related but secondary uncontrolled step is the numerical interpolation used to reconstruct l(alpha) for models (L1), (L2), (D1), and (D2) in Sections 3.1 and 3.5. The analytical length-independence elsewhere in the paper is internally consistent, but the exponential simplification is an additional, uncertified step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents one-dimensional mechanical responses of phase-field cohesive models constructed in Part II [33] for linear, bilinear, exponential, hyperbolic, and Dugdale traction-separation laws. It derives closed-form expressions for the material functions l and w, computes phase-field profiles, crack openings, and global responses through analytical formulas, and claims that the global response is independent of the internal length and that several distinct models produce identical cohesive responses. The central mathematical construction is delegated to the companion papers [32,33]; for the exponential law, an admissible modified version from [33] is replaced by the unmodified law (3.8) without a quantitative error estimate.","tokens_in":27845,"tokens_out":4149,"duration_ms":44360,"significance":"If the companion results and the exponential simplification are valid, the paper provides a practically useful catalog: exact (or near-exact) phase-field realizations of common cohesive laws with length-insensitive macroscopic response, and explicit evidence that phase-field profile evolution is not unique. The analytical formulas (2.16), (2.23), and (2.25) are internally consistent, and the claimed independence from the internal length follows from the scaling of the material functions. The paper also correctly identifies the non-monotone displacement jump in the Dugdale models and the associated snap-back response, which is a useful observation for engineering applications. However, the main claims are conditional on theorems stated only in the submitted companion paper [33], and the exponential example is not yet certified.","major_comments":[{"comment":"The exponential model (E) is the one example for which the construction in [33] is not used directly: the text states that [33, Sec. 3.3.6] treats a 'slightly modified version' of (3.9) needed to satisfy the hypotheses of [32, Theorem 1.1], and that neglecting this modification the response 'is well captured'. No error bound, no small parameter, and no explicit modified law are given, and hypotheses (Hp1)-(Hp4) and (Hp6)-(Hp8) are not checked for (E). Because the paper's headline claim is exact reproduction of a prescribed cohesive law, this uncontrolled approximation must be either removed (by using the modified law from [33]) or quantified (by an explicit estimate on the difference between the responses of (E) and law (3.8)).","section":"Sec. 3.3 and Eq. (E)"},{"comment":"The reconstruction of the functions {l,w} for every example is delegated to Theorems 3.1, 3.2, and 2.18 of the unpublished companion paper [33]. These theorems are not stated, and their hypotheses are only partially quoted: (Hp6)-(Hp8) are listed in Sec. 2.3, while (Hp1)-(Hp4) are cited by reference. As a result, the reader cannot verify that the displayed functions (L1), (L2), (L12), (B), (H1), (H2), (D1), and (D2) indeed correspond to the prescribed laws. The authors should state the relevant theorems and either verify the hypotheses for each example or include the verification in an appendix.","section":"Sec. 2.3 and Sec. 3"},{"comment":"For models (L1), (L2), (D1), and (D2), the function l(alpha) is obtained by numerical interpolation of an inverse that has no explicit analytic form, but the interpolation method and its error are not reported. Since the plotted traction-separation and global responses are computed through (2.16) and (2.23) using these interpolated functions, the claimed identity of the responses is only as accurate as the interpolation. The authors should specify the interpolation scheme and provide an error estimate or a convergence check with respect to the interpolation parameter.","section":"Secs. 3.1 and 3.5"}],"minor_comments":[{"comment":"The sentence 'the tree models (L1)-(L12) are describing the same cohesive fracture response' contains a typo: 'tree' should be 'three'.","section":"Sec. 3.1, first paragraph"},{"comment":"The phrase 'the linear traction-separation law within the mathematical framework (3.8)' refers to an exponential law; 'linear' appears to be a typo and should be corrected.","section":"Sec. 3.3, text before Eq. (3.9)"},{"comment":"The coefficient 0.170 in the expression for w(alpha) is introduced without derivation; it should be traced to condition (2.12) and the stated value k0 approx 0.386/k, and the computation should be shown explicitly.","section":"Sec. 3.4, Eq. (H1)"},{"comment":"The caption states 'for (L12) and ell = 10 mm' but also highlights displacement profiles for smaller internal lengths; please clarify which curves correspond to which value of ell.","section":"Fig. 9 caption"},{"comment":"The notation sigma-bar is used for both the critical stress in (2.17) and the argument of the limit stress function sigma(alpha-bar) in (2.16); the distinction should be made explicit to avoid confusion.","section":"Sec. 2.2, Eqs. (2.16) and (2.17)"}],"recommendation":"major_revision","confidential_remarks":"The paper's central claim depends substantially on two submitted companion papers [32,33]. If these papers are not yet accepted or available to the referees, the present submission is not verifiable in its current form; the editor may wish to obtain them or require the authors to state the essential theorems and hypotheses in an appendix. I did not find an internal inconsistency in the scaling argument for length independence; the main risk is verification rather than a fundamental contradiction. The exponential-law example in Sec. 3.3 is the clearest gap and should be fixed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a useful engineering-oriented demonstration of the framework built in Parts I and II, with genuinely new closed-form material functions for bilinear, exponential, hyperbolic, and Dugdale laws. The 1D computations are standard and internally consistent; the ℓ-independence is exactly what you'd expect from the scaling. But the paper's central claim—that these phase-field models reproduce prescribed cohesive laws—rests on theorems in the submitted-but-not-yet-available companions [32,33], and Section 3.3 deliberately drops an admissible modification without error control. Neither is fatal, but both need attention.\n\nWhat's good: the many-to-one message is clearly demonstrated. Taking three different (l,w) pairs that give the same linear cohesive response and showing they produce identical load-displacement curves while having very different phase-field profiles is a nice, concrete illustration of the flexibility of the construction. The closed-form exponential w(α) with cosh⁻¹ is new relative to [27]'s fitted polynomials, and the bilinear model appears genuinely absent from the literature. The paper is also honest about what is and isn't proven: it flags the exponential modification explicitly, which is more than many papers would do.\n\nWhere it's soft: the construction itself is entirely outsourced. A referee cannot check that (L1), (B), (H1) etc. actually satisfy hypotheses (Hp1)–(Hp8) without seeing [32,33]. That is acceptable for a trilogy, but the paper should be explicit that it is a companion-dependent demonstration, and the editors need to verify the companions exist and are consistent. Second, the exponential step in Sec. 3.3 is the one place where the authors consciously leave the proven path: the theorem requires a slightly modified law, and they then say the modification is negligible. No quantitative bound, no display of the modified law, no check of (Hp) for (E). This is an unverified approximation for one of five laws; it doesn't kill the paper, but it should be fixed or honestly reworded. Third, the \"interpolation functions\" used to reconstruct l(α) for (L1), (L2), (D1), and (D2) are unspecified. That's a reproducibility gap, though a minor one.\n\nWho should read it: engineers and applied mathematicians working on phase-field cohesive fracture who want ready-to-use material functions or want to understand how the Part II construction plays out in practice. It deserves a serious referee, with the explicit requirement that the companion papers be provided and the exponential simplification justified or replaced.","headline":"Useful closed-form phase-field cohesive models, but the core construction lives in unpublished companions and the exponential example carries an unquantified simplification.","tokens_in":28302,"tokens_out":2070,"would_cite":true,"duration_ms":22363,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["74R10","74G65","49J45"],"pacs":[],"model":"deepseek-v4-flash","headline":"Phase-field models built from a prescribed cohesive traction-separation law can look different internally yet deliver the same macroscopic fracture response, independent of the regularization length.","keywords":["phase-field fracture","cohesive zone model","traction-separation law","Γ-convergence","gradient damage","internal length","softening laws","one-dimensional bar"],"falsifier":"Take the model (E) for the exponential law, compute the traction-separation curve up to very large crack openings in the one-dimensional bar, and compare the area under $\\sigma(\\delta)$ with $G_c$ and the tail with the target exponential law: if the deviation grows beyond the small-parameter regime, the claim that the modification is negligible fails. Equivalently, run the same test for (L1) with the internal length set to a sizable fraction of the bar length; the claimed independence from $\\ell$ predicts no change in the global response, so a visible shift would falsify it.","tokens_in":27227,"feed_emoji":"🧩","tokens_out":6017,"duration_ms":65131,"temperature":0.7,"pith_summary":"This paper is the third part of a programme that turns prescribed cohesive traction-separation laws into phase-field fracture models with proven variational underpinnings. It works out, in a one-dimensional tensile bar, the mechanical response of models built for five standard softening laws: linear, bilinear, exponential, hyperbolic, and Dugdale. The main claim is that the global stress–displacement response reproduces the target cohesive law and does not depend on the internal length, even though the phase-field and displacement profiles differ between models and vary with that length. For the linear law, three different phase-field models are shown to give one and the same cohesive response. The paper's purpose is to show that the abstract construction of Part II yields directly usable engineering models.","feed_headline":"One cohesive law, three phase-field models, same fracture curve","feed_subtitle":"Construction recipes from the companion theory yield models whose softening response ignores the internal length.","key_machinery":"The central objects are the pair of material functions $(l(\\alpha), w(\\alpha))$ — the local phase-field dissipation potential and the auxiliary function defining the degradation through $g_\\ell(\\alpha) = 1/(1+(2G_c E/(\\ell \\bar{\\sigma}^2)) w(\\alpha)/l(\\alpha))$ — together with the closed-form parameterization of the response by the maximum phase-field value $\\bar{\\alpha}$. The limit stress $\\sigma(\\bar{\\alpha})=\\sqrt{K w(\\bar{\\alpha})/h_\\ell(\\bar{\\alpha})}$ and the crack opening $\\delta(\\bar{\\alpha})$ obtained by integrating the complementary degradation over the localization profile are independent of $\\ell$ whenever the degradation has the specific smooth form used throughout the paper. This is the mechanism that makes models with very different phase-field evolutions deliver the same traction-separation law: the construction theorems of Part II choose $(l,w)$ so that the parameterized curve $(\\delta(\\bar{\\alpha}), \\sigma(\\bar{\\alpha}))$ traces the target law, and the length cancels out.","core_discovery":"For each of the five considered cohesive laws, the reconstruction procedure of Part II produces phase-field energy functionals whose one-dimensional uniaxial response coincides with the prescribed traction-separation law. The paper exhibits explicit material functions for each law: for the linear law three distinct triples (L1), (L2), and (L12), and for Dugdale two models (D1) and (D2). Through the parameterization by the maximum phase-field value, it shows that the limit stress and the crack opening are independent of the internal length $\\ell$; only the width of the phase-field localization is controlled by $\\ell$. The surface fracture energy recovered from the model has area $G_c$ matching the fracture toughness, with the expected exception that Dugdale's law produces a snap-back response in which the dissipated energy exceeds $G_c$. The paper also notes that a linear term in the local dissipation potential is sufficient but not necessary for an explicit elastic stage.","pith_inferences":["The multiplicity result suggests a design principle: within the family of models realizing a given cohesive law, one can search for a member that satisfies side constraints such as monotone growth of the localization support, which is exactly the route the paper proposes for enforcing irreversibility.","The length independence shown in one dimension plausibly explains why the smooth degradation form avoids the numerical regularization and $\\ell$-dependent critical stress reported for truncated-degradation models; a direct finite-element comparison would test this transfer.","For the exponential law, dropping the small-parameter modification is stated without a quantitative error bound; a targeted numerical check of the tail of $\\sigma(\\delta)$ would reveal how large an opening is needed before the deviation matters.","The Dugdale construction shows the method can handle traction-separation laws whose crack opening is non-monotone in the phase-field variable, which may extend to other plateau-type or snap-back laws."],"forward_implications":["The global response of each constructed model reproduces the target cohesive response, so the internal length can be chosen for mesh resolution without altering the macroscopic softening curve.","Because many phase-field models realize the same cohesive law, a model can be selected for auxiliary properties: for example, (L12) has a constant finite localization support and thus automatically satisfies the irreversibility condition, while (L2) has unbounded support.","Closed-form material functions are available for the exponential law, avoiding the fitted polynomials used previously; cohesive forces never vanish because $w(\\alpha)\\to\\infty$ as $\\alpha\\to1$.","A phase-field model for a bilinear softening law is constructed, and its response is again independent of the internal length.","For Dugdale's law the construction yields a snap-back response: at a critical opening the phase-field jumps to its fully damaged profile, and the dissipated energy exceeds $G_c$ although the area under the traction-separation curve is still $G_c$."],"supporting_citations":[{"why":"Supplies the construction theorems (3.1, 3.2, 2.18) that define the material functions $(l,w)$ for each assigned cohesive law.","marker":"[33]"},{"why":"Supplies the $\\Gamma$-convergence result that justifies the phase-field functionals as approximations of the cohesive fracture energy.","marker":"[32]"},{"why":"Provides the integral relation linking degradation and dissipation to the traction-separation law that this work extends and validates.","marker":"[29]"},{"why":"Provides the earlier polynomial/rational phase-field cohesive models and the exponential law reference considered in the examples.","marker":"[27]"},{"why":"Supplies the one-dimensional analytical formulas for limit stress, crack opening, and localization profile used throughout Section 2.2.","marker":"[40]"},{"why":"Supplies the gradient-damage interpretation, crack density function, and the normalization condition used to define the engineering model.","marker":"[21]"},{"why":"Provides the earlier rigorous variational cohesive phase-field model and the first treatment of the Dugdale reconstruction problem.","marker":"[25]"}],"fun_headline_variants":["Distinct phase-field models reproduce the same cohesive fracture","Same fracture curve from different phase-field constructions","Internal length only sets localization width, not fracture stress","Multiple phase-field models share one cohesive response"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the construction theorems and hypotheses of the two companion papers are correct and apply to these five laws, and that for the exponential law the small modification made to satisfy the hypotheses can be neglected without affecting the response.","fun_headline_variants_meta":{"raw":{"variants":["Distinct phase-field models reproduce the same cohesive fracture","Same fracture curve from different phase-field constructions","Internal length only sets localization width, not fracture stress","Multiple phase-field models share one cohesive response"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000572,"raw_usage":{"total_tokens":2665,"prompt_tokens":867,"completion_tokens":1798,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":483,"completion_tokens_details":{"reasoning_tokens":1739}},"tokens_in":483,"tokens_out":1798,"duration_ms":15659,"temperature":1.0,"reasoning_tokens":1739,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:29:52.601858+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the model (E) for the exponential law, compute the traction-separation curve up to very large crack openings in the one-dimensional bar, and compare the area under $\\sigma(\\delta)$ with $G_c$ and the tail with the target exponential law: if the deviation grows beyond the small-parameter regime, the claim that the modification is negligible fails. Equivalently, run the same test for (L1) with the internal length set to a sizable fraction of the bar length; the claimed independence from $\\ell$ predicts no change in the global response, so a visible shift would falsify it.","supporting_citations":[{"cited_title":"Alessi, F","cited_arxiv_id":null,"evidence_quote":"Supplies the construction theorems (3.1, 3.2, 2.18) that define the material functions $(l,w)$ for each assigned cohesive law."},{"cited_title":"Alessi, F","cited_arxiv_id":null,"evidence_quote":"Supplies the $\\Gamma$-convergence result that justifies the phase-field functionals as approximations of the cohesive fracture energy."},{"cited_title":"Wu, A unified phase-field theory for the mechanics of damage and quasi- brittle failure, Journal of the Mechanics and Physics of Solids 103 (2017) 72–","cited_arxiv_id":null,"evidence_quote":"Provides the earlier polynomial/rational phase-field cohesive models and the exponential law reference considered in the examples."},{"cited_title":"Pham, J.-J","cited_arxiv_id":null,"evidence_quote":"Supplies the one-dimensional analytical formulas for limit stress, crack opening, and localization profile used throughout Section 2.2."},{"cited_title":"Marigo, C","cited_arxiv_id":null,"evidence_quote":"Supplies the gradient-damage interpretation, crack density function, and the normalization condition used to define the engineering model."},{"cited_title":"Phase field approximation of cohesive fracture models","cited_arxiv_id":"1405.6883","evidence_quote":"Provides the earlier rigorous variational cohesive phase-field model and the first treatment of the Dugdale reconstruction problem."}],"review_version":1}