Fatigue rate in the heterogeneous region of a welded joint

The authors say plainly that they left crack closure, welding residual stress and load history out of the model and will treat them in a separate paper, so this is their own deferred plan, and what it leaves unspecified is the comparator, the control condition and what would count as the correction making no difference.

Question

Does the parabolic heterogeneous-region model still beat the Paris law once crack closure and welding residual stress are measured and removed from the driving force?

Hypothesis

Replacing the applied stress intensity range with a closure-corrected effective range collapses the as-welded and stress-relieved heterogeneous-region curves onto one parabola.

Attempt type

wet-lab
Vocabulary note: None of the five values covers a mechanical fatigue test on metal. I used wet-lab because it is the only value that means a physical bench experiment, but nothing in this design is wet, biological or chemical. The field needs a value such as mechanical-test or bench-experiment.

Design

Cut E-CT specimens to the ASTM E647 geometry the source paper used, 10 mm thick, from the same flux-cored-arc weld-repaired Bisplate80 plate, with the notch placed so the crack runs through the heterogeneous region. Three arms: as-welded; post-weld heat treated to relieve residual stress; and as-welded with the residual stress profile along the crack path measured before testing. Run each arm at 25 kN and at 30 kN, stress ratio 0, haversine waveform, 5 Hz, room temperature, matching the source conditions. Record load against crack opening displacement periodically and take the crack opening force from the compliance offset procedure in ASTM E647 Appendix X2. Fit the parabola coefficients on the 25 kN stress-relieved data only, then predict 30 kN for every arm and score against measurement with the same correlation coefficient and mean squared error the source paper reports. The test is whether the arms agree once the effective range replaces the applied range, and whether the fitted quadratic coefficient stays negative.

Controls

• designated A post-weld heat treated specimen from the same plate, with the residual stress along the crack path measured and shown to be near zero before the fatigue test starts.
  Rules out: That the concave-down shape of the heterogeneous-region curve is produced by welding residual stress rather than by the change in fatigue properties across the weld.
  Vocabulary note: designated is the closest value, but in the vocabulary it means a designated positive control sample in an assay. Here the designated reference is a specimen condition, not a reagent or a known-active sample.
• known-positive-task Reproduce the published 25 kN as-welded growth curve for weld-repaired Bisplate80 from Zhang, van der Vyver and Hu 2011 on the new specimens before any new arm is scored.
  Rules out: That disagreement between arms comes from a different frame, specimen preparation or crack length measurement rather than from closure and residual stress.
• no-treatment The as-welded arm scored with the uncorrected applied stress intensity range, exactly as the source paper did.
  Rules out: That the closure and residual stress correction changes nothing, which would make agreement between the other arms trivial.
  Vocabulary note: no-treatment maps onto no stress relief and no closure correction only loosely. There is no treatment in the biological sense, only a specimen left in the condition the 2016 paper tested.
• within-subject Within each specimen, crack opening force is tracked against crack length, so early growth and late growth in the same specimen provide an internal comparison.
  Rules out: That a change in the measured opening level is specimen-to-specimen weld variation rather than residual stress redistributing as the crack advances, which Luan and colleagues report happens.
  Vocabulary note: A constant-amplitude fatigue test cannot carry two load amplitudes in one specimen, so this is a within-specimen comparison across crack length rather than across the condition being varied.

Sensitivity

This cannot be set from what is published, and saying so is the honest answer. The source paper tested one specimen per amplitude and reports no scatter band, no confidence interval and no power statement, so there is no variance estimate for log(da/dN) at a fixed stress intensity range. The only quantitative anchor is the margin the paper itself reports on the held-out 30 kN data, a mean squared error of 2.8964 x 10^-13 for the proposed model against 1.0295 x 10^-12 for the Paris model, about one order of magnitude, so a correction that moves the held-out error by much less than that margin is a null. Three things would settle a real sensitivity and none of them is in the record: the within-condition standard deviation of log(da/dN) from replicate as-welded specimens at 25 kN, the resolution of the compliance offset method for crack opening force on this specimen size, and the stated uncertainty of the residual stress measurement along the crack path.

Would count as

Supported → positive · opposed → refuted
Inconclusive if: The crack opening force cannot be separated from the compliance record because the offset sits inside the clip gauge noise, or the post-weld heat treatment changes the weld microstructure as well as the residual stress, in which case the comparison tests heat treatment rather than residual stress and decides nothing about the model.

Already done

• Plasticity-induced crack closure was identified and quantified as a shielding effect that lowers the effective driving force, and an effective range based on the crack opening level was proposed. (Elber 1971, ASTM Special Technical Publication 486, doi:10.1520/stp26680s)
  So: The correction the authors defer has been standard practice since 1971 and the source paper cites it without using it, so this design has to show the parabola carries information beyond a closure correction rather than merely re-deriving one.
• A loading protocol that changes the applied maximum and minimum loads as the crack grows can hold the total stress intensity range and the effective stress ratio constant, separating the residual stress effect from the microstructure effect on measured growth rate in welded compact tension specimens, without assuming any empirical growth law. (Xu, Bao, Liu and Zhang 2017, Fatigue and Fracture of Engineering Materials and Structures, doi:10.1111/ffe.12617)
  So: This is the protocol that separates the two effects the source paper lumps together, and the residual fluctuation it leaves is attributed to microstructure change, which is exactly the heterogeneity term the parabola claims to capture, so it should be an arm of this design rather than a footnote.
• Comparing as-welded against heat-treated specimens, with notch tips at gradually decreased distances from the weld centreline, showed that residual stress redistribution depends mainly on the distance from crack tip to weld centreline, that residual stress at the crack tip stays compressive through fatigue cycling, and that the contacted crack wake lengthens early in growth, so closure and compressive residual stress together lower the growth rate. (Luan, Zhang and Zhang 2023, Materials and Design, doi:10.1016/j.matdes.2023.112282)
  So: Chunguo Zhang is the first author of the 2016 paper, so the deferred work has already been done by the same group; the as-welded against heat-treated pair is the control design to reuse, and the mechanism it reports predicts the direction of the shape change this design would see.
• Redefining the crack opening stress intensity factor from the load against crack opening displacement curve shows that the residual stress effect on growth rate disappears once the minimum stress intensity factor exceeds the opening value, and that the opening value rises with the maximum stress intensity factor through cycling. (Luan, Zhang and Lu 2026, Fatigue and Fracture of Engineering Materials and Structures, doi:10.1111/ffe.70186)
  So: It supplies the measurable quantity and a stopping criterion: above that crossover there is nothing to correct, so the design must place at least one arm on each side of it or risk a null that only means the correction was not needed.
• A crack tip field model for welded structures that carries both welding residual stress and plasticity-induced closure has been formulated and validated. (Zhang, Yang and Wang 2024, Engineering Fracture Mechanics, doi:10.1016/j.engfracmech.2024.110297)
  So: It is a mechanistic competitor, so the parabola should be scored against it on the same held-out amplitude rather than only against the Paris law, which is the weakest available baseline.
• Welding residual stress in butt-welded joints relaxes towards an asymptotic distribution under fatigue loading rather than staying at its as-welded value. (Ferro, Berto and James 2016, Theoretical and Applied Fracture Mechanics, doi:10.1016/j.tafmec.2016.02.002)
  So: The load history part of this issue is not a static offset, so a residual stress profile measured once before testing is not the profile acting during growth, and the design needs either repeated measurement or an explicit relaxation assumption that it states.
• The standard test method for fatigue crack growth rates notes that residual stress and shielding effects such as closure can significantly influence interpretation, especially at low stress intensity and low force ratio, and gives a recommended practice for determining the fatigue crack opening force from compliance. (ASTM E647, Standard Test Method for Measurement of Fatigue Crack Growth Rates, Appendix X2, url:https://www.astm.org/standards/e647)
  So: The source paper already follows this standard for the E-CT geometry, so the closure measurement needs no new method, only the compliance records the 2016 tests did not report; which edition is current should be checked before a clause is cited.

Searched: Searched OpenAlex for crack closure and residual stress effects on fatigue crack growth rate in welded joints, and for effective stress ratio treatments of weld residual stress, both filtered to 2016 onwards; pulled the source paper's own reference list through OpenAlex; web searched for compliance-based crack opening measurement on welded specimens and for contour method or diffraction residual stress mapping on compact tension specimens, and found no study combining the two. The notable result is that the corresponding author's own group has published this deferred work twice since, in 2023 and 2026, so the issue is already part answered; I looked for and did not find any study testing whether the parabolic heterogeneous-region form survives a closure correction.

Feasibility

A structural integrity or welding laboratory with a servo-hydraulic fatigue frame, clip gauge instrumentation and access to a residual stress measurement facility. The Chang'an University and University of Western Australia groups behind the source paper already hold the weld procedure, the plate and the specimen geometry. · 9 to 18 months · moderate, because welding, machining and residual stress measurement dominate the cost while the fatigue frame time is long but not specialised
Blockers: Post-weld heat treatment changes the microstructure as well as the residual stress, so the two cannot be varied independently within one specimen and the stress-relieved arm is not a clean single-factor control; Residual stress measurement along the crack path is destructive by the contour method and slow by neutron or synchrotron diffraction, and beam time is competitive; The source paper tabulates no raw growth rate data, so the 2016 result can only be compared through figures digitised from the publication; Residual stress relaxes under cycling, so a single pre-test measurement may not describe the field acting during growth

Strongest objection

A fatigue specialist would say the premise is backwards. If the concave-down shape is largely a closure and residual stress artefact, the parabola has no separate heterogeneity content, and the right comparison is not parabola against Paris but closure-corrected parabola against closure-corrected Paris, with a fourth arm of parent plate away from the weld. The 2023 and 2026 results from the same group, which show compressive residual stress at the crack tip and a lengthening contacted crack wake both suppressing the rate, make that the likely outcome, so as written this design risks confirming that the 2016 model was fitting a shielding effect it never measured, and it would have been cheaper to say so from the literature.

Not derivable from what is published

• The number of replicate specimens per arm, because no within-condition scatter in log growth rate is published for this weld
• The resolution of the compliance offset method for crack opening force on a 10 mm E-CT specimen of this size, which sets whether a null is meaningful
• The stress intensity range over which the source paper declared the heterogeneous region to start and stop; the record does not say how that interval was bounded
• The post-weld heat treatment schedule that would relieve the residual stress without altering the weld microstructure of this particular flux-cored-arc repair
• Whether the Paris prediction in equation 12 was fitted to the 30 kN data or carried over from 25 kN, which the record flags as ambiguous and which decides whether the Paris baseline here is a prediction or a fit
• The units of the growth rate, the stress intensity range and the mean squared errors in the source equations, which the record notes are absent

The model is free of artificially adjustable parameters only in the sense that its three coefficients come from a curve fit rather than from tuning, so one real fatigue test at a known amplitude is still needed before any other amplitude can be predicted, and the open question is whether those three coefficients can instead be got from properties that are cheaper to measure.

Question

Can the three parabola coefficients be predicted from independently measured weld and material properties instead of from a fatigue test at one load amplitude?

Hypothesis

The three coefficients are a function of measurable properties such as tensile strength, fatigue limit and the hardness profile across the weld, so a joint never fatigue tested can be predicted.

Attempt type

computational
Vocabulary note: This is one study with two halves, a literature compilation and a regression on the compiled data, which map to evidence-synthesis and computational respectively. The vocabulary forces a single value. The schema's pathType has a mixed value but attemptType does not, so the mixture is lost.

Design

Build a table of heterogeneous-region growth curves for welded steel joints from the published literature, keeping only joints reported at two or more load amplitudes or stress ratios so that one can be held out. For each joint record whatever independent properties the source gives: parent and weld tensile strength, fatigue limit, hardness across the heat affected zone, weld process, plate thickness, joint configuration and notch position. Fit the parabola coefficients on the lowest available amplitude for each joint. Then regress the three coefficients on the recorded properties and test by leaving one joint out at a time: predict its coefficients from properties alone, predict its held-out amplitude curve from those coefficients, and score against measurement with the correlation coefficient and mean squared error the source paper uses. Compare three predictors on identical held-out points: the property regression, a direct fit at the low amplitude as the source paper did, and the corpus-mean coefficients.

Controls

• known-positive-task Run the source paper's own 25 kN to 30 kN prediction for weld-repaired Bisplate80 through the pipeline and check that it reproduces the reported correlation coefficient of 0.9316 and mean squared error of 2.8964 x 10^-13.
  Rules out: That a poor leave-one-out score comes from an error in the digitising, fitting or scoring code rather than from the properties failing to predict the coefficients.
• adversarial-label Permute the joint labels so each joint's measured properties are paired with another joint's fitted coefficients, then refit and rescore.
  Rules out: That the leave-one-out score is produced by the shape of the parabola family or by the narrow amplitude range rather than by any real link between properties and coefficients.
• background A predictor that ignores every property and uses the corpus mean of each coefficient, scored on the same held-out points.
  Rules out: That the property regression looks good only because welded steel joints all have similar coefficients, so that any constant would have scored as well.
• none No positive control on the coefficients themselves.
  Rules out: Nothing; this entry records an absence that the reader should hold against the design.
  Vocabulary note: There is no positive control in the assay sense available here, because no joint has a published, independently established set of true coefficients to recover. The known-positive-task entry above is the nearest substitute and it only checks the pipeline, not the quantity of interest.

Sensitivity

Partly derivable. The source paper's own held-out numbers set the bar a property-based predictor has to clear at a new amplitude: a mean squared error below 1.0295 x 10^-12 to beat the Paris law, and near 2.8964 x 10^-13 to match a direct fit at one amplitude. What cannot be derived is how many joints the corpus needs, because that depends on the between-joint variance of the three coefficients and no published source reports those coefficients for more than the three joints shown here, and those only as figures. A pilot that fits the parabola to ten or more digitised heterogeneous-region curves and reports the spread of each coefficient would settle it. For the sibling problem of stress ratio, the 2026 model from the same group reports a goodness of fit up to 0.98 on Q345qD steel plus 23 datasets, and that is the comparator to beat.

Would count as

Supported → positive · opposed → negative-no-association
Inconclusive if: Only a handful of published joints turn out to report both a heterogeneous-region growth curve at two amplitudes and the independent property measurements, so the leave-one-out error is dominated by digitising noise and the property regression cannot be told apart from the corpus mean either way.

Already done

• The growth rate follows a power law in the stress intensity range, with two constants fitted per material and condition. (Paris and Erdogan 1963, Journal of Basic Engineering, doi:10.1115/1.3656900)
  So: The baseline in the source paper is itself a two-parameter per-condition fit, so claiming freedom from adjustable parameters needs care: the honest comparison is how many fitted numbers each model needs per joint, which this design should report alongside the error.
• A growth law in which the stress ratio enters explicitly removes part of the need to refit for each load condition. (Forman, Kearney and Engle 1967, Journal of Basic Engineering, doi:10.1115/1.3609637)
  So: Sixty years of attempts to remove per-condition refitting means a new one has to be scored against the established forms, not only against the plain power law, or the result will not be believed.
• A modified Goodman mean-stress formulation yields an equivalent crack driving force that unifies stress intensity ranges at different stress ratios into an equivalent value at a stress ratio of 0 without introducing fitting parameters beyond the Paris constants, relying only on the fatigue limit and the tensile strength, with goodness of fit up to 0.98 across Q345qD steel and 23 datasets. (Wu, Zhang, Yang and Dong 2026, Applied Sciences, doi:10.3390/app16010547)
  So: This is the closest existing answer and it comes from the same corresponding author as the 2016 paper: it shows the route from basic properties to amplitude independence is viable, but it still needs the Paris constants from one dataset, so the part of this issue about eliminating the single calibration test is still open and this design should be framed against it rather than against the 2016 paper alone.
• An intrinsic driving force built from energy considerations and equivalent displacement accounts for stress ratio effects without relying on crack closure assumptions or closure measurements, and the ratio of stress intensity ranges at constant growth rate distinguishes single from multiple growth mechanisms. (Kujawski 2026, Fatigue and Fracture of Engineering Materials and Structures, doi:10.1111/ffe.70470)
  So: A second competitor that removes the per-condition refit by a different route, and its constant-growth-rate ratio gives a cheap diagnostic for whether a single mechanism holds across the heterogeneous region, which is worth running on the corpus before any regression.
• The UniGrow two-parameter driving force model predicts growth under spectrum loading from cyclic stress and strain properties rather than from growth data refitted per load history. (Mikheevskiy, Bogdanov and Glinka 2015, Theoretical and Applied Fracture Mechanics, doi:10.1016/j.tafmec.2015.06.010)
  So: It is the existing example of coefficients derived from independently measured material properties, so the property columns in this corpus should include the cyclic stress-strain quantities UniGrow uses, not only monotonic strength.
• UniGrow with plasticity-induced crack closure has been applied to S355 structural steel, a grade for which welded joint growth data are comparatively plentiful. (Pedrosa, Correia, Lesiuk and colleagues 2022, International Journal of Fatigue, doi:10.1016/j.ijfatigue.2022.107120)
  So: S355 is the one grade with enough data to support a leave-one-out test, so it should anchor the corpus even though the source paper's joints are higher strength.
• Several machine learning algorithms have been compared for calculating fatigue crack growth, establishing what a purely data-driven predictor achieves and roughly how much data it needs. (Wang, Zhang and Sun 2017, Materials, doi:10.3390/ma10050543)
  So: It sets the data-driven floor the property regression must beat to be worth its physical interpretation, and its data requirements are a warning about corpus size.

Searched: Searched OpenAlex for unified and parameter-free growth models that handle stress ratio or load history without refitting, for the UniGrow two-parameter driving force, and for machine learning prediction of growth rate across stress ratios, all filtered to 2015 onwards, and pulled the source paper's own reference list through OpenAlex to see which of these it already cites. The closest match is a 2026 paper from the same corresponding author that removes the stress ratio refit using only the fatigue limit and tensile strength, which answers the sibling question while still needing the Paris constants from one dataset. I searched for and did not find any work that predicts the parabola coefficients themselves from material properties, and no compilation that reports those coefficients for more than the three joints in this paper.

Feasibility

A structural integrity group, a standards committee working group, or any data holder with access to the welded joint fatigue literature and a figure digitiser. No new testing is needed for either half. · 4 to 8 months · low, because the work is literature extraction and regression, and the real cost is careful digitising and provenance rather than equipment
Blockers: Most published heterogeneous-region curves exist only as figures, so every point carries its own digitising error that propagates into the fitted coefficients; Papers that report growth rate rarely report the fatigue limit or a hardness traverse for the same joint, so the property columns will be sparse and the regression may be unidentifiable; A 2026 database paper reports that growth data for welded high strength steels are sparse compared with S355, especially outside air, so the corpus may be too small and too narrow in grade; Joints reported at two or more amplitudes with the same specimen geometry are a small subset of a small literature

Strongest objection

A reviewer would say the three coefficients are not physical quantities, so there is nothing for material properties to predict. In the source paper's own formulation they absorb the shape of whatever part of the heterogeneous region the specimen happened to sample, which depends on notch position, joint geometry and where the weakest site sits, not on bulk strength or hardness. If that is right the regression will fail for a reason that says nothing about materials, and the honest fix is to reparameterise the model in terms of the crack position relative to the weakest site before asking what predicts it, which is a different and larger piece of work.

Not derivable from what is published

• The between-joint spread of the three coefficients, which decides how many joints the corpus needs and therefore whether this is a study or an anecdote
• The values of the coefficients for the three joints in the source paper, which are shown only as fitted curves in figures and never tabulated
• Whether equation 11's coefficient of transformation, which the record says rests on uncited experimental data from welded joints, is itself a fourth fitted quantity; if it is, the model needs four numbers per joint, not three
• A standard definition of the heterogeneous region, without which coefficients fitted by different groups are not comparable across the corpus
• Digitising uncertainty for the published figures, which has to be propagated before any leave-one-out score can be read as a win or a null

The authors concede that what they fitted may be only a segment of a parabola, and that where the segment sits depends on the joint configuration, its dimensions and where the weakest point of the joint happens to be, so the full-parabola claim rests on three joints compared graphically.

Question

Beyond the three joints in this paper, is the heterogeneous-region growth curve reliably concave down, and does it beat the Paris and two-stage laws out of sample?

Hypothesis

The fitted quadratic coefficient is negative in most welded steel joints, and the parabola beats both the Paris law and the BS 7910 two-stage law on held-out load amplitudes.

Attempt type

evidence-synthesis
Vocabulary note: The corpus arm is evidence synthesis, but this design also has a prospective bench arm in which new specimens are cut with the notch at different distances from the weld centreline. No single value covers a study with both a synthesis arm and a laboratory arm, and splitting it into two attempts would break the shared pre-registered decision rule that makes it a test rather than two descriptions.

Design

Two arms. In the corpus arm, assemble every retrievable heterogeneous-region growth dataset for welded steel joints, recording joint configuration, plate thickness, notch position relative to the weld centreline and which zone the crack ran through. For each dataset fit three forms: the Paris law, the proposed parabola, and the two-stage bilinear law that BS 7910:2019 uses for welded steel. Score by prediction at a held-out load amplitude rather than by fit, and penalise the parabola's extra parameter. Pre-register the decision rule before fitting: the full-parabola claim stands only if the quadratic coefficient is negative and the fitted vertex falls inside the measured stress intensity range for a stated majority of joints. In the bench arm, cut E-CT specimens from one welded plate with the notch at three distances from the weld centreline, as Luan and colleagues did in 2023, so the crack samples different fractions of the heterogeneous region, and test whether the quadratic coefficient changes with notch position.

Controls

• background Parent plate Bisplate80 taken away from the weld, a homogeneous material where the Paris law is expected to hold, put through the identical fitting and scoring pipeline.
  Rules out: That the parabola wins everywhere because a quadratic absorbs ordinary scatter; if it also beats the Paris law on homogeneous material, the heterogeneity interpretation collapses.
  Vocabulary note: background in the vocabulary means an assay background signal. Here it is a material condition where the effect is expected to be absent, which is the same logic but not the same thing.
• known-positive-task The three joints the source paper already fitted, welded TMCP API 5L X65, EH36 TMCP and weld-repaired Bisplate80, must come out concave down through the same pipeline.
  Rules out: That a failure on new joints is caused by the digitising and fitting pipeline rather than by the joints themselves.
• adversarial-label Shift the declared boundaries of the heterogeneous region along the stress intensity axis by a fixed amount, in both directions, and refit all three forms.
  Rules out: That the parabola's advantage is an artefact of where the heterogeneous region was declared to start and stop, which the source paper appears to set by eye from the same curve it then fits.
• within-subject In the bench arm all three notch positions are machined from the same welded plate and the same weld pass.
  Rules out: That a change in the quadratic coefficient reflects plate-to-plate or weld-to-weld variation rather than how much of the heterogeneous region the crack traverses.

Sensitivity

The per-joint decision threshold is derivable from the source paper, which reports roughly one order of magnitude between the two models' held-out mean squared errors, 2.8964 x 10^-13 against 1.0295 x 10^-12, so a joint counts as a parabola win when the held-out error falls by about that much and as a null when it does not. The number of joints needed to support or refute a generality claim is not derivable, because the between-joint spread of the quadratic coefficient has never been reported: the three joints here are shown graphically with no coefficients given. A pilot that fits and reports the coefficient for ten digitised curves would set the required count. For the bench arm no sensitivity can be stated at all, because no study reports how much the quadratic coefficient moves per millimetre of notch offset, and that number would have to come from the pilot's first two specimens.

Would count as

Supported → positive · opposed → negative-not-replicated
Inconclusive if: Too few retrievable datasets carry two load amplitudes for the same joint, so the scoring falls back on goodness of fit, where the extra parameter wins by construction and the comparison stops being informative about model form.

Already done

• Microstructures and fatigue crack growth were characterised across EH36 TMCP steel weldments, with growth rate differing between weld zones. (Tsay and colleagues 1999, International Journal of Fatigue, doi:10.1016/s0142-1123(99)00021-3)
  So: One of the three joints the source paper refitted, so its raw data are the first entries in the corpus; the record also flags that the source paper may have attributed this dataset to the wrong reference number, which has to be resolved before reuse.
• Fatigue crack growth was measured in welded thermomechanically controlled processed API 5L X65 pipeline steel alongside sulfide stress corrosion cracking. (Tsay, Chen and Chan 2001, International Journal of Fatigue, doi:10.1016/s0142-1123(00)00081-5)
  So: The second of the three joints, and a reminder that environment was a variable in its original study, so the corpus must record test environment or it will mix air and corrosive data.
• Fatigue crack growth was characterised in weld-repaired high strength low alloy steel, the weld-repaired Bisplate80 specimen family and E-CT geometry that the 2016 paper reuses. (Zhang, van der Vyver and Hu 2011, Engineering Fracture Mechanics, doi:10.1016/j.engfracmech.2011.03.004)
  So: It is the third joint and the provenance of the bench arm's specimen design, and the same first author as the 2016 paper, so the corpus arm is partly a reanalysis of this group's own data and should say so.
• Microstructure and fatigue properties of welded high strength low alloy steel differ with and without a buffer layer in the weld, changing where the weakest material sits in the joint. (Zhang, Yang and Hu 2012, Materials Science and Engineering A, doi:10.1016/j.msea.2012.03.048)
  So: It is direct evidence that the weakest-point site the issue names is a design variable the welder controls, so buffer layer presence belongs in the corpus columns and would make a fourth notch condition worth adding to the bench arm.
• Growth rate changes as a fatigue crack approaches a weld interface in alloy and maraging steels, so the rate depends on crack position relative to the interface rather than on the stress intensity range alone. (Ukadgaonker, Bhat and Jha 2007, International Journal of Fatigue, doi:10.1016/j.ijfatigue.2007.05.005)
  So: It supports the issue's own worry that the curve shape is positional, which is why the bench arm varies notch distance rather than only adding more joints.
• A database of fatigue crack growth data for high strength steels extracted from the literature was built and analysed with single-stage and two-stage linear regression, the two-stage transition points of BS 7910:2019 were examined, and the analysis found a distinct lack of data in the corrosive offshore environment while suggesting existing growth laws may extend to higher strength steels. (Devine, Stiven, Asbury and colleagues 2026, Fatigue and Fracture of Engineering Materials and Structures, doi:10.1111/ffe.70457)
  So: It is both a ready-made corpus and the incumbent model form: the engineering standard answer to a non-linear growth curve is two straight lines with a transition, not a parabola, so that is the comparator the parabola has to beat, and this paper's own conclusion that existing laws may extend upward is evidence against needing a new form at all.
• Residual stress redistribution during growth depends mainly on the distance from the crack tip to the weld centreline, demonstrated with two groups of specimens whose notch tips sat at gradually decreased distances from the centreline. (Luan, Zhang and Zhang 2023, Materials and Design, doi:10.1016/j.matdes.2023.112282)
  So: It is the published template for the bench arm's notch-offset series, and it also means notch offset changes residual stress as well as how much heterogeneous region the crack samples, so the bench arm confounds the two unless it carries stress-relieved companions.

Searched: Searched OpenAlex for non-linear deviation from the Paris law in heterogeneous weld microstructure, for variation of growth rate across weld zones and the heat affected zone hardness gradient, for retardation and acceleration as a crack approaches the weld interface, for open compilations of welded joint growth data, and for model selection criteria applied to growth rate curves, mostly filtered to 2014 onwards; then web searched for any independent test of this paper's parabolic model. I found a 2026 literature-extracted high strength steel database that fits single-stage and two-stage linear forms rather than a parabola, and I searched for and did not find any study that has tested the parabolic form on joints beyond the three here. Two searches, on digital image correlation across weld zones and on model selection criteria for growth curves, could not be completed because the OpenAlex free daily quota for this network ran out.

Feasibility

A structural integrity group or a standards committee working group for the corpus arm; a welding laboratory with a servo-hydraulic frame for the bench arm. The two can be done by different groups if the decision rule is registered first. · 6 to 12 months for the corpus arm alone, 12 to 18 months with the bench arm · low for the corpus arm and moderate once the bench arm is added, because that needs new plate, welding, machining and frame time while the synthesis needs only digitising effort
Blockers: The heterogeneous region has no standard definition, so joints cannot be compared until the corpus adopts one and applies it uniformly, which may change the published conclusions; Datasets reported at two or more amplitudes with comparable geometry are scarce, which is what the decision rule's majority threshold depends on; Notch offset changes residual stress as well as the fraction of heterogeneous region sampled, so the bench arm needs stress-relieved companion specimens and roughly doubles; Digitising from figures across many papers introduces inconsistent error, and a two-stage law with a fitted transition point is sensitive to exactly that

Strongest objection

A specialist would object that the heterogeneous region is not objectively defined. The source paper appears to pick it by eye from where the curve bends, so a pre-registered rule that fits a parabola inside an interval chosen from the same curve is circular, and the adversarial shift control will probably just destroy the effect without telling anyone why. The design would then have to define the region from something independent of the growth data, such as a hardness traverse or a metallographic zone boundary, and that redefinition is a larger and different paper than a model comparison. A second objection is cheaper: engineering practice already handles a bending growth curve with two straight lines and a transition point, so a parabola has to earn its extra parameter against that, not against a single straight line.

Not derivable from what is published

• The majority threshold in the pre-registered decision rule, because no published spread of the quadratic coefficient exists to set it against
• How the heterogeneous region was bounded in the source paper and in each corpus entry, which the records do not state
• How much the quadratic coefficient shifts per millimetre of notch offset, so the bench arm's three positions are a guess until a two-specimen pilot reports it
• Which zone the crack actually ran through in most published datasets, since many report a growth curve without a fractography or a crack path map
• The transition point values BS 7910:2019 specifies for the steels in the corpus, which have to be read from the standard rather than inferred
• Whether the source paper's three literature fits used all the published points or a visually selected subset, since the comparison is graphical only