Momentum-Transfer Framework Unifies High-Velocity Impact and Failure Across Materials, Geometries, and Scales

Ara Kim; Jizhe Cai; Nicholas Jaegersberg; Ramathasan Thevamaran; Yasara Dharmadasa

REVIEW 5 major objections 5 minor 4 references

Momentum-Transfer Framework Unifies High-Velocity Impact and Failure Across Materials, Geometries, and Scales

T0 review · 5 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Momentum transfer peaks at the ballistic limit in all high-velocity perforation data, unifying materials, geometries, and scales.

desk verdict A serious momentum-based re-reading of ballistic perforation with strong new micro-projectile data, but the universality claim currently leans on literature v50 values that may have been generated by the very curve fits the paper criticizes. read the letter →

arxiv 2510.26360 v3 pith:JPCA6CO2 submitted 2025-10-30 physics.class-ph cond-mat.mtrl-sci

Yasara Dharmadasa , Nicholas Jaegersberg , Ara Kim , Jizhe Cai , Ramathasan Thevamaran This is my paper · ORCID

classification physics.class-phcond-mat.mtrl-sci

keywords momentumtransferballisticlimitperforationspecificenergyabsorptionhigh-velocityimpactscalingcapacityterminalballistics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that high-velocity impact perforation is governed by momentum transfer, not energy absorption. The central discovery is a universal bound: in every perforating impact, the momentum transferred to the target, normalized by the momentum the projectile has at the ballistic limit, stays below one. That means momentum transfer reaches its maximum exactly at the ballistic-limit velocity. The claim is supported by hundreds of micro-projectile and macroscale experiments plus 642 cases from prior studies spanning metals, polymers, composites, sandwich panels, and concrete. If true, it gives a physically grounded definition of the ballistic limit as a target's momentum capacity, explains why specific energy absorption inflates thin targets, and supplies a fairer cross-scale comparison metric.

What carries the argument

The central object is normalized momentum transfer ΔP̃ = (v_i - v_r)/v_bl, where ΔP = m_p(v_i - v_r) and P_bl = m_p v_bl. Normalizing every impact by its own ballistic-limit momentum collapses arrested cases onto ΔP̃ = ṽ_i and organizes perforated cases beneath the universal bound ΔP̃ < 1. Substituting the equivalent residual-velocity inequality v_r > v_i - v_bl into the energy-loss definition produces the energy envelope Ẽ_a < 2ṽ_i - 1, which appears in dimensional form as the tangent line E_a = ΔP_bl v_i - E_bl. A second construction, the inertial minimum ΔP̃_min = ζ ṽ_i with ζ = m_plug/(m_plug + m_p), accounts for rising momentum-transfer trends at high velocities and marks the regime

What would settle it

Find one perforation dataset with directly measured ballistic limit and paired incident/residual velocities in which v_i - v_r exceeds v_bl, equivalently ΔP̃ > 1; the paper reports none. Equally decisive would be an audit showing that the compiled literature ballistic-limit values are outputs of fitted curves of the form v_r = (v_i^p - v_bl^p)^(1/p) rather than independent measurements, which would transfer the bound from physics to curve-fitting convention.

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Extended reading notes

Core claim

The paper's central claim is that normalized momentum transfer, (v_i - v_r)/v_bl, is always less than 1 for perforating impacts, so the maximum momentum a target can absorb from a given projectile occurs at the ballistic limit. The equivalent inequality v_i - v_r <= v_bl means the ballistic limit sets the maximum possible velocity reduction in any single perforation event. Converting this momentum bound into energy space yields the envelope E_a < ΔP_bl v_i - E_bl, or Ẽ_a < 2ṽ_i - 1, so the energy absorbed at perforation is bounded but not fixed by the ballistic-limit energy. The paper consequently redefines the ballistic limit as a momentum-capacity threshold rather than an energy threshol

Load-bearing premise

The universal bound is only as universal as the ballistic-limit velocities used to normalize each case; if many literature values were produced by standard residual-velocity curve fits instead of direct measurement, the inequality could be baked into the fitting equation rather than discovered in the data.

Editorial extensions

If this is right

Since v_i - v_r <= v_bl, no projectile can be slowed by more than the ballistic-limit velocity in a single perforating impact, so raising v_bl becomes the primary route to improved perforation resistance.
The energy bound E_a < ΔP_bl v_i - E_bl means perforated impacts can absorb either more or less energy than at the ballistic limit; energy therefore cannot serve as the defining quantity for the ballistic limit.
Specific momentum capacity ΔP_bl* is proposed as a velocity- and layer-invariant metric, making it a fairer basis for comparing thin films, bulk plates, composites, and concrete than specific energy absorption.
For layered targets, stacking n identical layers raises the total energy-absorption bound linearly with n while lowering specific energy absorption by (n-1)E_bl*, so energy-based rankings can reverse the actual performance of layered designs.
The momentum-centered perspective is extended to other strongly dissipative impact processes, including micrometeoroid shielding, cold spray, shot peening, and particle abrasion, where energy-based descriptions may obscure the governing physics.

Reading between the lines

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

If many of the 642 compiled literature ballistic-limit values were obtained from standard residual-velocity curve fits rather than direct measurement, the apparent universality could be inherited from the fitting formula; re-analyzing those cases using raw v_i-v_r pairs would settle this.
The paper's own inertial model permits ΔP̃ > 1 when v_i exceeds v_bl/ζ; a targeted experiment with a very light projectile and a thick, low-density target could test whether the bound is an exact law or a practical low-velocity theorem.
Framing the ballistic limit as a momentum capacity suggests measuring ΔP_bl* across material families as a function of wave speed, toughness, and thermal softening, potentially producing a constitutive momentum-capacity map analogous to fracture-toughness charts.
A direct re-ranking of published thin-film energy-absorption records using specific momentum capacity would test the paper's claim that many nanomaterial performance advantages are geometric inflation rather than genuine material enhancement.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear X,

You should know two things about arXiv:2510.26360. First, the paper is a serious attempt to replace energy-based comparisons of ballistic perforation with a momentum-based one, and it brings a substantial amount of new micro-projectile data to bear. Second, the central 'universal bound' claim is plausible but not yet proven as broadly as stated, because the literature portion of the evidence may inherit the bound from the fitting models used to obtain v50.

What is actually new: the authors show across their own LIPIT experiments (polystyrene films, three D/h ratios, three projectile diameters) and matching macroscale gas-gun tests that normalized momentum transfer peaks at the ballistic limit. The translation of that bound into the energy envelope Ẽ_a < 2ṽ_i - 1 is elegant, and the proposed specific momentum capacity ΔP*_bl = γ v_bl is a genuinely useful comparative metric that avoids the geometry inflation of specific energy absorption. The multilayered-target analysis—showing that layering lowers specific energy absorption even when each layer performs identically—is a nice concrete demonstration of why energy-based metrics mislead. Credit is due: the paper cites Hetherington's earlier steel result and is candid about the limits of Recht–Ipson and Lambert–Jonas fits in Supplementary Note 11.

The soft spots are real, though not fatal. The collapse of arrested cases onto ΔP̃ = ṽ_i and Ẽ_a = ṽ_i² is definitional once data are normalized by v_bl; it is not independent evidence. That part can be dropped or clearly labeled as normalization. The bigger issue is the 642 literature cases in Fig. 3a. Many of the cited studies report v50 from Lambert–Jonas or Recht–Ipson fits, whose functional form v_r = (v_i^p - v_bl^p)^(1/p) mathematically enforces v_i - v_r ≤ v_bl. If those v50 values are fit parameters, then Fig. 3a partly confirms the fitting model rather than discovering a physical bound. The authors need to disclose, case by case, how each ballistic-limit velocity was determined, and where possible plot raw residual-velocity data. The same ambiguity affects their own v_bl estimates: the methods section describes velocity measurement but not how v_bl was extracted from the data. If it came from fitting a residual-velocity curve, circularity enters the own-data collapse as well.

The paper is not incoherent; the central observation for the authors' own experiments is likely correct. But the 'universal' framing outruns the evidence until the provenance of the ballistic-limit values is documented.

Who should read it: anyone working on ballistic impact, armor design, or LIPIT testing. It deserves a serious referee, but the authors should be asked to provide per-case data provenance and to tone down the universality claim accordingly.

Referee Report

5 major / 5 minor

Summary. The paper proposes that in high-velocity perforation, the momentum transferred to the target, normalized by the projectile momentum at the ballistic limit, never exceeds unity: ΔP̃ = (v_i − v_r)/v_bl < 1 for all perforating impacts. The authors support this with LIPIT experiments on polystyrene thin films over a range of projectile diameters, target thicknesses, and velocities, plus macroscale gas-gun tests, and a compilation of 642 literature cases across metals, polymers, composites, and concrete. They further argue that this momentum bound translates into an energy-absorption envelope, Ẽ_a < 2ṽ_i − 1, and propose 'specific momentum capacity' ΔP*_bl = γ v_bl as a more physical performance metric than specific energy absorption. The paper also critiques traditional Recht–Ipson and Lambert–Jonas models for embedding assumptions that make them unsuitable for interpreting energy trends.

Significance. If the central claim is established with truly independent data, it would be a notable unification: a simple, parameter-free inequality governing all ballistic penetration events, with practical implications for armor design and cross-scale comparison. The authors' own LIPIT data and the transparent derivation of the energy bound are strengths, as is the explicit recognition that arrested-case collapse is definitional. However, the universality claim currently rests on a literature compilation whose v50 values may have been obtained from fitting models that mathematically enforce the same bound, making the central empirical discovery vulnerable to circularity. The paper's proposed metric (specific momentum capacity) is interesting and could be useful, but its value depends on the solidity of the underlying bound.

major comments (5)

[§3, Fig. 3a] The universality claim is potentially circular. If the compiled v50/v_bl values were obtained from Recht–Ipson or Lambert–Jonas fits, then v_r = (v_i^p − v_bl^p)^(1/p) mathematically enforces v_i − v_r ≤ v_bl, which is exactly ΔP̃ ≤ 1. Supplementary Note 11 criticizes these fits, but the manuscript gives no evidence that the 642 literature cases avoided them. Many of the cited sources (e.g., Børvik, Dey, Holmen, Forrestal) commonly report v50 from such fits. Without a provenance table for each dataset's v50 determination, Fig. 3a may be confirming the fitting model's shape rather than discovering a physical bound. This is load-bearing for the universality claim.
[Methods and §2 (Fig. 1e)] The determination of v_bl for the authors' own experiments is not specified. The text states 'ballistic limit velocity ... is the maximum arresting v_i' but does not explain how this is extracted from the data — whether from direct observation of arrested/perforated boundaries or from a curve fit. If a Recht–Ipson-like fit was used, the own-data collapse in Fig. 2b would also inherit the bound. The manuscript should state the exact procedure, include the raw v_i–v_r data for all configurations, and report uncertainties in v_bl.
[Supplementary Note 4, Eqs. (9)–(10)] The collapse of arrested cases onto ΔP̃ = ṽ_i and Ẽ_a = ṽ_i² is definitional: for v_r = 0, the normalized definitions reduce identically to these relations. It therefore provides no independent evidence for the framework's predictive content. The paper should explicitly separate this definitional collapse from the substantive observation that perforated cases lie below ΔP̃ = 1. Currently, the text presents the arrested-case collapse as a supporting result, which overstates its evidentiary weight.
[§3, Eq. (16) and Fig. 3b] The energy bound Ẽ_a < 2ṽ_i − 1 is derived directly from the momentum bound. Consequently, it inherits any circularity in the momentum bound. If the literature v50 values are fit-derived, the energy-bound 'envelope' is similarly model-imposed rather than a new observation. The paper should clarify that the energy bound is a mathematical consequence and that its empirical support is no stronger than the independent measurement of v_bl.
[Fig. 3a, Data availability] The central claim rests on 642 compiled literature cases, but the compiled dataset and per-dataset provenance are not provided; 'data available from the corresponding author upon request' is insufficient for a claim of universal behavior. To allow verification of the no-violation claim and the v50 provenance, the authors should deposit the full dataset (material, geometry, v_i, v_r, v_bl, and source) as supplementary material.

minor comments (5)

[Main text, near Fig. 2c] Typo: 'saturation in energy absorption is communed with thermally softened' should likely be 'accompanied by'.
[§4 (minimum inertial momentum)] In the paragraph discussing the inertial bound, the phrase '(ΔP̃_min < 1) when ṽ_i > 1/ζ' appears algebraically inconsistent; the correct condition is ΔP̃_min > 1 when ṽ_i > 1/ζ. Please correct.
[Affiliation 2] 'F AMU-FSU College of Engineering' appears to have a spacing error; should be 'FAMU-FSU'.
[Fig. 2 and Fig. 3] The manuscript does not show error bars for the literature data in Fig. 3a. At least stating that error bars are omitted for clarity — or reporting them for a subset — would improve transparency.
[Supplementary Note 2] The air-drag correction assumes a known drag coefficient and spherical particles. The sensitivity of the extracted v0 to the drag-coefficient model should be reported, as systematic errors in v_i and v_r could influence the normalized bound.

Circularity Check

1 steps flagged · score 5.0 of 10

Literature v50 values may stem from Recht–Ipson/Lambert–Jonas fits, making the reported universality of ΔP̃<1 partly circular rather than empirically discovered.

fitted input called prediction [Fig. 3a and 'Unifying bounds and reimaging the ballistic limit using momentum transfer'; Supplementary Note 5 Eq. (12); Supplementary Note 11]
"A broad survey of independent studies across metals, polymers, composites, sandwich panels, and even reinforced concrete slabs reveals the same behavior: the momentum transferred to the target peaks at the ballistic limit and never exceeds the ballistic-limit value ... We did not encounter a single perforation dataset in which the momentum transfer exceeded the ballistic-limit value ... The figure includes 234 impact cases from the present study and 642 cases compiled from the literature."

Normalization of the 642 literature cases uses v50/v_bl values whose provenance is not disclosed. The paper itself (Supp. Note 11) gives the Recht–Ipson/Lambert–Jonas residual-velocity forms v_r=(v_i^p−v_bl^p)^{1/p} and notes these 'prescribe' v_r–v_i behavior. For any such fitted curve, v_i−v_r≤v_bl automatically (for p≥1), which is exactly the paper's Eq. (12) and hence ΔP̃≤1. If the compiled v50 values were obtained from such fits, Fig. 3a's 'universal bound' is an identity of the fitting model, not an empirical discovery. The paper reports no check that the 642 values were measured thresholds. Its own LIPIT v_bl is a measured threshold, so the authors' polystyrene data are independent; the circularity is limited to the load-bearing cross-study universality claim.

full rationale

The paper's own LIPIT and gas-gun experiments are not circular: v_bl is treated as a measured arresting threshold, and the observation that all perforated impacts satisfy ΔP̃<1 is an empirical statement about those data. The same holds for the derived energy bound Ẽ_a<2ṽ_i−1, which is merely an algebraic transformation of the momentum bound. The main circularity risk lies in the literature compilation that supports the paper's universal claim. The paper compiles 642 cases from 15 independent studies and normalizes them by v50/v_bl, but it gives no evidence that those v50 values were experimentally measured rather than obtained from Recht–Ipson or Lambert–Jonas fits. Supplementary Note 11 explicitly states that these fitting forms prescribe the residual-velocity curve and that using them to infer energy evolution 'leads to circular reasoning.' If the literature v50 values came from such fits, the plotted inequality ΔP̃<1 is mathematically guaranteed by the fitting form, so Fig. 3a would largely confirm the fit rather than discover a new universal physical bound. Since the paper's own data are independent but the headline cross-material, cross-scale universality depends on this unverified input, a moderate partial-circularity score is appropriate.

Assumptions & free parameters 2 free parameters · 4 assumptions · 2 invented entities

The central empirical bound for the authors' own data rests on careful velocity and mass measurements, which carry documented uncertainties. The universal claim, however, rests on external v50 values whose provenance is not documented, and on the assumption that those values are not already constrained by fitting models. The proposed metric introduces a new normalization but no new physical entity with independent evidence beyond its definition.

free parameters (2)

Literature ballistic-limit velocities v50/v_bl = per study, not tabulated in this paper
Used as normalization constants for the 642 compiled literature cases. If these values were obtained from Recht-Ipson or Lambert-Jonas fits, the functional form already enforces the momentum bound, making the universality claim inherited rather than measured.
Air-drag fitting parameter v0 = per projectile trajectory
In Supplementary Note 2, v0 is obtained as a fitting parameter when matching measured positions to the drag model; this is a measurement calibration, but it affects all incident and residual velocities used in the central claim.

assumptions (4)

domain assumption Air-drag correction assumes a constant drag coefficient C_D for smooth spheres over the full velocity range; closed-form solution Eqs. (5)-(6) treats B = C_D ρ_air A / 2m_p as constant.
Used to extract incident and residual velocities in Supplementary Note 2. If C_D varies significantly across 100–1000 m/s, the extracted velocities and hence v_bl could shift.
ad hoc to paper The compiled literature ballistic-limit velocities are independent experimental values, not curve-fit intercepts whose functional form already enforces ΔP ≤ m v_bl.
Supplementary Note 11 shows Recht-Ipson and Lambert-Jonas fits prescribe v_r-v_i curves satisfying the bound; the paper gives no evidence that the 642 literature cases avoided this, yet the universality claim rests on it.
domain assumption The ideal plug mass m_plug = ρ_t π D^2 h /4 represents the target mass participating in minimum inertial momentum transfer.
Used in Supplementary Note 7 for ΔP_min and in the specific momentum capacity metric. The actual deformed region of influence is estimated but not directly measured.
domain assumption Residual velocities of perforating projectiles are nonnegative and no rebound or projectile-reversal cases are considered.
The bound ΔP ≤ m v_bl is only meaningful for forward-perforating cases; the paper does not discuss whether ricochet or rebound could produce negative v_r and violate the framing.

invented entities (2)

Critical material point
purpose: Conceptual failure-threshold point used to rationalize why momentum transfer drops sharply just after the ballistic limit.
This is an explanatory device, not directly measured; the paper does not identify the point experimentally.
Specific momentum capacity ΔP*_bl = γ v_bl independent evidence
purpose: Proposed intrinsic performance metric separating imposed geometry (γ) from emergent material response (v_bl).
Although introduced as a new metric, it is computable from measured v_bl and plug/projectile masses, so it has a falsifiable handle: different targets can be ranked and the ranking can be tested against new perforation data.

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Cite this review

Pith. "Pith review of Momentum-Transfer Framework Unifies High-Velocity Impact and Failure Across Materials, Geometries, and Scales." pith.science (2026). https://pith.science/paper/JPCA6CO2

@misc{pith2026251026360,
  author       = {Pith},
  title        = {Pith review of: Momentum-Transfer Framework Unifies High-Velocity Impact and Failure Across Materials, Geometries, and Scales},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JPCA6CO2}},
  note         = {Machine review of arXiv:2510.26360}
}

read the original abstract

Materials that dissipate energy efficiently under high-speed impacts, from micrometeoroid strikes on spacecraft to ballistic penetration in protective systems, are essential for maintaining structural integrity in extreme environments. Yet, despite decades of study, predicting and comparing impact performance across materials, geometries, and length scales remains challenging because conventional projectile-impact models often rely on conservation-based or empirically partitioned descriptions that assume the projectile-target interaction is a closed system. Here, we relax this assumption and directly observe the momentum and energy transferred out of the projectile during impact. We find that the momentum transferred to the target consistently reaches its maximum at the ballistic-limit velocity, demonstrated through a coordinated suite of micro-projectile impact experiments spanning varied projectile diameters, target thicknesses, and impact velocities, and further supported by targeted macroscale tests. This behavior is reinforced across a broad range of independent studies encompassing metals, polymers, composites, sandwich panels, and reinforced concrete, with thicknesses ranging from nanometers to hundreds of millimeters and projectiles of spherical, blunt, ogive, and conical shape, under both normal and oblique impacts. Together, these observations reveal a consistent impact behavior across all available data: maximum momentum transfer occurs at the ballistic limit. Extending this bound into the energy absorption landscape addresses an entrenched misconception in the field by revealing that specific energy absorption inherently inflates the performance of thinner targets due to geometric normalization, rather than reflecting genuine material enhancement.

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Reference graph

Works this paper leans on

4 extracted references · 1 canonical work pages

[1]

Y ., Sundararajan, P

Yoon, D. Y ., Sundararajan, P. R. & Flory, P. J. Conformational Characteristics of Polystyrene. Macromolecules 8, 776–783 (1975)

1975
[2]

Recht, R. F. & Ipson, T. W. Ballistic Perforation Dynamics. Journal of Applied Mechanics 30, 384–390 (1963)

1963
[3]

Ipson, T. W. & Recht, R. F. Ballistic-penetration resistance and its measurement. Experimental Mechanics 15, 249–257 (1975)

1975
[4]

Lambert, J. P. & Jonas, G. H. Towards Standardization in Terminal Ballistics Testing: Velocity Representation: http://www.dtic.mil/docs/citations/ADA021389 (1976) doi:10.21236/ADA021389

work page doi:10.21236/ada021389 1976

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Y ., Sundararajan, P

Yoon, D. Y ., Sundararajan, P. R. & Flory, P. J. Conformational Characteristics of Polystyrene. Macromolecules 8, 776–783 (1975)

1975

[2] [2]

Recht, R. F. & Ipson, T. W. Ballistic Perforation Dynamics. Journal of Applied Mechanics 30, 384–390 (1963)

1963

[3] [3]

Ipson, T. W. & Recht, R. F. Ballistic-penetration resistance and its measurement. Experimental Mechanics 15, 249–257 (1975)

1975

[4] [4]

Lambert, J. P. & Jonas, G. H. Towards Standardization in Terminal Ballistics Testing: Velocity Representation: http://www.dtic.mil/docs/citations/ADA021389 (1976) doi:10.21236/ADA021389

work page doi:10.21236/ada021389 1976