{"id":"f9d162f9-f590-424d-9791-7683fbd7e7c2","arxiv_id":"2505.12850","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":11,"one_line_summary":"Large-scale DEM simulations confirm a three-stage compression process for hierarchical granular piles and yield a semi-analytical compression curve model.","lead":"This paper simulates the compression of piles made of smaller aggregates, using up to 26 million ice particles, and finds three distinct stages of deformation. It also offers a compact three-part formula that fits the simulated compression curve, which could help model the internal structure of comets and other small icy bodies.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Stage 1 is attributed to aggregate-aggregate friction, but the DEM is explicitly frictionless; the prefactor in Eq. 14 is instead rationalized by adhesive bond breaking, so the mechanism attached to the Stage 1 formula is internally inconsistent.","rationale":"The reader's weakest-assumption pick is the right one. The manuscript contains a direct internal contradiction: Section 2.1 removes tangential interactions and particle rotation from the DEM, but Section 5.1 and Conclusion item (2) attribute Stage 1 resistance to friction for aggregate-aggregate rotation. The prefactor that actually enters Eq. 14, Pmid in Eq. 15, is fit to the simulations, and the theoretical estimate in Eq. 18 is built from adhesive contact-breaking work during rolling, which is a rolling resistance rather than a tangential Coulomb friction. Thus the mechanistic story attached to the Stage 1 curve is not the physics implemented in the code unless the authors mean rolling friction in the adhesion sense and say so explicitly. This is load-bearing for the semi-analytical model as an explanation, though not for the raw three-stage observation: the stage boundaries are visible directly in Figures 7-8, 13, and 18, and the Stage 1-to-2 transition near phi_str approximately 0.5 is supported by coordination-number and velocity data. The concrete two-aggregate rolling test would settle whether Eq. 18's mechanism is real in the frictionless model and whether Pmid proportional to ragg^(-3/2) is mechanistic or just a two-point fit. The other flagged issues, including prior DEM studies reporting multi-step compression, dynamic pressure at phi below 0.16, the modest Nagg = 200, and the ragg = 16r* failure, are real but secondary: they constrain novelty or applicability rather than the internal consistency of the Stage 1 explanation. This supports keeping the CONDITIONAL verdict and requiring either a clarifying revision or a supporting simulation before the Stage 1 mechanism is presented as predictive.","tokens_in":19021,"tokens_out":7938,"duration_ms":84062,"concrete_test":"Perform a dedicated two-aggregate rolling test with the same frictionless JKR DEM: hold one aggregate fixed, compress a second aggregate against it to delta_agg approximately delta_agg,0, rotate the second around the first quasistatically, and integrate the torque-angle curve. Compare the total work per full rotation to N_rot * W_break from Eq. 18 and to a zero-torque control. If the measured work matches Eq. 18 within a factor of about 2, the Stage 1 mechanism should be relabeled adhesive rolling resistance and Eq. 14 is mechanistically supported; if the measured work is much smaller or does not scale as ragg^(-3/2), then Pmid in Eq. 15 is a numerical fit with no demonstrated micromechanical basis, and the Stage 1 explanation in Sections 5.1 and 7 should be withdrawn or substantially revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest point is the mechanism claimed for Stage 1. Section 2.1 states explicitly that the DEM does not include tangential interactions (frictional torques by rolling, sliding, and twisting between particles in contact) and that particle rotations are not solved. Yet Section 5.1 introduces the Stage 1 pressure by saying the yield strength would be proportional to the tangential friction between aggregates, and Conclusion item (2) says the friction for aggregate-aggregate rotation plays a key role. The derivation that actually produces the fitted prefactor is different: Eq. 18 counts the work to break adhesive monomer contacts during one aggregate rotation, divided by aggregate volume, giving Pmid about 200 (ragg/32r*)^(-3/2) Pa. This is an adhesion-induced rolling resistance, not Coulomb or tangential friction, and the text conflates the two. The inconsistency matters because Eq. 14 is first fit to the simulations (Pmid = 240 Pa in Eq. 15) and then explained by Eq. 18; if the stated friction mechanism is wrong, the quantitative explanation attached to Stage 1 is not supported by the simulation physics, and the scale dependence Pmid proportional to ragg^(-3/2) rests on an unvalidated rolling-contact argument. This does not invalidate the three-stage observation, which is visible in the raw compression curves and in the aggregate velocity data, but it undercuts the semi-analytical model as a mechanistic prediction. The authors should either replace friction by adhesive rolling resistance throughout, or demonstrate that the resistance actually present in the frictionless DEM matches Eq. 18.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"Using the DEM code DEPTH with adhesive JKR normal forces and viscous damping (and explicitly neglecting tangential interactions), the authors simulate quasi-static compression of hierarchical granular piles made of porous spherical aggregates of 0.1 µm ice monomers, for aggregate radii of 32 and 64 monomer radii (up to 26 million monomers). They also compress isolated aggregates. From wall-pressure curves and microstructural diagnostics (aggregate coordination, aggregate compression length, rotational/translational aggregate velocities, interparticle coordination), they identify three compression stages: aggregate-packing rearrangement, plastic deformation of aggregates, and elastic compression of monomers. They propose piecewise semi-analytical expressions for the pressure–filling-factor curve (Eqs. 14, 19, and 20) and argue these reproduce the simulations. The paper claims this is the first numerical confirmation of the experimentally inferred multi-step compaction of hierarchical granular piles, with applications to comet internal structure.","tokens_in":19303,"tokens_out":7031,"duration_ms":73302,"significance":"If the three-stage picture holds, the paper provides a useful numerical confirmation and a compact parameterization of an important compaction law for hierarchical granular materials. The strengths are the scale of the simulations, the use of a standard and carefully described contact model, the convergence of several independent diagnostics on the stage transitions, and the availability of the numerical data. The weaknesses are that the semi-analytical model is substantially an empirical fit with many free constants, and that the mechanistic explanation of the Stage 1 prefactor is inconsistent with the frictionless DEM model. These issues affect the quantitative-model claim, but not the core three-stage observation.","major_comments":[{"comment":"The Stage 1 mechanism is internally inconsistent. Section 2.1 explicitly states that tangential interactions (sliding, rolling, twisting friction) and particle rotations are not included in the DEM, and Equation (18) derives the Stage 1 prefactor from the work needed to break adhesive monomer contacts (NrotWbreak) during one aggregate rotation. Yet Section 5.1 states that the yield strength 'would be proportional to the tangential friction between aggregates' and Conclusion item (2) attributes Stage 1 to 'friction for aggregate–aggregate rotation.' This conflation matters because the prefactor in Eq. (14) is first fit to the simulations (Eq. 15, Pmid = 240 Pa) and then rationalized by an order-of-magnitude estimate that is actually an adhesion-induced rolling resistance, not Coulomb friction. Please either rephrase the mechanism as adhesive rolling resistance throughout, or add tangential interactions to the DEM to substantiate a friction-based explanation; the current text does not support the mechanistic claim attached to Eq. (14).","section":"§5.1 (Eqs. 14–18), §2.1, §7"},{"comment":"The semi-analytical model is substantially empirical, and this should be stated clearly. The prefactor Pmid (Eq. 15), the aggregate compression law (αagg, δagg,0, k32, F32, δoffset, Eq. 12), the correction factors c2 and c3 (Eqs. 19 and 20), the interparticle coordination fit (ϕ3, Eq. 21), and the compression-length coefficient α (Eq. 22) are all fixed by matching the simulation output; the Stage 2 and Stage 3 expressions then reproduce those same curves by construction. With the exception of the order-of-magnitude estimates for Pmid (Eq. 18) and Yagg (Eq. 13), the model does not provide independent predictions from material constants. I recommend labeling the model an empirical parameterization and tempering Conclusion item (1)'s claim of a 'quantitative semi-analytical model,' while retaining the useful compact representation.","section":"§5.1–5.3 (Eqs. 14–22)"},{"comment":"The paper demonstrates that the proposed Stage 1/Stage 2 model fails for ragg = 16r*, since the two fitted curves do not intersect and the transition cannot be captured. This limitation is acknowledged in Section 6.2, but its consequence for the generality of the central claim should be more prominent: the three-stage parameterization is validated only for ragg/r* ≳ 32, and the abstract and conclusions should state this applicability range explicitly.","section":"§6.2, Fig. 20"}],"minor_comments":[{"comment":"The text states that for a homogeneous distribution of contact directions, the fraction of contacts with angles between θ1 and θ2 is sinθ2 − sinθ1; the correct expression is (cosθ1 − cosθ2)/2. Please correct the formula and re-check the white dashed lines in Figure 12.","section":"§3.3, Fig. 12"},{"comment":"There are several small presentation errors: 'particls' should be 'particles' in Section 2.2; 'an semi-analytical fit' should be 'a semi-analytical fit' in the Figure 16 caption; and Section 3.4 labels the aggregate rotational velocity as Vtra immediately before defining Vrot.","section":"§2.2, §3.4, Fig. 16 caption"},{"comment":"The Figure 4(c) caption describes 'an aggregate consisting of 200 spheres,' which conflicts with the stated monomer counts Npar = 131,072 for ragg = 64r*; please rephrase to clarify that the RBD step produces a template pile of 200 mono-size spheres that are then replaced by aggregates.","section":"Fig. 4 caption"},{"comment":"Given the ragg = 16r* failure described in Section 6.2, the abstract and Conclusions should state that the three-stage parameterization is demonstrated for ragg/r* ≳ 32 rather than for hierarchical granular piles in general.","section":"Abstract and Conclusions"}],"recommendation":"major_revision","confidential_remarks":"The three-stage observation is credible and well supported by multiple diagnostics, but the Stage 1 mechanism is misattributed to friction in a frictionless model, and the semi-analytical model is overparameterized relative to the claim of quantitative prediction. These issues are fixable with restructuring and reframing, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth your time. It does the first large-scale DEM demonstration that adhesive JKR spheres reproduce the experimentally proposed three-stage compression of hierarchical granular piles, and it packages the result in a compact semi-analytical curve (Eqs. 14, 19, 20). The simulations are careful: 26 million particles, quasi-static checks, wall-pressure coupling, and multiple independent diagnostics (coordination, compression length, aggregate velocities) that all point to the same stage transitions. The single-aggregate compression tests and the plastic-sphere interpretation (Eq. 12, Yagg ≈ 1 MPa) are clean and useful. The authors are also transparent about limitations: Nagg = 200, dynamic pressure at low phi, and the failure of the model for ragg = 16r*. That is real, reproducible work—code and data are archived.\n\nThe soft spots are real but not fatal. The Stage 1 mechanism is mislabeled. Section 2.1 explicitly excludes tangential interactions and particle rotation, yet Section 5.1 and Conclusion item (2) say the pressure is controlled by \"friction for aggregate–aggregate rotation.\" What Eq. 18 actually estimates is the energy to break adhesive monomer contacts during rolling—an adhesion-induced rolling resistance, not Coulomb friction. The text conflates the two. That is an internal inconsistency, but it does not invalidate the three-stage observation, which is visible in the raw curves and velocity data. It does mean the scaling Pmid ∝ ragg^(-3/2) rests on an unvalidated rolling-contact argument rather than on demonstrated simulation physics.\n\nThe bigger wobble is circularity. Pmid, alpha_agg, k32, F32, c2, c3, phi_3, and alpha are all tuned to match the simulations they then reproduce. The Stage 1 prefactor does get an order-of-magnitude rationalization (Eq. 18) that lands close, which is something, but Stages 2–3 are largely empirical fits. Calling the result a \"semi-analytical model\" is fair; calling it predictive is not, at least not yet.\n\nAlso, the \"first numerical validation\" claim is overstated—prior DEM studies of crushable aggregates (Ishihara et al., Chen et al.) already reported multi-step behavior. The genuinely new bit is the quantitative curve at this scale, not the qualitative picture.\n\nWho is this for? Planetary scientists and granular mechanics people who want a compact compression law for comet interior modeling. It is a useful engineering fit with a plausible microphysical story, not a derivation from first principles. It deserves a serious referee: the core observation is solid, the data are open, and the issues are addressable in revision. I would send it to review, and I would ask the authors to either replace \"friction\" with adhesive rolling resistance throughout or test that the frictionless DEM actually matches Eq. 18.","headline":"A solid, honest DEM study that convincingly reproduces the three-stage compression of hierarchical granular piles and offers a compact semi-analytical curve, though the Stage 1 mechanism is mislabeled as friction in a frictionless model and several parameters are fit rather than predicted.","tokens_in":19895,"tokens_out":719,"would_cite":true,"duration_ms":9027,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["45.70.-n"],"model":"deepseek-v4-flash","headline":"A pile built from smaller sticky clusters — the structure thought to fill comet interiors — compresses in three stages: rearrangement, cluster crushing, then elastic grain contact, each with its own pressure formula.","keywords":["asteroids","comets","discrete element method","earthquakes","fault gouges","granular mechanics","hierarchical granular piles","compression curve"],"falsifier":"Run the same compression setup twice, once with and once without tangential (rolling and sliding) friction between the constituent particles, and compare the Stage-1 branch of the pressure curve; if the two curves nearly coincide, the friction explanation of Stage 1 is refuted and the adhesive-tearing estimate (Eq. 18) is the operative mechanism. As a separate probe, repeat the Stage-1 compression with a changed surface energy (for example $\\gamma=0.2\\,\\mathrm{J\\,m^{-2}}$) and check whether the measured prefactor $P_{\\rm mid}$ follows the $\\gamma^{5/3}$ scaling implied by $F_{\\rm crit}\\delta_0$; a different scaling would show that Eq. (15) is a fitting constant rather than a consequence of the stated mechanism.","tokens_in":18752,"feed_emoji":"☄️","tokens_out":27394,"duration_ms":245261,"temperature":0.7,"pith_summary":"This paper sets out to establish that a 'hierarchical' granular pile — a pile whose grains are themselves loose clusters (aggregates) of much smaller sticky particles — compresses in three distinct stages, each with its own quantitative pressure law. Laboratory experiments on dust and ice aggregates had suggested such a multi-step evolution, but it had not been confirmed by simulation. Using large particle-resolved discrete-element simulations of up to 26 million adhesive ice particles, the authors reproduce the three stages: first the aggregates roll past one another without changing shape, then the aggregates deform plastically once their packing can no longer rearrange, and finally the pile behaves like a jammed packing of the individual particles, which deform elastically. The authors fit each stage with a semi-analytical formula, giving a recipe for the pressure a comet-like body can support at a given density — the input needed to predict internal density profiles and the early thermal history of small icy bodies.","feed_headline":"Simulation confirms three-stage compression of comet-like piles","feed_subtitle":"26-million-particle simulations reproduce the predicted stages and yield one pressure formula per stage.","key_machinery":"The machinery is a two-scale contact description of the pile. All elastic forces come from the JKR adhesive contact law for elastic spheres with surface energy, which fixes the pull-off force $F_{\\rm crit}$, the equilibrium compression $\\delta_0$, and the force–displacement relation $F_E/F_{\\rm crit}=4(a/a_0)^3-4(a/a_0)^{3/2}$; a viscous dissipation term proportional to the contact radius and compression rate damps the motion. The pile is described at two levels — aggregate–aggregate contacts and inter-particle contacts — each characterized by a filling factor (the aggregate-packing filling factor is $\\phi_{\\rm str}\\equiv\\phi/\\phi_{\\rm agg}$, the pile filling factor divided by the aggregate internal packing fraction), an average coordination number ($\\langle Z_{\\rm str}\\rangle\\simeq(6/0.64)\\phi_{\\rm str}$ and $\\langle Z\\rangle\\simeq20(\\phi-\\phi_3)$), and an average compression length. The Stage-2 and Stage-3 formulas feed these network statistics, together with the appropriate force–displacement law, into a Rumpf-type relation that converts average contact forces into macroscopic pressure; the Stage-1 prefactor is instead derived from an energy-bookkeeping argument, $N_{\\rm rot}W_{\\rm break}/(4\\pi r_{\\rm agg}^3/3)$, where $N_{\\rm rot}\\sim2.4(r_{\\rm agg}/r_{\\bullet})^{3/2}$ counts the monomer contacts torn during one full rotation of an aggregate and $W_{\\rm break}\\approx1.3F_{\\rm crit}\\delta_0$ is the work to break a single contact.","core_discovery":"On the paper's own terms, the central claim is that the experimentally suspected multi-step compression of hierarchical granular piles is real and numerically reproducible: the compression curve divides into (i) rearrangement of the aggregate packing structure, (ii) plastic deformation of the small aggregates, and (iii) elastic deformation of the constituent particles. In stage (i) the pressure follows a modified polytropic law in the aggregate-structure filling factor, $P_{S1}=P_{\\rm mid}\\left((\\phi_{\\rm str}-\\phi_{\\rm min})/(\\phi_{\\rm max}-\\phi_{\\rm str})\\right)^{1+1/n}$ with index $n=1$, and the prefactor scales as $P_{\\rm mid}\\propto r_{\\rm agg}^{-3/2}$; the authors trace that scaling to the work of tearing adhesive monomer contacts as aggregates roll over one another, estimating $P_{\\rm rot}\\approx 200\\,(r_{\\rm agg}/32r_{\\bullet})^{-3/2}$ Pa against the fitted value of 240 Pa. In stage (ii), once the aggregate–aggregate coordination number crosses about 4 near $\\phi_{\\rm str}\\simeq0.45$, the aggregates can no longer rotate freely and deform plastically, with the pressure set by the average contact force between aggregates. In stage (iii), past the jamming filling factor 0.64, the pile compresses as a packing of the individual particles with JKR elastic contacts. The three formulas reproduce the simulated compression curves for aggregate radii $32r_{\\bullet}$ and $64r_{\\bullet}$.","pith_inferences":["The explanation attached to Eq. (14) names friction for aggregate–aggregate rotation as the Stage-1 resistance (Section 5.1), yet the model explicitly excludes tangential interactions (Section 2.1), and the numerically successful prefactor estimate (Eq. 18) is computed from the energy to break adhesive contacts. My reading is that adhesive tearing is the operative mechanism and the friction langua","A quantitative test presents itself: vary the surface energy $\\gamma$ in the simulations and check whether the measured Stage-1 prefactor follows $F_{\\rm crit}\\delta_0\\propto\\gamma^{5/3}$, as the adhesive-tearing estimate implies. The paper does not perform this sweep, so the prefactor's status as a derived quantity rather than a fitting constant is not yet settled.","The same three-stage structure should be observable beyond ice: soil crumbs bound by organic matter and rock fragments in fault gouge share the same hierarchy, and existing compression data for submicron silica grains offer a ready comparison. The paper notes that the silica data lie above its ice-based curve for $\\phi>0.3$; silica-parameter simulations would show whether that gap is a material-pr","Because the Stage 1 to Stage 2 transition is tied to the aggregate coordination number crossing the just-rigid value 4 rather than to any material constant, the transition filling factor $\\phi_{\\rm str}\\approx0.45$ may be near-universal for random packings of monodisperse aggregates, with material properties setting the pressure scale but not the transition density."],"forward_implications":["The three formulas give a direct route to the internal density profile of a comet or other small icy body: self-gravity sets the pressure at each depth, and the model converts that pressure into a filling factor stage by stage.","The same model yields the inter-aggregate contact area as a function of pressure, and since thermal conductivity inside the pile is controlled by those contact areas, the formulas feed heat-conduction models of comet interiors and their early thermal evolution.","Larger aggregates make a much softer pile in Stage 1 — the prefactor falls as $r_{\\rm agg}^{-3/2}$ — and for sufficiently large aggregates the transition between Stages 1 and 2 sits at $\\phi_{\\rm str}\\approx0.45$, where the aggregate coordination number crosses the just-rigid (isostatic) value 4 and plastic deformation of aggregates begins.","The model works within stated limits: it fails for aggregates as small as $r_{\\rm agg}=16r_{\\bullet}$, where the single-aggregate force law develops spike-like finite-size effects, and pressure data below $\\phi\\approx0.16$ are contaminated by the dynamic compression term.","In Stage 3 the hierarchy is erased: past the jamming point the pile compresses like a packing of its constituent monomers, so the three-stage structure is a low- to moderate-pressure phenomenon."],"supporting_citations":[{"why":"Supplies the JKR adhesive contact law from which all elastic inter-particle forces, the pull-off force, and the contact-radius relations in the simulations are computed.","marker":"Johnson et al 1971"},{"why":"The discrete element method itself, the simulation framework the entire study is built on.","marker":"Cundall and Strack 1979"},{"why":"Provides the random ballistic deposition procedure that builds the porous aggregates, and the value phi_min = 0.15 used in the Stage-1 polytropic formula.","marker":"Blum and Schräpler 2004"},{"why":"Supplies the polytropic compression law that Stage 1 adapts to hierarchical piles, plus the exponential compression protocol used in the simulations.","marker":"Kataoka et al 2013"},{"why":"The experimental compression study whose reported multi-step packing evolution is the hypothesis the simulations are designed to confirm.","marker":"Schräpler et al 2015"},{"why":"The plastic-sphere contact theory used to interpret the single-aggregate force–displacement law and to extract the aggregate yield stress of about 1 MPa.","marker":"Andrews 1930"},{"why":"Source of the work-to-break-a-contact estimate W_break approximately 1.3 F_crit delta_0 used in the Stage-1 prefactor derivation.","marker":"Wada et al 2007"},{"why":"The relation between average inter-contact force and macroscopic pressure into which the Stage-2 and Stage-3 formulas feed.","marker":"Rumpf 1970"},{"why":"Fixes the maximum aggregate-packing fraction phi_max = 0.64 and the jamming point used in the Stage-1 and Stage-3 formulas.","marker":"Berryman 1983"}],"fun_headline_variants":["Three-stage compression of granular piles confirmed by simulation","26M-particle simulation validates three-stage pile compression","Comet pile compression: three stages reproduced numerically","Granular pile crush: simulation confirms three steps","Simulation shows hierarchical piles compress in three stages"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The Stage-1 pressure formula stands on the premise that a pile of clusters resists the first stage of compression through one microscopic cost — the energy to tear adhesive contacts between the tiny sticky particles — because friction between clusters, the mechanism the paper also invokes for this stage, does not exist in its simulations.","fun_headline_variants_meta":{"raw":{"variants":["Three-stage compression of granular piles confirmed by simulation","26M-particle simulation validates three-stage pile compression","Comet pile compression: three stages reproduced numerically","Granular pile crush: simulation confirms three steps","Simulation shows hierarchical piles compress in three stages"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000269,"raw_usage":{"total_tokens":1651,"prompt_tokens":1002,"completion_tokens":649,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":618,"completion_tokens_details":{"reasoning_tokens":578}},"tokens_in":618,"tokens_out":649,"duration_ms":6569,"temperature":1.0,"reasoning_tokens":578,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:24:49.091517+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same compression setup twice, once with and once without tangential (rolling and sliding) friction between the constituent particles, and compare the Stage-1 branch of the pressure curve; if the two curves nearly coincide, the friction explanation of Stage 1 is refuted and the adhesive-tearing estimate (Eq. 18) is the operative mechanism. As a separate probe, repeat the Stage-1 compression with a changed surface energy (for example $\\gamma=0.2\\,\\mathrm{J\\,m^{-2}}$) and check whether the measured prefactor $P_{\\rm mid}$ follows the $\\gamma^{5/3}$ scaling implied by $F_{\\rm crit}\\delta_0$; a different scaling would show that Eq. (15) is a fitting constant rather than a consequence of the stated mechanism.","supporting_citations":[{"cited_title":"Physical Review Letters 93(11):115503","cited_arxiv_id":null,"evidence_quote":"Provides the random ballistic deposition procedure that builds the porous aggregates, and the value phi_min = 0.15 used in the Stage-1 polytropic formula."},{"cited_title":"Theory of collision of spheres of soft metals","cited_arxiv_id":null,"evidence_quote":"The plastic-sphere contact theory used to interpret the single-aggregate force–displacement law and to extract the aggregate yield stress of about 1 MPa."},{"cited_title":"Chemie Ingenieur Technik 42(8):538--540","cited_arxiv_id":null,"evidence_quote":"The relation between average inter-contact force and macroscopic pressure into which the Stage-2 and Stage-3 formulas feed."}],"review_version":1}