{"id":"bde8cd3d-ef85-4d08-b922-8b5c3f2e2620","arxiv_id":"2608.04781","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A coupled phase field model derives dislocation nucleation from a single energy functional, yielding grain-size-dependent thresholds and showing that dissipationless shielding provides no toughening.","lead":"This paper builds a single phase field energy for a crack plus dislocations, so that whether the crystal cleaves or emits dislocations is decided by energy minimization alone. The model derives an integral nucleation criterion with size effects and shows that in a dissipationless limit dislocations shield a crack but do not toughen it.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The nucleation criterion and the d^-1 size effect rest entirely on the unverified degree-one dislocation energy in Eq. (5); if the true low-density dislocation energy is smooth, the model's threshold and its main predictions vanish.","rationale":"The central derivation is internally sound: the chord criterion follows from convex duality, the ADMM treatment is exact for the nonsmooth term, and the Griffith-limit and nucleation-threshold validations provide strong support. The concern is not internal consistency but the physical status of the degree-one energy. The reader's weakest assumption points to the same term, and I agree. If the true dislocation energy is smooth at zero density, the model's integral threshold, the d^-1 size effect, and the dissipationless zero-toughening result all cease to hold, because the threshold is created by the kink in ψ_d rather than by any physical mechanism in the elastic/damage part. I additionally note that the 'logarithmic energy' justification is not straightforward: a logarithmically divergent self-energy does not have a well-defined linear coefficient without specifying its cutoff, and if that cutoff is the grain size, Eq. (15) acquires a logarithmic factor. The paper's numerical certificates, reproducibility checks, and explicit acknowledgment of the unproved grazing limit are commendable and keep the failure mode specific rather than fatal. The proposed atomistic check would directly test the foundational premise; until then the conditional verdict remains appropriate.","tokens_in":21294,"tokens_out":28537,"duration_ms":321861,"concrete_test":"Perform atomistic or discrete-dislocation statics on a periodic array of edge dislocations at fixed line density ρ in a cell of size d, and extract the excess energy per unit area E(ρ,d) for small ρ. If E(ρ,d) is asymptotically linear in ρ with a coefficient that is flat in d, Eq. (5) is supported; if E is quadratic in ρ, or is ρ ln(d/b) to leading order, the degree-one premise fails and the derived threshold and d^-1 scaling would need revision. As an internal cross-check, replace ψ_d in Eq. (5) with the regularized smooth function q_c sqrt((∂_s β)^2 + ε^2) and solve the constrained-shear problem; if the nucleation threshold (14) vanishes as ε→0 at fixed q_c, it confirms that the nonsmoothness alone carries the central predictions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing premise is the degree-one dislocation energy q_c|∂_s β| in Eq. (5). This term is the only source of a nonzero nucleation threshold: in the subdifferential condition (11), the elastic driving term is linear in the trial slip, so if ψ_d were smooth (e.g., quadratic), an infinitesimal admissible δβ would always lower the energy and the dislocation-free state would not be an energetic minimizer under any load. Consequently the integral chord criterion, the constrained-shear threshold (14), the d^-1 grain-size yield stress (15), and the shielding-without-toughening computation all depend on this nonsmoothness. The paper adopts the term from Berdichevsky's continuum dislocation theory but does not derive it from a microscopic Hamiltonian or test it here. Moreover, the justification as \"the expansion of a logarithmic energy about ρ=0\" is not automatic: a logarithmic self-energy has a divergent slope at zero density and its linear coefficient can carry a grain-size cutoff ln(d/b), in which case the clean d^-1 scaling of Eq. (15) would become d^-1 ln(d/b). The grazing-limit statement (Section 3.4) is explicitly left unproved, but the deeper dependency is the form of ψ_d.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a variational phase-field model of a cracked single crystal in which a damage field, a slip field, and geometrically necessary dislocations all descend from a single energy functional. The central results are: (i) an integral dislocation-nucleation criterion evaluated in closed form along slip chords, with a grain-size-dependent yield stress scaling as d^-1; (ii) numerical validation of the Griffith limit and of the nucleation threshold to 4% by a nonsmooth solver; (iii) coupled computations showing a two-stage response—dislocation band emission and blunting at loads an order of magnitude below cleavage, followed by crack growth with a traveling dislocation cluster; and (iv) a 'zero toughening' result: in the dissipationless (reversible-slip) limit, emission shields the crack but the dissipated fracture resistance equals the elastic one to two parts per thousand, implying that toughening requires dissipative slip resistance. The paper includes detailed numerical certificates: equipartition-based mesh diagnostics, a posteriori primal-gap bounds, and positive-lag checks for surfing measurements.","tokens_in":21548,"tokens_out":18898,"duration_ms":215843,"significance":"If correct, the paper makes a significant contribution to coupling fracture and dislocation physics. The chord criterion is a clean analytical result that converts a nonsmooth but convex energy into a closed-form nucleation condition, and the numerical methodology—especially the a posteriori certificates for measure-valued slip—is careful and reproducible. The model yields falsifiable predictions (emission loads, standoff, d^-1 scaling, surface-source asymmetry) and gives a precise statement of why dissipation is needed for toughening. The validation is genuinely two-sided: the Griffith limit is recovered with convergence diagnostics, and the nucleation threshold matches a full nonsmooth solver to 4%. The strength of the paper lies in its clarity and the care with which its numerical claims are certified.","major_comments":[{"comment":"The degree-one dislocation energy q_c|∂_s β| in Eq. (5) is the unique source of the nucleation threshold, as the paper itself states in Section 3.1 ('the threshold is governed by the competition between the elastic driving term and the degree-one BV term alone'). However, the justification that this term is 'the expansion of a logarithmic energy about ρ=0' is not mathematically sound: a logarithmic self-energy density such as |ρ| ln(1/(b²ρ)) has a divergent slope at ρ=0, and any regularized linear coefficient carries an outer cutoff (typically ln(d/b)). If the true energy were smooth at zero density, the first variation of the energy at β=0 would not vanish and the dislocation-free state would be unstable at any load; the thresholds of Section 3 and the d^-1 scaling of Eq. (15) would not follow. The paper should either derive the degree-one term as a controlled limit of a regularized logarithmic energy, stating the dependence of q_c on the cutoff, or present it explicitly as a phenomenological postulate. If q_c carries a cutoff ln(d/b), Eq. (15) becomes d^-1 ln(d/b), which changes the claimed size-effect scaling.","section":"Section 2.2, Eq. (5); Section 3.3, Eq. (15)"},{"comment":"The corollary that a bare surface tangent to the slip plane acts as a dislocation source with vanishing threshold is explicitly unproved ('A rigorous analysis of the grazing limit is beyond the scope of this paper'). This result is load-bearing for the notched-disk computations: it predicts the surface micro-slip at loads two orders of magnitude below the band scale (Section 5.2) and the broad emission interval of Section 6.1. The numerical signatures (the √C divergence of the penalized multiplier and the activation of surface microslip) are suggestive, but the 1D chord reduction of Appendix A relies on chords with two distinct ends; at the tangency point the chord degenerates to a point and the reduction is not justified. I recommend replacing the claim of a vanishing threshold with a clearly labeled conjecture supported by the numerics, or supplying a rigorous asymptotic analysis for a model geometry such as a half-plane with a tangent characteristic.","section":"Section 3.4 and Appendix A"},{"comment":"The statement 'emission shields the crack but does not toughen it' is made for the dissipationless limit. In the actual finite-grain computation, the stored dislocation energy slope is dE_disl/da = 0.17 G_c (Eq. (20)), so the total energy release rate exceeds the elastic resistance by about 16%. The paper attributes this to a finite-grain effect that would vanish in an unbounded domain. This is plausible, but it is not demonstrated; the measured slope is also stated to be the 'least certain number reported' (15% variation between solver schedules). The zero-toughening result should be formulated as a limit statement (rigid translation of the dislocation pattern in an infinite domain) and the finite-grain apparent toughening should be reported as a separate quantity, not folded into the 'zero toughening' claim.","section":"Section 6.3, Eq. (20)"},{"comment":"The comparison of Eq. (15) to the experimental d^-1 scaling of Li, Conrad, and Dunstan & Bushby is only qualitative. The model's prefactor 2μk/(bρ_s) contains the free parameter k; without a quantitative fit to a specific dataset, the prediction is not falsifiable in the strict sense claimed in Section 7 ('the model is predictive in the strict sense'). The paper should identify a specific experimental dataset (material, grain-size range, temperature) against which the predicted prefactor can be tested, or at least state the range of k over which the d^-1 scaling is consistent with the cited data.","section":"Section 3.3, Eq. (15); Section 7"}],"minor_comments":[{"comment":"The phrase 'expansion of a logarithmic energy about ρ=0' appears repeatedly; since this premise is the basis for the major concern above, I suggest replacing the phrase with a more precise statement of the regularized logarithmic energy and the sense in which the degree-one term approximates it.","section":"Introduction and Section 2.2"},{"comment":"The resolved shear stress τ is defined as τ:=σ:P with σ=g(α)C:ε(u), but the definition appears after the first use; please move the definition before Eq. (11) for clarity.","section":"Section 3.1, Eq. (11)"},{"comment":"The nucleation test uses k=0.42, far from the physical calibration k=0.03; the departure is flagged, but a sentence explaining why this choice places the threshold in the Rice–Thomson window and whether the identified chord (longest crack-free chord versus crack-tip dipole) is robust to variations in k would help the reader.","section":"Table 1 and Section 5.2"},{"comment":"The statement 'coarse-graining N≈32 discrete dislocations in 8–9 walls within a region of extent ≈7ℓ' interprets the continuum density field; please state the criterion (e.g., integrated density threshold) used to count discrete dislocations from the density.","section":"Section 6.2"},{"comment":"The notation ∂_s β is typeset inconsistently as '∂ sβ' in several places; please ensure uniform mathematical formatting.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a serious, self-aware paper, and the central derivation is clean, but the headline results are only as strong as the degree-one dislocation energy term, which is adopted from the authors' own prior theory rather than independently derived or tested here. Read the quantitative claims with that conditional in mind.\n\nWhat is genuinely new: the chord evaluation of the nucleation criterion, the way a nonsmooth degree-one defect energy turns a pointwise strength condition into an integral, geometry- and size-dependent threshold, and the explicit computation that in a dissipationless, reversible-slip limit emission shields the crack but does not toughen it. The last result is not a platitude in this setting; it is derived from the functional and computed to two parts per thousand. The numerical work is unusually careful for this literature: Griffith limit recovered with mesh-convergence diagnostics, nucleation load matching a closed-form criterion to 4%, an a posteriori primal-gap certificate for measure-valued slip, and a solution-based equipartition check for mesh adequacy. These are the right instruments, and they make the paper's internal claims credible.\n\nThe soft spots are concentrated. The degree-one term q_c|∂_s β| is the load-bearing premise: if the true low-density dislocation energy were smooth, the dislocation-free state would not be a minimizer under any load, so the chord criterion, the d^-1 size effect, and the shielding-without-toughening computation would not follow. The paper adopts this term from Berdichevsky's continuum dislocation theory and justifies it as the first term of a logarithmic expansion; that is reasonable as a modeling choice, but the expansion's linear coefficient could plausibly carry a grain-size cutoff ln(d/b), which would turn the clean d^-1 law into d^-1 ln(d/b). The paper should say more about this, and ideally test the prediction against micro-pillar or in-situ TEM data. The grazing-limit surface-source prediction is also explicitly left without a rigorous proof; the authors record it as an open point, which is honest, and the numerical signatures are suggestive but not conclusive. The free parameter k is a real uncertainty, though only the product q_c enters thresholds and the authors flag the validation setting.\n\nOverall: the math is solid, the assumptions are flagged, and the paper is transparent about its limits. It is not a closed case; the degree-one premise needs independent support. But it is a well-built, falsifiable framework, aimed at researchers in phase-field fracture and continuum dislocation theory, and it deserves serious referee time. I would bring it to a reading group.","headline":"A carefully built variational model whose headline results hang on a degree-one dislocation energy the paper adopts rather than derives; worth engaging seriously, but read the d^-1 law and the zero-toughening claim with that premise in mind.","tokens_in":22051,"tokens_out":3085,"would_cite":true,"duration_ms":37822,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["61.72.Lk","62.20.mm"],"model":"deepseek-v4-flash","headline":"A single energy functional, minimized alone, decides whether a stressed crack cleaves or emits dislocations, and in the dissipationless limit emission shields a crack but does not toughen it.","keywords":["phase-field fracture","continuum dislocation theory","dislocation nucleation","crack-tip blunting","shielding versus toughening","grain-size yield stress","brittle-to-ductile transition"],"falsifier":"The most direct check is the small-grain yield stress: measure initial yield as a function of grain size $d$ in passivated single-crystal samples; the model predicts $\\tau_y \\sim 2\\mu k/(b\\rho_s d)$, an exact $d^{-1}$ law with no fitted exponent, and a crossover away from the classical $d^{-1/2}$ Hall\\textendash Petch scaling at larger sizes. Data showing only $d^{-1/2}$ everywhere would refute the energetic nucleation premise. A second decisive check is the closed-form nucleation load: in the paper's pinned square crystal the criterion predicts first slip at a dimensionless load of 0.00951, and the full nonlinear solver agrees within about 4%.","tokens_in":21097,"feed_emoji":"💥","tokens_out":9059,"duration_ms":97246,"temperature":0.7,"pith_summary":"The paper proposes that a single energy functional, minimized over displacement, slip, and damage, decides whether a stressed crack in a single crystal cleaves or emits dislocations, with no separate yield criterion imposed. Minimizing that functional yields a nucleation threshold of integral form along slip chords, which naturally produces a grain-size-dependent yield stress: smaller grains are stronger, with a $\\tau_y \\sim 2\\mu k/(b\\rho_s d)$ scaling that pointwise strength conditions cannot express. With slip suppressed the model reproduces Griffith fracture, and with fracture suppressed its closed-form nucleation criterion matches the full nonsmooth computation to about 4%. Coupled computations show two stages: dislocation bands emitted an order of magnitude below cleavage blunt and shield the tip, raising the initiation load; after growth begins the bands heal and a compact cluster of like-signed dislocations travels with the tip, with dissipated fracture resistance equal to the elastic one to roughly two parts per thousand. If correct, energy minimization alone decides the emission-cleavage competition, and the classical shielding-versus-toughening distinction becomes a derived result: toughening requires dissipative slip resistance.","feed_headline":"Dislocation emission shields cracks but does not toughen them","feed_subtitle":"A single energy functional decides emission vs. cleavage; fracture resistance stays elastic in the dissipationless limit.","key_machinery":"The load-bearing object is the nonsmooth dislocation energy density $\\psi_d = q_c |\\partial_s \\beta| + (c^2/2)(\\partial_s \\beta)^2$, where $\\partial_s \\beta$ is the slip gradient along the slip direction and, up to $1/b$, the signed density of geometrically necessary edge dislocations. The degree-one term $q_c |\\partial_s \\beta|$ is the physical heart: it gives dislocation content a finite energetic cost per unit density even at vanishing density, turning the stability of the dislocation-free state into an integral criterion evaluated along one-dimensional slip chords, with sharp thresholds such as $\\operatorname{osc}_{[0,L]} Q \\le 2q_c$ for pinned ends. This same term is what converts a pointwise strength condition into a size-dependent nucleation criterion, and it is carried numerically by an ADMM soft-threshold step on the slip gradient.","core_discovery":"The paper's central claim is that cleavage and dislocation emission are not competing constitutive outcomes but two terms of one energy functional $J[u, \\beta, \\alpha]$: elastic energy degraded by damage, a dislocation energy $\\psi_d = q_c |\\partial_s \\beta| + (c^2/2)(\\partial_s \\beta)^2$ for the geometrically necessary dislocations, and the Ambrosio\\textendash Tortorelli fracture energy. Because the dislocation energy is not smooth at zero density, the stability condition for the dislocation-free state takes the integral form $|\\int_\\Omega \\tau \\, \\delta\\beta \\, da| \\le q_c \\int_\\Omega |\\partial_s \\delta\\beta| \\, da$, which the paper evaluates in closed form along slip chords: a pinned\\textendash pinned chord yields the threshold $\\operatorname{osc}_{[0,L]} Q \\le 2q_c$, and a grain of size $d$ gives the yield stress $\\tau_y \\sim 2\\mu k/(b\\rho_s d)$. The coupled minimization then produces, without any fitted yield surface, the predicted order of events: surface microslip and dislocation bands at loads far below cleavage, blunting and shielding that elevate the initiation load by a factor of 1.2\\textendash 1.7 in stress intensity, healing of the bands during growth, a traveling cluster of about 32 like-signed dislocations at standoff $1.4\\ell$, and a steady-state dissipated fracture resistance equal to the elastic resistance to two parts per thousand. In the purely energetic limit, emission shields the crack but does not toughen it.","pith_inferences":["A sharp experimental test of the model is the small-grain yield stress: if initial yield in the few-hundred-nanometer regime follows $d^{-1/2}$ rather than $d^{-1}$, the energetic nucleation premise would be contradicted.","The grazing-chord prediction that a bare surface tangent to a slip plane is an inexhaustible low-threshold dislocation source could be tested directly by comparing coated and uncoated notches; the model predicts a finite threshold only in the passivated case.","If the zero-toughening result is robust, measured fracture resistance in experiments must be attributed to dissipative wake mechanisms; separating stored from dissipated dislocation energy would change how R-curves in ductile fracture are interpreted.","The two-stage picture suggests a concrete microstructure to look for in in situ crack-growth experiments: a compact like-signed dislocation cluster traveling at a standoff of the order of the damage length, with opposite-sign walls pinned at the grain boundary."],"forward_implications":["The emission\\textendash cleavage competition requires no separate yield or nucleation criterion: a single energy functional decides it by minimization.","Smaller grains are stronger with the specific $d^{-1}$ scaling $\\tau_y \\sim 2\\mu k/(b\\rho_s d)$, a size effect that no pointwise strength surface can produce; the paper also derives an inverse-thickness threshold for constrained shear.","Bare free surfaces tangent to the slip plane are predicted to act as dislocation sources with vanishing threshold, while passivated surfaces and grain boundaries restore a finite threshold.","In the coupled response, dislocation bands blunt and shield the tip, raising the initiation load by a factor of 1.2\\textendash 1.7 in stress intensity; during growth a compact cluster travels with the tip, and the dissipated fracture resistance equals the elastic resistance.","Toughening is dissipative: the dissipationless limit gives the exact zero intercept of the toughening curve, $\\Gamma(0) = \\Gamma_{\\text{elastic}}$."],"supporting_citations":[{"why":"Poses the emission-versus-cleavage question and supplies the blunting, emission, and shielding picture the model is built on.","marker":"(Rice and Thomson, 1974)"},{"why":"Supplies the continuum dislocation theory of single crystals and the logarithmic dislocation energy whose degree-one expansion is central.","marker":"(Berdichevsky, 2006a,b)"},{"why":"Provides the energetic nucleation thresholds of constrained shear that the chord criterion generalizes.","marker":"(Berdichevsky and Le, 2007; Le and Sembiring, 2008)"},{"why":"Identifies the dislocation density as the incompatible gradient of slip, making it the state variable that carries free energy.","marker":"(Nye, 1953; Bilby, 1955; Kröner, 1955)"},{"why":"Establishes the variational energy-minimization framework for brittle fracture and the alternate-minimization algorithm.","marker":"(Francfort and Marigo, 1998; Bourdin et al., 2000)"},{"why":"Supplies the AT2 elliptic regularization of the fracture energy used for the damage field.","marker":"(Ambrosio and Tortorelli, 1990)"},{"why":"Supplies the surfing protocol used to measure steady-state fracture resistance.","marker":"(Hossain et al., 2014)"},{"why":"States the shielding-versus-toughening distinction that the paper derives rather than postulates.","marker":"(Ritchie, 2011)"},{"why":"Documents the small-grain $d^{-1}$ yield-stress scaling that the energetic threshold reproduces and contrasts with $d^{-1/2}$ Hall\\textendash Petch scaling.","marker":"(Li, 1963; Conrad, 1963; Dunstan and Bushby, 2014)"},{"why":"Defines the experimental tungsten benchmark and rate-dependent brittle-to-ductile transition that the sequel targets.","marker":"(Gumbsch et al., 1998)"}],"fun_headline_variants":["Emission shields cracks, but toughening requires dissipation","Shielding without toughening: dislocations alone fall short","One energy functional decides between cleavage and dislocation","Crack blunting raises load, yet fracture resistance stays elastic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"A finite energetic cost per unit of newly created dislocation content must persist even at vanishing content; if the dislocation energy were smooth at zero density, the integral nucleation threshold, the inverse-grain-size yield stress, and the shielding-without-toughening result would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Emission shields cracks, but toughening requires dissipation","Shielding without toughening: dislocations alone fall short","One energy functional decides between cleavage and dislocation","Crack blunting raises load, yet fracture resistance stays elastic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000185,"raw_usage":{"total_tokens":1389,"prompt_tokens":1083,"completion_tokens":306,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":699,"completion_tokens_details":{"reasoning_tokens":240}},"tokens_in":699,"tokens_out":306,"duration_ms":4185,"temperature":1.0,"reasoning_tokens":240,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:43:35.196924+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The most direct check is the small-grain yield stress: measure initial yield as a function of grain size $d$ in passivated single-crystal samples; the model predicts $\\tau_y \\sim 2\\mu k/(b\\rho_s d)$, an exact $d^{-1}$ law with no fitted exponent, and a crossover away from the classical $d^{-1/2}$ Hall\\textendash Petch scaling at larger sizes. Data showing only $d^{-1/2}$ everywhere would refute the energetic nucleation premise. A second decisive check is the closed-form nucleation load: in the paper's pinned square crystal the criterion predicts first slip at a dimensionless load of 0.00951, and the full nonlinear solver agrees within about 4%.","supporting_citations":[{"cited_title":"Ductile versus brittle behaviour of crystals","cited_arxiv_id":null,"evidence_quote":"Poses the emission-versus-cleavage question and supplies the blunting, emission, and shielding picture the model is built on."},{"cited_title":"Dislocation nucleation and work hard- ening in anti-plane constrained shear","cited_arxiv_id":null,"evidence_quote":"Provides the energetic nucleation thresholds of constrained shear that the chord criterion generalizes."},{"cited_title":"Some geometrical relations in dislocated crystals","cited_arxiv_id":null,"evidence_quote":"Identifies the dislocation density as the incompatible gradient of slip, making it the state variable that carries free energy."},{"cited_title":"Revisiting brittle fracture as an energy minimization problem","cited_arxiv_id":null,"evidence_quote":"Establishes the variational energy-minimization framework for brittle fracture and the alternate-minimization algorithm."},{"cited_title":"Approximation of functionals depend- ing on jumps by elliptic functionals viaΓ-convergence","cited_arxiv_id":null,"evidence_quote":"Supplies the AT2 elliptic regularization of the fracture energy used for the damage field."},{"cited_title":"Effective toughness of heterogeneous media","cited_arxiv_id":null,"evidence_quote":"Supplies the surfing protocol used to measure steady-state fracture resistance."},{"cited_title":"The conflicts between strength and toughness","cited_arxiv_id":null,"evidence_quote":"States the shielding-versus-toughening distinction that the paper derives rather than postulates."},{"cited_title":"Effect of grain size on the lower yield and flow stress of iron and steel","cited_arxiv_id":null,"evidence_quote":"Documents the small-grain $d^{-1}$ yield-stress scaling that the energetic threshold reproduces and contrasts with $d^{-1/2}$ Hall\\textendash Petch scaling."},{"cited_title":"Control- ling factors for the brittle-to-ductile transition in tungsten single crystals","cited_arxiv_id":null,"evidence_quote":"Defines the experimental tungsten benchmark and rate-dependent brittle-to-ductile transition that the sequel targets."}],"review_version":1}