{"id":"c6b79b67-1b36-4fe9-b3d4-9bc35863e97b","arxiv_id":"2507.15813","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A chosen hyperbolic hyperelasticity maps the linear Drucker-Prager yield criterion into a quadratic Hoek-Brown type criterion in stress space.","lead":"The paper shows that a linear Drucker-Prager yield criterion, written in terms of a thermodynamic 'plastic force', transforms into a quadratic Hoek-Brown type yield criterion in observable stress space when the elastic bulk modulus depends hyperbolically on volume change. This gives rock mechanics a new route from a simple linear failure law plus a chosen hyperelasticity to the widely used empirical Hoek-Brown envelope.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The derivation is internally sound, but the central claim is untested: the same β_m governs both the hyperbolic bulk law (12) and the yield-surface curvature in (19), and the paper never checks whether an independently measured elastic β_m reproduces the observed Hoek-Brown curvature.","rationale":"The reader's weakest assumption is that the hyperbolic moduli in Eq. (12) are chosen rather than derived, and that no experimental or micromechanical justification is offered. I agree that this is the main soft spot, but it can be sharpened into a concrete, falsifiable prediction: because β_m appears both in the elastic hydrostatic stress-strain relation and in the curvature of the stress-space yield criterion, the model can be checked by comparing an independently measured β_m from hydrostatic loading with the β_m needed to fit triaxial yield stresses. The paper does not perform this check; its single experimental comparison fits β_m from the same uniaxial curve it validates, and that comparison visibly fails on pre-peak dilatancy. This does not invalidate the algebra: Eq. (19) does follow from Eqs. (12) and (16), and the authors are careful to call the connection 'specific'. It does, however, mean that the central claim is a conditional modelling insight rather than an established material property. Since the reader already assigned CONDITIONAL and the concern does not point to an internal contradiction, I recommend no change to the verdict; the paper should be accepted conditionally on a clearer framing and on an independent test of the β_m cross-constraint.","tokens_in":17786,"tokens_out":16283,"duration_ms":185307,"concrete_test":"Perform an independent two-stage test on an intact rock. First, measure hydrostatic compression and unload-reload bulk stiffness as a function of volumetric strain, fit κ_i and β_m in Eq. (12), and verify that the shear modulus μ is approximately constant. Second, measure the yield surface in triaxial compression and fit the coefficients of Eq. (19). Then compare the β_m obtained from the elastic hydrostatic response with the β_m implied by the quadratic coefficient of the yield criterion. If the two values disagree by more than experimental error, the claimed hyperelastic derivation of the quadratic yield criterion is falsified for that material. The same test can be run numerically by refitting the basalt example of Sec. 4.5 with β_m fixed from a hypothetical hydrostatic measurement and checking whether the predicted yield stress still matches the triaxial data.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The mathematical step from (12) and (16) to (19) is correct, so the concern is not internal inconsistency. The load-bearing scientific step is the interpretation that quadratic Hoek-Brown-type yield criteria are the stress-space images of linear Drucker-Prager criteria under hyperelasticity. This interpretation requires that Eq. (12) be a real material property, not merely a convenient ansatz. Eq. (12) is selected in Sec. 3.1 precisely to make the mapping between the plastic force X and the stress σ strain-independent, and no independent evidence is given that rock elasticity follows this hyperbolic law. More importantly, the model makes a sharp cross-constraint that the paper does not test: the same β_m that controls the nonlinear bulk modulus in the hydrostatic law (22) also fixes the quadratic coefficient of the yield criterion in (19). In Sec. 4.5, β_m is calibrated from the same uniaxial test that is later used to show agreement, so the constraint is never checked independently. If a hydrostatic experiment yields a different β_m than the value needed to fit the yield-surface curvature, the proposed hyperelastic origin of Hoek-Brown does not hold for that material. The paper's own comparison also fails to reproduce the observed pre-peak dilatancy, further weakening the empirical case. Thus the central claim is best read as a plausible modeling equivalence, not an established material law.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a nonlinear elasto-plastic model in which the bulk modulus has a specific hyperbolic dependence on the volumetric strain, Eq. (12). Under this assumption, the mapping between the thermodynamic plastic force X and the observable stress σ becomes strain-independent, Eq. (15). For a linear Drucker-Prager yield criterion in X-space, Eq. (16), the stress-space yield criterion becomes a quadratic expression, Eq. (19), which the authors identify with the Hoek-Brown/Pan-Hudson type. The model is integrated with an analytical return-mapping scheme, tested on material-point simulations including cyclic triaxial loading, and demonstrated on mesh-converged finite element shear band simulations. The algebra from Eqs. (7), (12), and (16) to Eq. (19) is internally consistent and clearly traceable, and the numerical scheme in Appendix B is complete with a proof of semi-definite positiveness of the tangent operator.","tokens_in":18102,"tokens_out":3091,"duration_ms":32376,"significance":"If the proposed interpretation is established, the paper offers a novel thermodynamic link between two classical yield criteria: the linear Drucker-Prager criterion in the plastic-force space and a quadratic Hoek-Brown-type criterion in stress space. The work provides a fully analytical integration scheme, a positive-definiteness proof for the elastic tangent, and reproducible structural simulations, which are concrete strengths. The significance is currently limited by two factors: the specific hyperbolic elasticity, Eq. (12), is an ad hoc constitutive assumption with no independent experimental or micromechanical support, and the empirical validation in Sec. 4.5 is confined to a single uniaxial test with parameters calibrated on the same data set. The claimed cross-constraint that the same βm controls both elastic nonlinearity and yield-surface curvature is never tested independently. Thus the contribution is best viewed as a plausible modeling equivalence rather than an established material law.","major_comments":[{"comment":"The hyperbolic dependence κ(ε) = κi/(1 + 2κiβmTrε) is chosen specifically to make the X-to-σ mapping (15) strain-independent, but the paper offers no independent experimental or micromechanical justification for this functional form. Since the derivation of the quadratic yield criterion (19) depends entirely on this constitutive assumption, the central claim that Hoek-Brown-type criteria arise from linear Drucker-Prager under hyperelasticity is conditioned on an untested postulate. The authors should either provide independent evidence for Eq. (12), for example from hydrostatic loading experiments, or explicitly reframe the result as a modeling equivalence valid within this constitutive class.","section":"Sec. 3.1, Eq. (12)"},{"comment":"The model parameters, including βm, are calibrated on the same uniaxial compression test that is subsequently compared with the model response, so the agreement in Fig. 7b is not an independent validation. More importantly, the same βm that sets the hydrostatic nonlinearity (Eq. 22) also fixes the quadratic coefficient in the yield criterion (Eq. 19); this cross-constraint is never checked against independent data, such as triaxial tests under different confining pressures or a separate hydrostatic test. The paper should include such an independent check or explicitly narrow the scope of the claimed connection.","section":"Sec. 4.5, Fig. 7"},{"comment":"As the text acknowledges, the model does not reproduce the experimentally observed onset of dilatancy before peak stress. The authors attribute this to the absence of hardening, but this recognized shortcoming weakens the physical interpretation that the quadratic yield criterion is a genuine consequence of rock hyperelasticity. The conclusions should be tempered accordingly, and ideally the fit should be quantified rather than presented as a visual comparison.","section":"Sec. 4.5, Fig. 7b"}],"minor_comments":[{"comment":"Several references contain typographical errors, including 'Hoek-Brown strenght' in the Li et al. entry and 'Internation Journal' in the Zhang et al. entry.","section":"References"},{"comment":"The convexity condition a − 2βmb ≥ 0 is stated but its physical interpretation and its compatibility with the calibrated values in Sec. 4.5 (where a = 2.8, b = 0) are not discussed.","section":"Sec. 3.2, Eq. (19)"},{"comment":"The calibrated parameter values are reported only in the figure legend; for reproducibility they should also appear in the text or in a table.","section":"Sec. 4.5, Fig. 7"}],"recommendation":"major_revision","confidential_remarks":"The paper is mathematically sound and well written, but the central scientific claim is supported more by an elegant derivation than by empirical evidence. The authors should be asked to either supply an independent test of the βm cross-constraint or clearly label the result as a modeling equivalence; without this, the claim risks overstatement. The contribution could fit the journal, but the empirical section needs strengthening."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing: the math is fine. If you accept the hyperbolic elastic moduli (12), the derivation from linear Drucker-Prager in the plastic-force space to the quadratic criterion (19) in stress space is correct and clearly presented, and the return-mapping with analytical tangent is a useful piece of work. That is the real contribution: a simple, explicit transformation (15) that maps a linear yield surface into a quadratic one via hyperelasticity, with the strain-independence requirement leading exactly to the hyperbolic form. The historical appendix and the FE shear-band simulations are also decent.\n\nThe soft spot is the title. Calling this the hyperelastic nature of Hoek-Brown is a stronger claim than the paper supports. The hyperbolic law (12) is chosen to make the mapping strain-independent, not derived from rock physics or microstructure. So the result is a modeling equivalence or a sufficient condition, not an explanation of why real rocks obey Hoek-Brown. The cross-constraint the stress-test highlights is real and important: the same beta_m controls both the hydrostatic nonlinearity (22) and the curvature of the yield surface (19). The paper calibrates beta_m from the same uniaxial test it later compares against, so the constraint is never checked independently. A hydrostatic test on the same rock would be the obvious check, and its absence is the main empirical gap.\n\nThe validation is otherwise thin: one basalt test, parameters fitted to it, and the model visibly misses the pre-peak dilatancy. The authors say so themselves, which I credit. The dilatancy saturation under cyclic loading is the most interesting qualitative behavior, but it is only shown in simulations, not matched to any cyclic test.\n\nNone of this invalidates the formal derivation. It is a neat and useful result for people working in hyperelastic-plastic coupling and computational geomechanics. It just needs to be framed as a constructed link between two yield-criteria families under a particular hyperelasticity, with the physical claim deferred until independent data come in.\n\nMy recommendation: send it to review. The derivation deserves referee time, and the referees can push the authors to tone down the title and add the missing experiments or at least an explicit discussion of the cross-constraint.","headline":"A clean formal mapping from Drucker-Prager to Hoek-Brown via hyperbolic hyperelasticity, but the \"hyperelastic nature\" claim rests on an untested cross-constraint and thin validation.","tokens_in":18638,"tokens_out":2359,"would_cite":true,"duration_ms":26738,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["74C05","74B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper derives the quadratic Hoek-Brown yield criterion from the linear Drucker-Prager criterion by adding one hyperbolic elastic law, showing that the curved rock strength envelope is a hyperelastic image of an underlying linear…","keywords":["hyperelasticity","elasto-plastic model","Hoek-Brown criterion","Drucker-Prager criterion","Generalized Standard Materials","yield surface","dilatancy saturation","finite elements"],"falsifier":"Measure the elastic unloading branch in cyclic hydrostatic compression: the model predicts $\\bar\\sigma=\\frac{1}{4\\beta_m}\\left(1-(1+2\\kappa_i\\beta_m\\,\\mathrm{Tr}\\varepsilon)^{-2}\\right)$ and a finite limiting volumetric strain $\\varepsilon_0=-1/(2\\kappa_i\\beta_m)$. Observing reversible compression beyond that strain, or measuring a stress-space yield surface that is not parabolic in a material whose plastic-force surface is linear, would refute the central transformation.","tokens_in":17520,"feed_emoji":"🪨","tokens_out":11360,"duration_ms":116155,"temperature":0.7,"pith_summary":"The paper sets out to show that a famously empirical piece of rock mechanics, the curved, quadratic Hoek-Brown yield criterion that says shear strength rises nonlinearly with compression, is not an independent law. It builds an elasto-plastic model with a hyperelastic bulk modulus that softens hyperbolically as the material compresses, and shows that under this law the linear Drucker-Prager yield criterion, written in the thermodynamic force space, maps exactly to a quadratic Hoek-Brown-type criterion in observable stress space. If correct, the curvature of a geomaterial's yield surface and its elastic nonlinearity are two views of the same effect, and one extra elastic parameter converts a linear failure rule into the empirically fitted quadratic one. The paper also reports that this model produces dilatancy saturation under cyclic triaxial loading, a behavior linear elasto-plastic models miss, and demonstrates stable finite element simulations, so the construction is not only formal.","feed_headline":"Elastic law turns a linear rock-failure rule into the Hoek-Brown curve","feed_subtitle":"Rocks' curved strength envelope is the stress-space image of the linear Drucker-Prager criterion.","key_machinery":"The load-bearing object is the hyperbolic strain-dependent bulk modulus (12), $\\kappa(\\varepsilon)=\\kappa_i/(1+2\\kappa_i\\beta_m\\,\\mathrm{Tr}\\varepsilon)$, with the shear modulus held constant. Because the inverse of this modulus is linear in $\\mathrm{Tr}\\varepsilon$, the derivative term in the stress expression becomes strain-independent, which makes the map between plastic force and stress exact and strain-independent: $\\sigma=X-\\beta_m X_m^2 I$. Substituting this algebraic identity into the linear cone $f_X$ is what produces the quadratic $f_\\sigma$; the same identity keeps the stress-space yield surface fixed during elastic loading, so the perfect-plasticity description remains well posed. The associated flow rule supplied by the Generalized Standard Materials framework gives the plastic strain evolution.","core_discovery":"The central claim is equation (19). Start with the standard linear Drucker-Prager yield criterion $f_X(X)=\\frac{1}{\\sqrt 6}\\|X_D\\|+aX_m-b$ in the space of the plastic force $X$, and choose the hyperbolic elasticity of eq. (12): a bulk modulus $\\kappa(\\varepsilon)=\\kappa_i/(1+2\\kappa_i\\beta_m\\,\\mathrm{Tr}\\varepsilon)$ with a constant shear modulus. Then stress and plastic force are linked by the strain-independent relation $\\sigma=X-\\beta_m X_m^2 I$, and substituting it into $f_X$ gives, exactly, $f_\\sigma(\\sigma)=\\frac{\\beta_m}{6}\\|\\sigma_D\\|^2+\\frac{a-2\\beta_m b}{\\sqrt 6}\\|\\sigma_D\\|+a^2\\sigma_m-b(a-\\beta_m b)$. Thus what experimentalists see as the curved Hoek-Brown-type stress criterion is the image of a linear plastic-force criterion under hyperelasticity; the quadratic shape is derived, not assumed. The paper further claims that the same model yields dilatancy saturation and strain accommodation in cyclic triaxial tests, and that it remains a practical constitutive model for finite element computations.","pith_inferences":["If the mapping idea generalizes, any yield criterion in plastic-force space, not only a linear cone, could be reinterpreted as the stress-space shadow of a simpler thermodynamic criterion; the case $\\beta_D>0$, which the paper leaves open, would add deviatoric-volumetric coupling and change the octahedral cross-section of the resulting surface.","Because the derivation makes yield-surface curvature a property of the elastic branch, experimental programs that assume linear elasticity while inferring rock strength parameters may be systematically misattributing some observed nonlinearity; re-analyzing existing triaxial data with the curvature parameter $\\beta_m$ as a free variable would test this directly.","The paper's own comparison with alkali basalt shows the model captures the initial nonlinear volumetric strain but not the dilatancy onset before peak stress, a shortfall the authors attribute to missing hardening; the unification claim should therefore be read as a statement about yield-surface shape, not as a complete constitutive law for rocks.","The hyperbolic elasticity route offers a thermodynamically consistent alternative to non-associated flow rules for controlling dilatancy, since it achieves dilatancy saturation through the elastic stiffness evolution rather than by weakening the flow-rule normality condition."],"forward_implications":["A single extra parameter $\\beta_m$ controls the curvature of the stress-space yield surface, so calibrating a Hoek-Brown-type criterion reduces to fitting a linear Drucker-Prager criterion plus one elastic nonlinearity parameter.","Under cyclic triaxial compression the model shows progressive accommodation of volumetric strain and saturation of dilatancy, phenomena the linear elasto-plastic Drucker-Prager model cannot reproduce, giving a thermodynamic explanation for a well-known geomaterial observation.","Hydrostatic compression under the model has a finite volumetric strain limit $\\varepsilon_0=-1/(2\\kappa_i\\beta_m)$ while the compressive stress grows without bound, so the elastic law predicts a maximum compactable strain.","The implicit return-mapping integration is fully analytical and the consistent tangent operator is symmetric positive semidefinite, allowing robust finite element calculations of shear-band formation.","The derivation provides a new status for the Hoek-Brown criterion: rather than a purely empirical fit, it is the observable stress-space form of a linear criterion in the thermodynamic force space whenever the bulk modulus follows the hyperbolic law."],"supporting_citations":[{"why":"Supplies the linear yield criterion $f_X$ used as the starting plastic-force surface.","marker":"(Drucker and Prager, 1952)"},{"why":"Supplies the empirical quadratic strength criterion that equation (19) is claimed to recover and explain.","marker":"(Hoek and Brown, 1980)"},{"why":"Supplies the rotational-invariant parabolic form (A.3) to which the derived stress-space criterion is compared.","marker":"(Pan and Hudson, 1988)"},{"why":"Provides the Generalized Standard Materials framework that justifies the associated flow rule and the role of the plastic force $X$.","marker":"(Halphen and Nguyen, 1975)"},{"why":"Provides the work-principle basis for the free-energy and dissipation formulation the model uses.","marker":"(Marigo, 1989)"},{"why":"Documents the excessive-dilatancy problem of associated linear criteria that motivates the cyclic-loading comparison.","marker":"(Vermeer and de Borst, 1984)"},{"why":"Provides the alkali-basalt uniaxial compression data used for calibration and partial experimental comparison.","marker":"(Heap et al., 2009)"}],"fun_headline_variants":["Hyperelastic law bends linear yield into Hoek-Brown curve","Derived, not assumed: quadratic rock failure from linear law","Linear plastic force becomes curved Hoek-Brown stress criterion","Dilatancy saturation traced to hyperelastic-plastic coupling","How hyperelasticity turns Drucker-Prager into Hoek-Brown shape"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation assumes the bulk modulus softens exactly as $\\kappa(\\varepsilon)=\\kappa_i/(1+2\\kappa_i\\beta_m\\,\\mathrm{Tr}\\varepsilon)$ while the shear modulus stays constant, a form chosen so that the stress-plastic-force map is strain-independent and not justified by experiment or microstructure; if real elastic stiffness follows another law, the exact quadratic Hoek-Brown-type stress criterion does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Hyperelastic law bends linear yield into Hoek-Brown curve","Derived, not assumed: quadratic rock failure from linear law","Linear plastic force becomes curved Hoek-Brown stress criterion","Dilatancy saturation traced to hyperelastic-plastic coupling","How hyperelasticity turns Drucker-Prager into Hoek-Brown shape"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000176,"raw_usage":{"total_tokens":1294,"prompt_tokens":957,"completion_tokens":337,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":573,"completion_tokens_details":{"reasoning_tokens":248}},"tokens_in":573,"tokens_out":337,"duration_ms":4146,"temperature":1.0,"reasoning_tokens":248,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:22:35.737448+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the elastic unloading branch in cyclic hydrostatic compression: the model predicts $\\bar\\sigma=\\frac{1}{4\\beta_m}\\left(1-(1+2\\kappa_i\\beta_m\\,\\mathrm{Tr}\\varepsilon)^{-2}\\right)$ and a finite limiting volumetric strain $\\varepsilon_0=-1/(2\\kappa_i\\beta_m)$. Observing reversible compression beyond that strain, or measuring a stress-space yield surface that is not parabolic in a material whose plastic-force surface is linear, would refute the central transformation.","supporting_citations":[{"cited_title":", author Prager, W","cited_arxiv_id":null,"evidence_quote":"Supplies the linear yield criterion $f_X$ used as the starting plastic-force surface."},{"cited_title":", author Brown, E.T","cited_arxiv_id":null,"evidence_quote":"Supplies the empirical quadratic strength criterion that equation (19) is claimed to recover and explain."},{"cited_title":", author Hudson, J","cited_arxiv_id":null,"evidence_quote":"Supplies the rotational-invariant parabolic form (A.3) to which the derived stress-space criterion is compared."},{"cited_title":", author Nguyen, Q.S","cited_arxiv_id":null,"evidence_quote":"Provides the Generalized Standard Materials framework that justifies the associated flow rule and the role of the plastic force $X$."},{"cited_title":", year 1989","cited_arxiv_id":null,"evidence_quote":"Provides the work-principle basis for the free-energy and dissipation formulation the model uses."},{"cited_title":", author de Borst , R","cited_arxiv_id":null,"evidence_quote":"Documents the excessive-dilatancy problem of associated linear criteria that motivates the cyclic-loading comparison."},{"cited_title":", author Vinciguerra, S","cited_arxiv_id":null,"evidence_quote":"Provides the alkali-basalt uniaxial compression data used for calibration and partial experimental comparison."}],"review_version":1}