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REVIEW 3 major objections 3 minor 53 references

Hyperelastic nature of the Hoek-Brown criterion

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2507.15813 v1 pith:2PZVPH6C submitted 2025-07-21 physics.app-ph cs.CEphysics.class-ph

Ilaria Fontana , Goustan Bacquaert , Daniele A. Di Pietro , Kyrylo Kazymyrenko This is my paper · ORCID
classification physics.app-phcs.CEphysics.class-ph MSC 74C0574B20
keywords hyperelasticityelasto-plasticmodelHoek-BrowncriterionDrucker-PragerGeneralizedStandardMaterialsyieldsurfacedilatancysaturationfiniteelements
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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

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.

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 (3)
  1. [Sec. 3.1, Eq. (12)] 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.
  2. [Sec. 4.5, Fig. 7] 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.
  3. [Sec. 4.5, Fig. 7b] 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.
minor comments (3)
  1. [References] Several references contain typographical errors, including 'Hoek-Brown strenght' in the Li et al. entry and 'Internation Journal' in the Zhang et al. entry.
  2. [Sec. 3.2, Eq. (19)] 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.
  3. [Sec. 4.5, Fig. 7] The calibrated parameter values are reported only in the figure legend; for reproducibility they should also appear in the text or in a table.

Circularity Check

1 steps flagged · score 3.0 of 10

Mathematical derivation is self-contained, but the uniaxial 'prediction' is a calibration of the same test and the physical claim rests on an untested ad hoc hyperelastic law.

  1. fitted input called prediction [Section 4.5 and Figure 7 caption]
    "we first calibrate the model's elastic parameters. This process is illustrated in Figure 7a by fitting the initial evolution of the axial stress with the deviatoric strain, and with the volumetric strain, whose relationships involve µi and (κi, βm) respectively. ... The parameter a is then estimated by aligning the yield stress with the experimentally observed failure stress of approximately 140 MPa ... Solid lines represent the model predictions, while dots correspond to experimental measurements."

    The plotted 'predictions' are generated with parameters κi, βm, µi, and a calibrated on the same uniaxial test: κi and βm from the initial volumetric-strain branch, µi from the deviatoric branch, and a from the observed failure stress. The model curve is therefore constrained to reproduce those calibration points, so the agreement in Figure 7 is partly forced by construction rather than being an out-of-sample test. This does not make the algebraic transformation from (12) and (16) to (19) circular, but it means the empirical 'comparison' does not independently validate the hyperelastic origin claim.

full rationale

The central derivation is not circular. The quadratic stress-space criterion (19) is obtained by eliminating the plastic force X between the assumed hyperbolic elasticity, leading to the strain-independent relation σ = X − βm Xm² I in (15), and the linear Drucker–Prager criterion (16). The Hoek–Brown quadratic shape is not an input: it emerges from the algebra, and the paper does not fit (19) to Hoek–Brown data. The hyperbolic bulk law (12) is chosen so that the elastic–plastic mapping is strain-independent, and the paper offers no independent experimental or micromechanical justification for that particular dependence; this is a limitation of the physical claim, not a circular step. The main circularity is confined to Section 4.5, where the model parameters are calibrated on the same uniaxial basalt test that is then displayed as 'model predictions'. The cross-constraint that the same βm controls both the hydrostatic law (22) and the quadratic coefficient in (19) is an untested falsifiable consequence, but failing to test it is a validation gap rather than a circular reduction. Self-citations in the paper (Bacquaert et al. 2024, Marigo and Kazymyrenko 2019, Fontana 2022) appear in contextual or perspective remarks and are not load-bearing for the derivation.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

All material parameters are fitted to the experimental test or set by the user; none are predicted by the model. The derivation of the quadratic yield criterion itself is parameter-free once the hyperbolic energy is assumed.

free parameters (3)
  • beta_m (nonlinear parameter) = beta_m kappa_i = 130 (Section 4.5); beta_m = 120/E in synthetic tests
    Controls the curvature of the transformed yield surface and the volumetric nonlinearity; fitted to the initial hydrostatic branch of the basalt response.
  • kappa_i, mu_i (initial bulk and shear moduli) = kappa_i = 1.2 GPa, mu_i = 4.5 GPa (Section 4.5); kappa_i = 5E/6, mu_i = 5E/13 for synthetic tests
    Fitted to the initial elastic loading curves of the basalt test (deviatoric and volumetric strain).
  • a (friction parameter), b (cohesion) = a = 2.8, b = 0 (Section 4.5); a = 1/9, b = E/3000 for synthetic tests
    Chosen so the yield stress matches the observed failure stress near 140 MPa; b set to 0 for the basalt comparison.
assumptions (4)
  • domain assumption Generalized Standard Materials framework with Helmholtz free energy and positive intrinsic dissipation.
    The model is built inside the GSM formalism; this supplies the thermodynamical force X and the normality flow rule (Section 2.1).
  • ad hoc to paper The yield criterion is fixed in the plastic force space and has the linear Drucker-Prager form f_X(X) = ||X_D||/sqrt(6) + a X_m - b (eq. 16).
    This is the central modeling choice; the linearity in X-space is what gets transformed into a quadratic criterion in sigma-space.
  • ad hoc to paper The isotropic elastic moduli follow the hyperbolic strain dependence kappa(epsilon) = kappa_i/(1 + 2 kappa_i beta_m Tr epsilon), mu = const (eq. 12).
    This is the load-bearing constitutive premise; it makes the plastic force to stress map strain-independent and produces the quadratic yield surface.
  • domain assumption Associated flow rule (normality) and perfect plasticity without hardening.
    Standard simplifying assumptions in plasticity; the paper explicitly states the model is a perfect plasticity model (Section 3.2).

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Pith. "Pith review of Hyperelastic nature of the Hoek-Brown criterion." pith.science (2026). https://pith.science/paper/2PZVPH6C

@misc{pith2026250715813,
  author       = {Pith},
  title        = {Pith review of: Hyperelastic nature of the Hoek-Brown criterion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2PZVPH6C}},
  note         = {Machine review of arXiv:2507.15813}
}
read the original abstract

We propose a nonlinear elasto-plastic model, for which a specific class of hyperbolic elasticity arises as a straight consequence of the yield criterion invariance on the plasticity level. We superimpose this nonlinear elastic (or hyperelastic) behavior with plasticity obeying the associated flow rule. Interestingly, we find that a linear yield criterion on the thermodynamical force associated with plasticity results in a quadratic yield criterion in the stress space. This suggests a specific hyperelastic connection between Mohr-Coulomb and Hoek-Brown (or alternatively between Drucker-Prager and Pan-Hudson) yield criteria. We compare the elasto-plastic responses of standard tests for the Drucker-Prager yield criterion using either linear or the suggested hyperbolic elasticity. Notably, the nonlinear case stands out due to dilatancy saturation observed during cyclic loading in the triaxial compression test. We conclude this study with structural finite element simulations that clearly demonstrate the numerical applicability of the proposed model.

Figures

Figures reproduced from arXiv: 2507.15813 by the authors.

Figure 1
Figure 1. Transformation of the yield criterion for different values of the nonlinear parameter [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Graphs of hydrostatic triaxial tests. While during hydrostatic compression, loading and unloading [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Graphs of the triaxial compression test. The different regimes of the response are shown with [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Graphs of the cyclic test. The different regimes of the response are shown with different colors: [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Response of the linear elasto-plastic model for a triaxial compression test. The different regimes of [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Response of the linear elasto-plastic model for a cyclic test. The different regimes of the response [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Comparison between the proposed nonlinear elasto-plastic model and experimental data from a [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Geometry with boundary condition and a mesh example for the compression test in plane strain [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: Responses of the vertical force (positive in traction) vs loading time ( [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Contour plots of the von Mises equivalent strain [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: Close–up of the shear band in case of on heterogeneously oriented mesh. [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.