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The corotational stability postulate: positive incremental Cauchy stress moduli for diagonal, homogeneous deformations in isotropic nonlinear elasticity

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Pith's one-line read The paper proves that a single rate-type stability condition, the corotational stability postulate, forces every standard diagonal deformation test in isotropic hyperelasticity—uniaxial, equibiaxial, planar, and hydrostatic tension—to…

desk verdict CSP implies positive incremental moduli in the four standard homogeneous tests, and the proof is basically right, with one small reparameterization gap that CSP itself closes. read the letter →

arxiv 2411.12552 v1 pith:LEN5AJOO submitted 2024-11-19 math.AP

Patrizio Neff , Nina J. Husemann , Aurélien S. Nguetcho Tchakoutio , Sergey N. Korobeynikov , Robert J. Martin This is my paper · ORCID
classification math.AP MSC 74B20
keywords corotationalstabilitypostulateincrementalCauchystressmoduliisotropichyperelasticitylogarithmicstrainHillinequalityDruckerBaker-Ericksentension-extension
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that one constitutive assumption, the corotational stability postulate (CSP), is strong enough to make Cauchy stress increase with stretch in every classical homogeneous test. For isotropic hyperelastic materials, when a homogeneous deformation keeps its principal axes fixed, the arbitrary corotational rate in CSP reduces to the ordinary material derivative, and the postulate becomes a monotonicity requirement along scalar loading curves. From that reduction the paper derives positive incremental moduli for uniaxial tension, equibiaxial extension, planar tension, and hydrostatic tension, in both compressible and incompressible response. A reader should care because CSP is a minimal rate-type stability condition that is independent of convexity and of local material stability, so these monotonicity consequences give it concrete physical content as a constitutive stability postulate.

What carries the argument

The load-bearing object is the corotational stability postulate (CSP): for every nonzero symmetric deformation rate tensor $D$, the inner product of $D$ with any corotational stress rate $\frac{\mathrm{D}^{\circ}}{\mathrm{D}t}[\sigma]$ is positive, where a corotational rate adds to the material derivative a commutator term $\sigma\Omega - \Omega\sigma$ with an arbitrary spin tensor $\Omega$. The decisive simplification is that for a diagonal, homogeneous family $F(t) = \mathrm{diag}(\lambda_1(t), \lambda_2(t), \lambda_3(t))$, the spin term vanishes and the material derivative becomes the ordinary time derivative of the principal stresses, so CSP reduces to a sum over principal rates. In each standard test the lateral boundary conditions eliminate lateral terms, leaving one term of the form $D_{\lambda_1} \tilde{\sigma}(\lambda_1)\, |\dot{\lambda}_1|^2 / \lambda_1$; because the factor $|\dot{\lambda}_1|^2/\lambda_1$ is positive, positivity of the inner product is exactly positivity of the scalar derivative. This identity is what converts a tensor inequality into four positive incremental moduli.

What would settle it

Search for an isotropic hyperelastic energy that satisfies the true-stress–true-strain monotonicity condition TSTS-M+ for all positive definite symmetric stretch tensors but whose compressible uniaxial path, obtained by solving $\sigma_2(\lambda_1,\lambda_2,\lambda_2)=\sigma_3(\lambda_1,\lambda_2,\lambda_2)=0$ for a smooth $\lambda_2(\lambda_1)$, has a negative slope $d\tilde{\sigma}_1/d\lambda_1$ at some $\lambda_1>1$; if such an energy exists, equation (4.4) and the central claim are contradicted.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is the reduction encoded in equation (4.4): for a diagonal homogeneous deformation family, the positivity condition $\langle \frac{\mathrm{D}^{\circ}}{\mathrm{D}t}[\sigma], D \rangle > 0$ collapses to a single term $D_{\lambda_1} \tilde{\sigma}(\lambda_1) \, |\dot{\lambda}_1|^2 / \lambda_1 > 0$, so the slope of the uniaxial Cauchy stress–stretch curve must be positive; the same calculation with different lateral constraints yields positivity of the incremental equibiaxial, planar tension, and bulk moduli, and their incompressible analogues with Kirchhoff stress in place of Cauchy stress. Since CSP is equivalent to monotonicity of the Cauchy stress as a function of logarithmic strain, the paper also recalls that CSP implies the Baker–Ericksen and tension–extension inequalities and local invertibility of the Cauchy stress–stretch relation. The conclusion is deliberately restricted to diagonal, homogeneous deformations, and the paper stresses that CSP neither implies nor is implied by convexity of the energy in the deformation gradient or the stretch tensor, nor by the local material stability condition known as LH-ellipticity.

Load-bearing premise

The proof assumes that in each loading protocol the unconstrained lateral stretches can be expressed smoothly in terms of the pulled stretch, so the lateral stress conditions can be eliminated; if no such smooth lateral branch exists at some stretch, the reduction of the CSP inner product to one scalar derivative, and hence the positivity conclusion there, is not established.

Editorial extensions

If this is right

  • Any isotropic hyperelastic material satisfying CSP has a monotone increasing Cauchy stress in uniaxial tension, so its incremental Young's modulus is never negative.
  • The same monotonicity holds for equibiaxial extension, planar tension, and hydrostatic tension, giving positive incremental equibiaxial, planar tension, and bulk moduli.
  • In incompressible response CSP coincides with Hill's inequality, so the incompressible uniaxial, equibiaxial, and planar tension stress–stretch curves are monotone.
  • CSP implies the Baker–Ericksen inequalities and the tension–extension inequalities, and it makes the Cauchy stress–stretch relation locally invertible.
  • CSP is compatible with energies that are neither convex nor LH-elliptic, so these positive-modulus conclusions are not consequences of convexity or local material stability.

Reading between the lines

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

  • The paper proves only the forward direction: CSP forces positive incremental moduli; it does not establish that these monotonicities are sufficient for CSP, and the compressible Neo-Hooke example showing monotone uniaxial response without CSP indicates they are not.
  • Because the proof uses only constancy of the principal axes, the same reduction plausibly extends to other homogeneous deformation families whose principal axes stay fixed beyond the four protocols named, whenever the lateral constraints are smooth.
  • The result gives a cheap screening test for proposed constitutive energies: if a computed uniaxial Cauchy stress–stretch curve has a negative slope at some stretch, that energy necessarily violates CSP.
  • The reliance on smooth lateral-stretch functions makes the conclusion local; a material whose lateral response bifurcates or loses smoothness in a test protocol could evade the positivity statement exactly at those stretches.
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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

2 major / 3 minor

Summary. The paper studies the corotational stability postulate (CSP) in isotropic nonlinear elasticity, defined by the requirement that the inner product of any corotational stress rate with the deformation rate be strictly positive for all nonzero symmetric D. It recalls the equivalence between CSP and the TSTS-M^+ monotonicity condition in the logarithmic strain, and then specializes to homogeneous, diagonal deformation families corresponding to uniaxial tension, equibiaxial extension, planar tension, and hydrostatic tension. The central technical content is Section 4, where the CSP inequality is reduced along these test paths to positivity of the relevant incremental Cauchy stress moduli (Eqs. (4.4)-(4.10)). The paper also recalls known consequences of CSP (Baker-Ericksen and tension-extension inequalities, local invertibility of the Cauchy stress-stretch relation) and illustrates the results with compressible and incompressible versions of exponentiated Hencky, Neo-Hooke, and quadratic Hencky energies.

Significance. If the central claim is accepted, CSP emerges as a clean and easily checkable sufficient condition for monotone, physically reasonable stress-strain response in the standard homogeneous tests, complementing the rank-one convexity/LH-ellipticity condition. The derivations in Section 4 are elementary and transparent, and the examples provide concrete, falsifiable predictions, e.g., that exp-Hencky energies have monotone uniaxial Cauchy stress while quadratic Hencky energies can have non-monotone response. The paper is also honest about what CSP does not imply: it does not imply convexity of the energy nor LH-ellipticity. The main weakness is that the admissible-path structure used in the key reductions is not stated as a formal assumption, and one of the worked incompressible examples contains a quantitative formula error.

major comments (2)
  1. [Section 4.1, Eq. (4.4)] The chain-rule reduction in Eq. (4.4) is not fully justified as written. The manuscript assumes λ2 = λ3 = λ2(λ1(t)) without proving that the lateral conditions σ2 = σ3 = 0 determine λ2 and λ3 as locally smooth functions of λ1, nor that the admissible path can be reparameterized by λ1. Since the derivation divides by |λ̇1|²/λ1 to conclude Dλ1σ̃(λ1) > 0, the existence of admissible nonzero paths with λ̇1 ≠ 0 is needed. A short lemma repairs this: along a uniaxial path with σ2 = σ3 = 0 identically, the CSP inner product equals σ̇1 λ̇1/λ1, so any nonzero admissible path must have λ̇1 ≠ 0, and local reparameterization by λ1 follows. Please add this lemma and state the corresponding standing assumptions on existence and regularity of the admissible paths for Eqs. (4.4)-(4.6) as well.
  2. [Section 4.2, Eqs. (4.8)-(4.10)] The incompressible reductions silently omit the pressure-rate term. Since σ = τ = -p 1 + τ_e when det F = 1, the material derivative contains -ṗ 1, and its inner product with D is -ṗ tr D. The reduction to ∑∂t[τ_i] λ̇_i/λ_i is valid only because tr D = 0 for isochoric motion. This one-line justification should be stated explicitly; as written, the derivation appears to ignore the pressure contribution without explanation.
minor comments (3)
  1. [Example 5.4, Eq. (5.16)] The principal stress formula in Eq. (5.16) contains an erroneous division by λ_i. For the incompressible exponentiated Hencky energy, τ_i = ∂W/∂log λ_i = 2μ exp(k‖log V‖²) log λ_i, so the displayed σ_i = τ_i = -p + 2μ (log λ_i)/λ_i e^{...} is inconsistent with Eq. (5.2) evaluated at det F = 1. The formulas in (5.17)-(5.18) and the corresponding curves should be corrected; the qualitative monotonicity conclusion is unaffected.
  2. [Footnote 1 and reference [25]] The equivalence (1.1) ⇔ (1.3) is quoted from a submitted article and an in-preparation article. Since Section 4 repeatedly invokes the implication TSTS-M^+ ⇒ ⟨∂tσ,D⟩ > 0, the paper would be more self-contained if this equivalence were either proved here or explicitly declared as an imported result whose proof is available elsewhere.
  3. [Title and abstract] The phrase "diagonal, homogeneous deformations" is broader than the class of standard tests (uniaxial, equibiaxial, planar, hydrostatic) analyzed in Section 4; a qualifier such as "in standard homogeneous tests" would make the scope of the claim more precise.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the positive-moduli results are direct consequences of the CSP inner product on diagonal paths; only minor same-group citations appear in the framing equivalences.

full rationale

The paper's central implication is derived, not assumed: for a diagonal homogeneous deformation family the corotational rate reduces to the material derivative, and with the uniaxial constraints σ2=σ3=0 the CSP inequality (4.3) collapses to Dλ1 σ̃ |λ̇1|²/λ1 > 0, giving Eincr = Dλ1 σ̃ > 0. The analogous computations for equibiaxial, planar, and hydrostatic loading repeat the same algebraic reduction. Nothing is fitted, and no incremental modulus is defined in terms of the CSP inner product; the positivity is an inequality consequence rather than a renaming. The only technical caveat is the implicit assumption that lateral-stretch functions such as λ2(λ1) exist and are differentiable and that λ̇1 ≠ 0 at the evaluation point; the strict CSP inequality forces the displayed product to be positive, so the nonzero-rate condition is automatic inside the argument, and the solvability/regularity of the admissible path is a domain assumption, not a circular one. The framing equivalences (CSP ⇔ TSTS-M++ and the extension to all reasonable corotational rates) are cited to same-group submitted or in-preparation works ([30], footnote 1 [25]), but Section 4 does not depend on those equivalences: it only uses the diagonal-path reduction, which is independent. Accordingly no load-bearing circular step is present; the score reflects only these minor self-citations in the paper's framing. The absent proof of the 'all reasonable corotational rates' equivalence is an acknowledged limitation, not a circularity.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

No fitted parameters are used in the derivation; the only hand-chosen numbers are illustrative material constants in the examples. The argument rests on the CSP postulate itself, the diagonal-homogeneous deformation assumption, hyperelastic response, existence of the lateral-stretch response, and previously established equivalences involving Hill's inequality and corotational rates. The main new result does not depend on the unpublished equivalence for 'all reasonable corotational rates' cited in footnote 1, but the overall framing of CSP as a general postulate does.

free parameters (1)
  • Illustrative material parameters in examples = e.g., μ=k= ̂k=1, λ=2 (Example 5.1); μ and κ (Example 5.2); E and ν (Examples 5.3-5.6)
    Chosen by hand for the plots only; they do not enter the general derivation and do not affect the conditional positivity conclusions.
assumptions (6)
  • domain assumption The corotational stability postulate (CSP) is assumed for the material response
    The paper's results are conditional: if CSP holds, then the incremental moduli are positive; CSP itself is not derived.
  • domain assumption Deformation family is diagonal and homogeneous with constant principal axes
    Section 1 and eq. (4.1): F(t) = diag(λ1,λ2,λ3) with ˙F F^{-1} diagonal and independent of position; this is the stated scope of the main theorem.
  • domain assumption Hyperelastic response
    Section 1: 'we assume hyperelastic response', used for the principal-stress expressions and the existence of a strain energy.
  • standard math Lateral-stretch response functions λ2(λ1), λ3(λ1) exist and are differentiable along the test paths
    Sections 4.1-4.2: the uniaxial path is written as λ2 = λ2(λ1(t)); this requires the implicit function theorem applied to the lateral free-stress conditions, with no invertibility guarantee discussed.
  • standard math For diagonal D, the corotational rate term is invisible in the inner product
    Used in eq. (4.3) to reduce ⟨D°σ/Dt,D⟩ to ⟨∂tσ,D⟩; follows from the trace identity tr((σΩ−Ωσ)D)=0 for skew Ω and diagonal D, though the paper cites [31].
  • domain assumption In the incompressible case, Hill's inequality on Kirchhoff stress is equivalent to CSP on Cauchy stress
    Section 4.2; cited to [30, 44, 13, 16, 17] and [5], not proved here; the pressure cancellation is not shown.

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Pith. "Pith review of The corotational stability postulate: positive incremental Cauchy stress moduli for diagonal, homogeneous deformations in isotropic nonlinear elasticity." pith.science (2026). https://pith.science/paper/LEN5AJOO

@misc{pith2026241112552,
  author       = {Pith},
  title        = {Pith review of: The corotational stability postulate: positive incremental Cauchy stress moduli for diagonal, homogeneous deformations in isotropic nonlinear elasticity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LEN5AJOO}},
  note         = {Machine review of arXiv:2411.12552}
}
abstract

In isotropic nonlinear elasticity the corotational stability postulate (CSP) is the requirement that \begin{equation*} \langle\frac{\mathrm{D}^{\circ}}{\mathrm{D} t}[\sigma] , D \rangle > 0 \quad \forall \ D \in \text{Sym}(3)\setminus \{0\} \, , \end{equation*} where $\frac{\mathrm{D}^{\circ}}{\mathrm{D} t}$ is any corotational stress rate, $\sigma$ is the Cauchy stress and $D = \text{Sym} \, L$, condition $L= \dot{F} \, F^{-1}$ is the deformation rate tensor. For $\widehat{\sigma}(\log V) := \sigma (V)$ it is equivalent to the monotonicity (TSTS-M$^+$) \begin{equation*} \langle \widehat{\sigma} (\log V_1) - \widehat{\sigma} (\log V_2) , \log V_1 - \log V_2 \rangle > 0 \quad \forall \ V_1, V_2 \in \text{Sym}^{++}(3), \ V_1 \neq V_2 \, . \end{equation*} For hyperelasticity, (CSP) is in general independent of convexity of the mapping $F \mapsto \mathrm{W}(F)$ or $U \mapsto \widehat{\mathrm{W}}(U)$. Considering a family of diagonal, homogeneous deformations $t \mapsto F(t)$ one can, nevertheless, show that (CSP) implies positive incremental Cauchy stress moduli for this deformation family, including the incremental Young's modulus, the incremental equibiaxial modulus, the incremental planar tension modulus and the incremental bulk modulus. Aside, (CSP) is sufficient for the Baker-Ericksen and tension-extension inequality. Moreover, it implies local invertibility of the Cauchy stress-stretch relation. Together, this shows that (CSP) is a reasonable constitutive stability postulate in nonlinear elasticity, complementing local material stability viz. LH-ellipticity.

Figures

Figures reproduced from arXiv: 2411.12552 by the authors.

Figure 1
Figure 1. Infinitesimal Drucker stability: Cauchy stress σ increases with infinitesimal strain ε for geometrically linear but physically nonlinear response, as e.g. in work-hardening small strain plasticity. ⟨σ, ˙ ε˙⟩ > 0 ⇐⇒ ⟨dσ, dε⟩ > 0 ⇐⇒ ⟨σ(ε1) − σ(ε2), ε1 − ε2⟩ > 0 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 4
Figure 4. Incremental Young’s modulus in the logarithmic representation according to (3.2). The physical content of both Eincr and Eincr log coincides, only their numerical values differ. 4 One constitutive condition to rule them all: positive incre￾mental moduli for (CSP) We are now showing that (CSP) in conjunction with a diagonal, homogeneous deformation family t 7→ F(t) leads to positive incremental moduli in uniaxial ten… view at source ↗
Figure 5
Figure 5. Three-dimensional example of a rectangular beam that is pulled in [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figures from the paper (8 more)
Figure 7
Figure 7. Figure 7: Exp-Hencky: convex energy in uniax￾ial tension and its monotone derivative. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Compressible Neo-Hooke: tensile Cauchy stress σ1 and TBiot-stress are still both monotone increasing while (CSP) is not satis￾fied. W (λ1 ) ( ) 1 1 d d W λ λ λ1 [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 11
Figure 11. Figure 11: Nonconvex quadratic Hencky energy in uniaxial tension and its non-monotone deriva￾tive. the LH-ellipticity domain (cf. [3, 35]). Our result in (4.4) is consistent with the non-monotone response of λ1 7→ σe1(λ1). 16 [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 13
Figure 13. Figure 13: Exp-Hencky incompressible: uniax￾ial convex energy and its monotone derivative. Due to det F = 1: λ2 = λ3 = √ 1 λ1 and this implies p = µ λ2 2 = µ 1 √ λ1 2 = µ 1 λ1 =⇒ τe1(λ1) = σ1(λ1) = µ (λ 2 1 − 1 λ1 ) = E 2 (1 + ν) (λ 2 1 − 1 λ1 ) ν= 1 2 = E 3 (λ 2 1 − 1 λ1 ) (λ1…
Figure 15
Figure 15. Figure 15: Neo-Hooke incompressible: convex uniaxial energy and its monotone derivative. We observe that λ1 7→ τe1(λ1) is monotone as it should be due to the satisfaction of Hill’s inequality in the incompressible case and our statement in (4.8). τ 1 (λ1 )  ( ) 1 1 T Biot λ λ1…
Figure 16
Figure 16. Figure 16: Quadratic Hencky incompressible: the uniaxial Kirchhoff-stress τ1 is monotone, while the T 1 Biot-stress remains non monotone. W (λ1 ) ( ) 1 1 d d W λ λ λ1 [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]
Figure 18
Figure 18. Figure 18: Illustration of the deformation φ(x, t) : Ωx → Ωξ and the velocity V (x, t) = v(ξ, t). Considering an arbitrary material quantity Q(x, t) on Ω, equivalently represented by q(ξ, t) on Ωξ, we obtain by the chain rule for the time derivative of Q(x, t) D Dt q(ξ, t) := d …
Figure 19
Figure 19. Figure 19: ): Ωξ Ωx x γ˙ (0) TxΩx γ(s) φ(x, t0) ξ d ds φ(γ(s), t0) [PITH_FULL_IMAGE:figures/full_fig_p024_19.png]

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Works this paper leans on

45 extracted references · 39 canonical work pages

  1. [1]

    Inequalities restricting the form of the stress-deformation relation for isotropic elastic solids and Reiner-Rivlin fluids

    M. Baker and J. L. Ericksen. “Inequalities restricting the form of the stress-deformation relation for isotropic elastic solids and Reiner-Rivlin fluids”. Journal of the Washington Academy of Sciences 44 (1954). Pp. 33–35

    work page 1954

  2. [2]

    On the general theory of elastic stability

    C. B. Biezeno and H. Hencky. “On the general theory of elastic stability.” Koninklijke Akademie van Wettenschappen te Amsterdam 31 (1928). Pp. 569–592

    work page 1928

  3. [3]

    Constitutive inequalities for an isotropic elastic strain energy function based on Hencky’s logarithmic strain tensor

    O. T. Bruhns, H. Xiao, and A. Mayers. “Constitutive inequalities for an isotropic elastic strain energy function based on Hencky’s logarithmic strain tensor”. Proceedings of the Royal Society of London A: Mathematical and Physical Sciences 457 (2001). Pp. 2207–2226

    work page 2001

  4. [4]

    Shear, pure and simple

    R. J. M. C. Thiel J. Voss and P. Neff. “Shear, pure and simple”. International Journal of Non-Linear Mechanics, arXiv: 1806.07749 112 (2018/9). Pp. 57–72

    work page Pith review arXiv 2018

  5. [5]

    A constitutive condition for idealized isotropic Cauchy elasticity involving the logarithmic strain

    M. V. d’Agostino, S. Holthausen, D. Bernardini, A. Sky, I. D. Ghiba, R. J. Martin, and P. Neff. “A constitutive condition for idealized isotropic Cauchy elasticity involving the logarithmic strain”. to appear in Journal of Elasticity, arXiv: 2409.01811 (2024)

    work page Pith review arXiv 2024

  6. [6]

    Nonlinear elasticity

    T. C. Doyle and J. L. Ericksen. “Nonlinear elasticity”. Advances in Applied Mechanics 4 (1956). Pp. 53–115

    work page 1956

  7. [7]

    Some implications of work hardening and ideal plasticity

    D. C. Drucker. “Some implications of work hardening and ideal plasticity”. Quarterly of Applied Mathematics 7.4 (1950). Pp. 411–418

    work page 1950

  8. [8]

    A more fundamental approach to plastic stress-strain relations

    D. C. Drucker. “A more fundamental approach to plastic stress-strain relations”. Proceedings of the First U. S. National Congress of Applied Mechanics (1951). Pp. 187–491

    work page 1951

Show all 45 references
  1. [9]

    Deformations possible in every compressible, isotropic, perfectly elastic material

    J. L. Ericksen. “Deformations possible in every compressible, isotropic, perfectly elastic material”. Studies in Applied Mathematics 34.1-4 (1955). Pp. 126–128

    1955

  2. [10]

    Major symmetry of the induced tangent stiffness tensor for the Zaremba-Jaumann rate and Kirchhoff stress in hyperelasticity: two different approaches

    S. Federico, S. Holthausen, N. J. Husemann, and P. Neff. “Major symmetry of the induced tangent stiffness tensor for the Zaremba-Jaumann rate and Kirchhoff stress in hyperelasticity: two different approaches”. submitted, arXiv: 2410.22163 (2024)

    2024 arXiv

  3. [11]

    I. D. Ghiba, R. J. Martin, and P. Neff. “Constitutive properties for isotropic energies in ideal nonlinear elasticity for solid materials: numerical evidence for invertibility and monotonicity in different stress-strain pairs”. in preparation ()

  4. [12]

    An ellipticity domain for the distortional Hencky logarithmic strain energy

    I.-D. Ghiba, P. Neff, and R. J. Martin. “An ellipticity domain for the distortional Hencky logarithmic strain energy”. Proceedings of the Royal Society of London A: Mathematical and Physical Sciences 471.2184 (2015). doi: 10.1098/ rspa.2015.0510. 20

    2015

  5. [13]

    On uniqueness and stability in the theory of finite elastic strain

    R. Hill. “On uniqueness and stability in the theory of finite elastic strain”. Journal of the Mechanics and Physics of Solids 5.4 (1957). Pp. 229–241

    1957

  6. [14]

    A general theory of uniqueness and stability in elastic-plastic solids

    R. Hill. “A general theory of uniqueness and stability in elastic-plastic solids”. Journal of the Mechanics and Physics of Solids 6.3 (1958). Pp. 236–249

    1958

  7. [15]

    Some basic principles in the mechanics of solids without a natural time

    R. Hill. “Some basic principles in the mechanics of solids without a natural time”. Journal of the Mechanics and Physics of Solids 7 (1959). Pp. 209–225

    1959

  8. [16]

    On constitutive inequalities for simple materials - I

    R. Hill. “On constitutive inequalities for simple materials - I.” Journal of the Mechanics and Physics of Solids 16.4 (1968)

    1968

  9. [17]

    Constitutive inequalities for isotropic elastic solids under finite strain

    R. Hill. “Constitutive inequalities for isotropic elastic solids under finite strain”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 314.1519 (1970). Pp. 457–472

    1970

  10. [18]

    Sur l’extension de la condition de Legendre du calcul des variations aux int´ egrales multiples a plusieurs fonctions inconnues

    L. C. P. van Hove. “Sur l’extension de la condition de Legendre du calcul des variations aux int´ egrales multiples a plusieurs fonctions inconnues”. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 50.1 (1947). Pp. 18–23

    1947

  11. [19]

    Conditions for the onset of elastic and material instabilities in hyperelastic materials

    C. S. Jog and K. D. Patil. “Conditions for the onset of elastic and material instabilities in hyperelastic materials”. Archive of Applied Mechanics 83.5 (2013). Pp. 661–684. issn: 0939-1533. doi: 10.1007/s00419-012-0711-8

    2013 doi

  12. [20]

    Families of Hooke-like isotropic hyperelastic material models and their rate formulations

    S. N. Korobeynikov. “Families of Hooke-like isotropic hyperelastic material models and their rate formulations.” Archive of Applied Mechanics 93 (2023). Pp. 3863–3893

    2023

  13. [21]

    The minimization of matrix logarithms: On a fundamental property of the unitary polar factor

    J. Lankeit, P. Neff, and Y. Nakatsukasa. “The minimization of matrix logarithms: On a fundamental property of the unitary polar factor”. Linear Algebra and its Applications 449 (2014). Pp. 28–42.doi: 10.1016/j.laa.2014.02.012

    2014 doi

  14. [22]

    A constitutive inequality for hyperelastic materials in finite strain

    J. B. Leblond. “A constitutive inequality for hyperelastic materials in finite strain”. European Journal of Mechanics - A/Solids 11.4 (1992). Pp. 447–466

    1992

  15. [23]

    Conditions de stabilit´ e et postulat de Drucker

    J. Mandel. “Conditions de stabilit´ e et postulat de Drucker”. In: J. Kravtchenko, P. M. Sirieys (eds) Rheology and Soil Mechanics/Rh´ eologie et M´ ecanique des Sols, International Union of Theoretical and Applied Mechanics(1966). Pp. 58–68

    1966

  16. [24]

    J. E. Marsden and T. Hughes. Mathematical Foundations of Elasticity . Courier Dover Publications, 1994

    1994

  17. [25]

    The corotational stability postulate is equivalent to the true stress-true strain monotonicity condition

    R. J. Martin, I. D. Ghiba, and P. Neff. “The corotational stability postulate is equivalent to the true stress-true strain monotonicity condition”. in preparation ()

  18. [26]

    Monotonicity of isotropic tensor functions on the set of symmetric matrices: Hill’s generalization of the Davis-Lewis convexity theorem revised

    R. J. Martin, J. Voss, I. D. Ghiba, M. V. d’Agostino, and P. Neff. “Monotonicity of isotropic tensor functions on the set of symmetric matrices: Hill’s generalization of the Davis-Lewis convexity theorem revised”. in preparation ()

  19. [27]

    A non-ellipticity result, or the impossible taming of the logarithmic strain measure

    R. J. Martin, I.-D. Ghiba, and P. Neff. “A non-ellipticity result, or the impossible taming of the logarithmic strain measure”. International Journal of Non-Linear Mechanics 102 (2018). Pp. 147–158

    2018

  20. [28]

    How to characterize a nonlinear elastic material? A review on nonlinear constitutive parameters in isotropic finite elasticity

    L. A. Mihai and A. Goriely. “How to characterize a nonlinear elastic material? A review on nonlinear constitutive parameters in isotropic finite elasticity”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473.2207 (2017). P. 20170607

    2017

  21. [29]

    A finite element implementation of the isotropic exponentiated Hencky-logarithmic model and simulation of the eversion of elastic tubes

    B. Nedjar, H. Baaser, R. J. Martin, and P. Neff. “A finite element implementation of the isotropic exponentiated Hencky-logarithmic model and simulation of the eversion of elastic tubes”. Computational Mechanics 62.4 (2018). Pp. 635–654, available at arXiv:1705.08381

    2018 arXiv

  22. [30]

    Hypo-elasticity, Cauchy-elasticity, corotational stability and monotonicity in the logarithmic strain

    P. Neff, S. Holthausen, M. V. d’Agostino, D. Bernardini, A. Sky, I. D. Ghiba, and R. J. Martin. “Hypo-elasticity, Cauchy-elasticity, corotational stability and monotonicity in the logarithmic strain”. submitted, arXiv:2409.20051 (2024)

    2024 arXiv

  23. [31]

    A natural requirement for objective corotational rates - on structure preserving corotational rates

    P. Neff, S. Holthausen, S. N. Korobeynikov, I. D. Ghiba, and R. J. Martin. “A natural requirement for objective corotational rates - on structure preserving corotational rates”. to appear in Acta Mechanica, arXiv: 2409.19707 (2024)

    2024 arXiv

  24. [32]

    An essay on constitutive stability in idealized isotropic nonlinear elasticity for universal deformations, positive incremental moduli and the onset of necking

    P. Neff, N. J. Husemann, S. Holthausen, A. S. N. Tchakoutio, I. D. Ghiba, and R. J. Martin. “An essay on constitutive stability in idealized isotropic nonlinear elasticity for universal deformations, positive incremental moduli and the onset of necking”. in preparation ()

  25. [33]

    The axiomatic deduction of the quadratic Hencky strain energy by Heinrich Hencky

    P. Neff, B. Eidel, and R. J. Martin. “The axiomatic deduction of the quadratic Hencky strain energy by Heinrich Hencky”. arXiv preprint, available at arXiv:1402.4027 (2014)

    2014 arXiv

  26. [34]

    Geometry of logarithmic strain measures in solid mechanics

    P. Neff, B. Eidel, and R. J. Martin. “Geometry of logarithmic strain measures in solid mechanics”. Archive for Rational Mechanics and Analysis 222.2 (2016). Pp. 507–572, available at arXiv:1505.02203.doi: 10.1007/s00205-016-1007- x

    2016 arXiv

  27. [35]

    The exponentiated Hencky-logarithmic strain energy. Part I: Constitutive issues and rank-one convexity

    P. Neff, I.-D. Ghiba, and J. Lankeit. “The exponentiated Hencky-logarithmic strain energy. Part I: Constitutive issues and rank-one convexity”. Journal of Elasticity 121.2 (2015). Pp. 143–234. doi: 10.1007/s10659-015-9524-7

    2015 doi

  28. [36]

    The exponentiated Hencky-logarithmic strain energy. Part II: coercivity, planar polyconvexity and existence of minimizers

    P. Neff, J. Lankeit, I.-D. Ghiba, R. J. Martin, and D. J. Steigmann. “The exponentiated Hencky-logarithmic strain energy. Part II: coercivity, planar polyconvexity and existence of minimizers”. Zeitschrift f¨ ur angewandte Mathematik und Physik 66.4 (2015). Pp. 1671–1693. doi:...

    2015 doi

  29. [37]

    On Grioli’s minimum property and its relation to Cauchy’s polar decomposition

    P. Neff, J. Lankeit, and A. Madeo. “On Grioli’s minimum property and its relation to Cauchy’s polar decomposition”. International Journal of Engineering Science 80 (2014). Pp. 209–217. doi: 10.1016/j.ijengsci.2014.02.026. 21

    2014 doi

  30. [38]

    A logarithmic minimization property of the unitary polar factor in the spectral and Frobenius norms

    P. Neff, Y. Nakatsukasa, and A. Fischle. “A logarithmic minimization property of the unitary polar factor in the spectral and Frobenius norms”. SIAM Journal on Matrix Analysis and Applications 35.3 (2014). Pp. 1132–1154. doi: 10.1137/130909949

    2014 doi

  31. [39]

    On the second-order work in plasticity

    H. Petryk. “On the second-order work in plasticity”. Archives of Mechanics (Warszawa) 43.2-3 (1991). Pp. 377–397

    1991

  32. [40]

    Das isotrope Elastizit¨ atsgesetz

    H. Richter. “Das isotrope Elastizit¨ atsgesetz”. Zeitschrift f¨ ur Angewandte Mathematik und Mechanik 28.7/8 (1948). Pp. 205–209, available at https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/richter_ isotrop_log.pdf

    1948

  33. [41]

    Verzerrungstensor, Verzerrungsdeviator und Spannungstensor bei endlichen Form¨ anderungen

    H. Richter. “Verzerrungstensor, Verzerrungsdeviator und Spannungstensor bei endlichen Form¨ anderungen”.Zeitschrift f¨ ur Angewandte Mathematik und Mechanik29.3 (1949). Pp. 65–75

    1949

  34. [42]

    Constitutive modeling of brain tissue: current perspective

    R. de Rooij and E. Kuhl. “Constitutive modeling of brain tissue: current perspective”. Applied Mechanics Review 68 (2016). P. 010801

    2016

  35. [43]

    The incremental bulk modulus, Young’s modulus and Poisson’s ratio in nonlinear isotropic elasticity: physically reasonable response

    N. H. Scott. “The incremental bulk modulus, Young’s modulus and Poisson’s ratio in nonlinear isotropic elasticity: physically reasonable response”. Mathematics and Mechanics of Solids 12 (2006). Pp. 526–542

    2006

  36. [44]

    Sur les restrictions ` a imposer ` a l’´ energie de d´ eformation d’un mat´ eriau hyper´ elastique

    R. Sidoroff. “Sur les restrictions ` a imposer ` a l’´ energie de d´ eformation d’un mat´ eriau hyper´ elastique.”Comptes Rendus de l’Acad´ emie des Sciences Paris279 (1974). Pp. 379–382

    1974

  37. [45]

    Universal deformations and inhomogeneities in isotropic Cauchy elasticity

    A. Yavari. “Universal deformations and inhomogeneities in isotropic Cauchy elasticity”. arXiv:2404.06235 (2024). A Notation The deformation φ(x, t), the material time derivative D Dt and the partial time derivative ∂t In accordance with [24] we agree on the following conventio...

    2024 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.