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Identification of second-gradient elastic materials from planar hexagonal lattices. Part I: Analytical derivation of equivalent constitutive tensors

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Pith's one-line read A hexagonal lattice of hinged, axially loaded bars is equivalent to a form I second-gradient elastic material, with closed-form condensed constitutive tensors.

desk verdict Closed-form condensed second-gradient tensors for the three-stiffness hexagonal lattice, honestly scoped; the main limitation—condensed projection—is disclosed, not hidden. read the letter →

arxiv 1908.01568 v1 pith:6CZZY42I submitted 2019-08-05 physics.class-ph

G. Rizzi , F. Dal Corso , D. Veber , D. Bigoni This is my paper · ORCID
classification physics.class-ph MSC 74Q0574B05 PACS 46.05.+b
keywords straingradientelasticitynon-localmaterialnon-centrosymmetricinternallengthhomogenizationhexagonallatticesecond-gradientenergyequivalence
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 attempts to establish that a periodic planar hexagonal lattice of hinged bars, with three different axial stiffnesses, behaves at larger scales as a homogeneous second-gradient elastic material of 'form I' type, not merely as an ordinary Cauchy elastic solid. The equivalence is obtained by matching the elastic energy stored in a lattice cell with that of a continuum cell under remote quadratic displacement fields augmented by an additional field that restores equilibrium. The paper derives closed-form expressions for the condensed constitutive tensors that encode the lattice's nonlocal response, including couplings between strain and curvature. If correct, these formulas make the lattice's size-dependent, anisotropic, and non-centrosymmetric behavior computable from the bar stiffnesses and the hexagon side length, with ordinary Cauchy elasticity recovered only as the side length goes to zero. The identification is explicitly 'condensed,' and completing it to a full positive-definite second-gradient material is deferred to Part II.

What carries the argument

The central object is a 'form I' second-gradient elastic material, that is, a continuum whose energy is quadratic in both the strain and the strain gradient. The machinery is energy equivalence under constrained quadratic displacement boundary conditions: a lattice cell is loaded by $u = \alpha x + (x\otimes x):\beta$ plus an additional linear displacement $\Delta u$ chosen so that all nodal resultants vanish, and the same equilibrium constraint reduces the six quadratic amplitudes $\beta$ to four independent ones. The key fact is that this constraint produces the same reduced kinematics for the lattice and for the second-gradient continuum, so the transformation matrix between constrained and free amplitudes coincides ($T_{\rm lat}=T_{\rm SGE}$). Matching the cell energies for every cell position and every pair of amplitudes then determines the condensed tensors $C$, $M^*$, and $A^*$; matching the purely linear part first reproduces the known isotropic Cauchy moduli of the lattice. The 'condensed' qualification means that only those combinations of the full fifth- and sixth-order tensors that enter the quadratic self-equilibrated tests are fixed by the procedure.

What would settle it

Deform a large hexagonal lattice under a set of independent cubic displacement fields, measure the cell energy, and test whether these data can be matched by a positive-definite completion of the condensed tensors $M^*$ and $A^*$; a mismatch, or a completion that violates positive definiteness for positive $k,\hat{k},\tilde{k}$, would show that the lattice is not the full 'form I' material claimed.

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Extended reading notes

Core claim

The central claim is that a hexagonal lattice with axially deformable bars can be identified with a 'form I' second-gradient elastic material: a continuum whose stored energy depends on strain and on the gradient of strain. The identification is made by imposing equality between the energy of a lattice unit cell and that of a hexagonal continuum cell for displacement fields with a dominant quadratic part, plus an extra linear displacement that makes the nodal forces vanish. This yields closed-form condensed constitutive tensors: the fourth-order Cauchy tensor $C$, the fifth-order strain-curvature coupling tensor $M^*$, and the sixth-order curvature tensor $A^*$ (Eqs. (87)--(90)). The equivalent material is generally nonlocal, anisotropic, and non-centrosymmetric, while its first-order Cauchy reduction is isotropic, local, and centrosymmetric; it reduces to a Cauchy material only as the hexagon side length $\ell$ tends to zero, when $M^*$ and $A^*$ vanish. Because the energy matching is restricted to self-equilibrated quadratic displacement fields, only a 'condensed' form of the material is identified, and the completion to the full set of second-gradient tensors is left to Part II.

Load-bearing premise

The load-bearing premise is that matching the energy only for displacement fields that already satisfy equilibrium is enough to pin down a real equivalent material: the missing higher-order stiffness constants must be fillable into a physically valid, positive-definite elastic solid, not just a description of the particular tests used.

Editorial extensions

If this is right

  • The equivalent continuum has an internal length $\ell$: the higher-order moduli $M^*$ scale linearly and $A^*$ quadratically with the hexagon side length, so size effects disappear as the lattice is refined.
  • The first-order (Cauchy) equivalent material is local, isotropic, and centrosymmetric, while the higher-order equivalent material is generally anisotropic and non-centrosymmetric; $M^*_{13}$ and $M^*_{14}$ vanish only on special curves in the stiffness-ratio plane.
  • The first-order identification recovers the known isotropic Cauchy moduli of the hexagonal lattice, so the second-gradient formulas extend the established effective elasticity rather than replacing it.
  • Because the identification is condensed, further independent tests or constraints are needed to determine the full second-gradient material; Part II is presented as supplying the missing completion, positive definiteness, and validation.
  • The closed-form dependence of the tensors on $k,\hat{k},\tilde{k}$ and $\ell$ gives a direct design rule for microstructured solids with tunable nonlocal response.

Reading between the lines

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

  • A natural extension, not made in the paper: if Part II produces a positive-definite completion, the condensed formulas become a ready homogenization rule, so that dispersion relations and gradient-dominated deformations of the lattice could be computed directly from the continuum tensors without solving the discrete network.
  • A testable consequence the authors do not draw: subjecting the lattice to remote displacement fields with cubic components would excite the higher-order moduli that the condensed identification leaves undetermined; measuring that energy would show whether a valid completion exists.
  • The same energy-matching-with-equilibrium-restoring-field scheme could be applied to other periodic truss topologies; the hexagonal case is special because the equilibrium constraint on the quadratic amplitudes coincides for the lattice and the continuum, which may not hold for other geometries.
  • The stiffness-ratio-dependent non-centrosymmetry at the condensed level implies that lattices with unequal bars could serve as planar mechanical analogues of non-centrosymmetric or odd-elastic solids, a directional-response consequence the paper mentions only as design relevance.
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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 / 6 minor

Summary. This paper (Part I of two) derives, by energy equivalence, a homogeneous second-gradient elastic ('form I' Mindlin) solid equivalent to an infinite periodic hexagonal lattice of hinged bars deforming in pure axial extension, with three distinct bar stiffnesses k, k-hat, and k-tilde arranged so as to preserve hexagonal symmetry. The method prescribes remote displacements with a dominant quadratic component, augmented by an 'additional field' chosen to make the lattice nodal resultants vanish, and matches the lattice cell energy to the energy of a form I Mindlin solid under the corresponding equilibrium-constrained quadratic fields. This yields closed-form expressions for the first-order Cauchy tensor C (Eq. 87), the condensed coupling tensor M* and condensed curvature tensor A* (Eqs. 89-90), together with the coincidence of the lattice and continuum equilibrium-constraint sets, T_lat = T_SGE (Eq. 88). The identification is explicitly 'condensed': it fixes only the projections of the full higher-order tensors onto the four-dimensional self-equilibrated quadratic subspace, a limitation that the abstract and Section 5 acknowledge. The equivalent response reduces to a Cauchy material only as the hexagon side length tends to zero. Positive definiteness and symmetry of the tensors, completion to a full second-gradient material, and numerical validation are deferred to Part II.

Significance. If correct, the result provides the first fully explicit, closed-form nonlocal constitutive law for the three-stiffness hexagonal axial-bar lattice, demonstrating a physically interesting effect: a lattice that is isotropic, local, and centrosymmetric at the Cauchy order becomes anisotropic, nonlocal, and non-centrosymmetric at the second-gradient order (Eq. 91, Fig. 5). Strong points are the systematic energy-matching framework, the explicit reporting of all coefficient matrices in Appendix A (which makes the closed-form results independently checkable), the clean l- and l^2-scaling of the higher-order terms, and a nontrivial internal consistency check: the self-equilibrium constraint sets of the lattice and of the second-gradient solid coincide once C is identified (Eq. 88). The derivation is parameter-free in the sense that the constitutive coefficients are explicit functions of (k, k-hat, k-tilde, l) with no fitted constants, and the first-order benchmark against the Day et al. moduli holds once Eq. (1)'s prefactor is read as sqrt(27/16). The condensed character of the identification is honestly disclosed.

major comments (2)
  1. [Sec. 3, Eqs. (41)-(45); Sec. 4.3, Eqs. (85)-(90)] The two central algebraic steps of the derivation are asserted rather than derived. The passage from the resultant equilibrium equations (41)-(43) to the solution (44)-(45) of the 'system of 30 linear equations in the 18 unknown components' is presented without any solution procedure, even though this solution defines the additional field used in every subsequent energy computation. Likewise, the energy matching (84) is a heavily overdetermined system (more than one hundred scalar identities G[r] = H[r] for r = 1,...,10 against roughly twenty-eight constitutive unknowns), and the paper asserts without demonstration that Eqs. (87)-(90) satisfy all of these identities simultaneously. Because the manuscript is explicitly an analytical derivation, please outline the solution procedure for the 30-equation system, or state clearly that Eqs. (44)-(45) and the identities (85) were verified by symbolic computation, and indicate which of the scalar identities are independent.
  2. [Secs. 1, 4.3, and 5] The headline claim that the lattice 'can be identified with a form I Mindlin elastic material' (Section 1) is stated unconditionally, whereas the derivation fixes only the condensed tensors M* = M T_SGE and A* = T_SGE^T A T_SGE, i.e., the projections of the full fifth- and sixth-order constitutive tensors onto the four-dimensional equilibrium-constrained quadratic subspace (Eqs. 81-90). Section 5 discloses this limitation, but the abstract and introduction do not carry the qualification, and no argument is given that an admissible (positive-definite, symmetry-respecting) completion M, A exists. Since a positive-definite algebraic extension of A always exists once A* is positive definite, the decisive open check is the positive definiteness of A* over the admissible stiffness range, which is deferred to Part II. I recommend restating the central claim as the identification of the condensed equilibrium-subspace response, with the full material identification explicitly conditional on the Part II completion, or, alternatively, verifying positive definiteness of A* within this paper.
minor comments (6)
  1. [Eq. (1)] The shear-modulus prefactor should be typeset as sqrt(27/16) (that is, 3 sqrt(3)/4) rather than (sqrt 27)/16, since only the former value is consistent with C33 in Eq. (87).
  2. [Sec. 3 and Sec. 5] There are typos in 'purely linear (beta = 0) didplacement' (near Fig. 3) and 'positivedefinitess' (Section 5).
  3. [Eqs. (36)-(44) and (58)] The (m,n|i) node notation, with its heavy sub- and superscripts, is difficult to parse; a nomenclature table or a consolidated statement of the Voigt index map (54)-(66) would substantially improve readability.
  4. [After Eq. (88)] Add a sentence showing that substituting C from Eq. (87) into the coefficients D1,...,D8 of Eq. (75) reproduces the entries of Tlat in Eq. (58); currently the claimed equality T_lat = T_SGE has to be re-derived by the reader.
  5. [Eqs. (49)-(50)] State explicitly that the condensed tensors are defined relative to the normalization that the additional field does not alter the mean displacement gradient (Eq. 40); a different normalization would in general change the lattice energy (59) and hence the identified M* and A*.
  6. [Sec. 4.4] The claim that the additional field plays a marginal kinematic role is illustrated with one stiffness ratio (k-hat/k = 2, k-tilde/k = 3) and no quantitative error measure; this is acceptable as an illustration but should be labeled as such.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the constitutive tensors are obtained from an explicit energy-equivalence derivation, not fitted to the target, and the condensed-form limitation is acknowledged rather than concealed.

full rationale

The derivation chain is self-contained. The lattice elastic energy (Eq. 59) is computed directly from bar elongations under equilibrium-constrained quadratic displacement fields, while the SGE energy (Eq. 83) is the integrated Mindlin form-I energy. Identification is imposed by the matching condition U_lat = U_SGE for every cell and every admissible pair (a, b*), Eq. (84), which gives the coefficient identities G[r] = H[r], Eq. (85). The first-order coefficients (Eq. 87) reproduce the external Cauchy moduli of Day et al., and the higher-order condensed tensors M* and A* (Eqs. 89-90) are solved from the same coefficient matching rather than fitted to any preselected outcome. The equality Tlat = TSGE, Eq. (88), is a consistency result following from the determined C, not an input assumption. The paper explicitly states that only a 'condensed' form is identified and that completion, positive definiteness, and validation are deferred to Part II; this is a transparent scope limitation rather than a circular dependency. Self-references to prior quadratic-field identification methods and to Part II are methodological or deferred-validation citations, not load-bearing premises that reduce the claimed result to its own inputs.

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

No data are fitted: the three bar stiffnesses (k, k-hat, k-tilde) and the cell side length l are inputs of the discrete model. The additional-field components are solved from equilibrium, not fitted; the condensed identification leaves some components undetermined rather than fitted. No new physical entities are introduced; the additional displacement field is a mathematical corrector, and the 'condensed' SGE is a restricted representation, not a new entity.

assumptions (6)
  • domain assumption Bars are linearly elastic, hinge-connected, and deform only axially (no bending).
    Defines the discrete lattice model, Sect. 2.1 and Eq. (15).
  • domain assumption The lattice is infinite and periodic, with a hexagonal unit cell as representative volume.
    Used throughout to define cell energy and periodic node identifications, Eqs. (9)-(10).
  • domain assumption The equivalent continuum belongs to the class of Mindlin 'form I' second-gradient elastic materials with energy quadratic in strain and curvature.
    Sect. 4.2, Eqs. (61)-(62); the target material class is postulated, not derived.
  • ad hoc to paper The equilibrium-restoring additional field is linear in nodal coordinates (Eq. 39).
    Sect. 3 states this as the outcome of solving the resultant equations; the algebra is summarized, not shown.
  • ad hoc to paper Energy matching over self-equilibrated quadratic fields is sufficient to identify the condensed tensors.
    Sect. 4.3, Eq. (84); this is the central identification hypothesis and restricts the result to a condensed form.
  • standard math Cell displacement fields are linearly interpolated along bars when computing averaged gradients.
    Sect. 2.2, Eq. (22); a conventional finite-element-type interpolation.

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Pith. "Pith review of Identification of second-gradient elastic materials from planar hexagonal lattices. Part I: Analytical derivation of equivalent constitutive tensors." pith.science (2026). https://pith.science/paper/6CZZY42I

@misc{pith2026190801568,
  author       = {Pith},
  title        = {Pith review of: Identification of second-gradient elastic materials from planar hexagonal lattices. Part I: Analytical derivation of equivalent constitutive tensors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6CZZY42I}},
  note         = {Machine review of arXiv:1908.01568}
}
read the original abstract

A second-gradient elastic (SGE) material is identified as the homogeneous solid equivalent to a periodic planar lattice characterized by a hexagonal unit cell, which is made up of three different linear elastic bars ordered in a way that the hexagonal symmetry is preserved and hinged at each node, so that the lattice bars are subject to pure axial strain while bending is excluded. Closed form-expressions for the identified non-local constitutive parameters are obtained by imposing the elastic energy equivalence between the lattice and the continuum solid, under remote displacement conditions having a dominant quadratic component. In order to generate equilibrated stresses, in the absence of body forces, the applied remote displacement has to be constrained, thus leading to the identification in a \lq condensed' form of a higher-order solid, so that imposition of further constraints becomes necessary to fully quantify the equivalent continuum. The identified SGE material reduces to an equivalent Cauchy material only in the limit of vanishing side length of hexagonal unit cell. The analysis of positive definiteness and symmetry of the equivalent constitutive tensors, the derivation of the second-gradient elastic properties from those of the higher-order solid in the \lq condensed' definition, and a numerical validation of the identification scheme are deferred to Part II of this study.

Figures

Figures reproduced from arXiv: 1908.01568 by the authors.

Figure 1
Figure 1. (Left) A planar lattice obtained as the periodic repetition of a hexagonal unit cell (with side length [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Resultant forces R(m,n|i) (left) and additional displacements ∆u(m,n|i) (right) associated with the node i (i = 0, ..., 6) belonging to the cell {m, n} within the lattice drawn in its undeformed configuration. Assuming quasi-static conditions, from property (10) the equilibrium of the whole lattice is attained when the three primary resultants R(m,n|0) , R(m,n|1), and R(m,n|2) vanish for every cell {m, n} R(m,n|0) =… view at source ↗
Figure 3
Figure 3. (Upper part) Deformed configurations for a lattice with bars of stiffness [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: (Left) Rectangular domain (having sides 25 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Nonlocal constitutive parameters M∗ 13 (left) and M∗ 14 (right) as functions of the bar stiffness ratios bk/k and ek/k. The red lines represent the stiffness ratios pairs for which a centrosymmetric response is attained, while in all the other cases the solid equivalen…

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    2019

  33. [42]

    + 27I[1]I[2]I[3] + 81I2 [3] )] / ( 64I2 [1]I3 [2] ) , H[10] 22 = 9I[3] 64I2 [1]I4 [2] [( −10 ( ˆk +~k )3 k 5 − ( ˆk +~k )2( 20ˆk2− 137~kˆk + 20~k2 ) k 4 + − ( ˆk +~k )( 10ˆk4− 53~kˆk3 + 219~k2ˆk2− 53~k3ˆk + 10~k4 ) k 3 + +2ˆk~k ( 12ˆk4− 45~kˆk3 + 349~k2ˆk2− 45~k3ˆk + 12~k4 ) k...

    2019

  34. [43]

    + 27I[1]I[2]I[3] + 81I2 [3] )] / ( 64I2 [1]I3 [2] ) , H[10] 44 = 9 64I2 [1]I4 [2] [( 2 ( ˆk +~k )3( 12ˆk2−~kˆk + 12~k2 ) k 6 + + ( ˆk +~k )2( 48ˆk4 + 260~kˆk3 + 103~k2ˆk2 + 260~k3ˆk + 48~k4 ) k 5 + + ( ˆk +~k )( 24ˆk6 + 286~kˆk5 + 583~k2ˆk4− 255~k3ˆk3 + 583~k4ˆk2 + 286~k5ˆk + ...

    2019

  35. [44]

    + D2 ( M∗ 21 + √ 3M∗ 31 ) + √ 3D5M∗ 12 + D5M∗ 32 + D6 (√ 3M∗ 11 + M∗ 31 ) + M∗ 12 ) , G[9] 13 = 9 2𝓁 ( D1M∗ 23 + √ 3D1M∗ 33 + D3M∗ 21 + √ 3D3M∗ 31 + √ 3D5M∗ 13 + D5M∗ 33 + D7 (√ 3M∗ 11 + M∗ 31 ) + M∗ 13 + M∗ 31 ) , G[9] 14 = 9 2𝓁 (√ 3 (D1M∗ 34 + D4M∗

  36. [45]

    + D1M∗ 24 + ( D4 + √ 3 ) M∗ 21 + √ 3D5M∗ 14 + D5M∗ 34 + D8 (√ 3M∗ 11 + M∗ 31 ) + M∗ 14 ) , G[9] 22 = 9 𝓁 ( D2M∗ 22 + √ 3 (D2 + 1) M∗ 32 + D6 (√ 3M∗ 12 + M∗ 32 )) , G[9] 23 = 9 2𝓁 ( D2M∗ 23 + √ 3D2M∗ 33 + D3M∗ 22 + √ 3D3M∗ 32 + D6 (√ 3M∗ 13 + M∗ 33 ) + D7 (√ 3M∗ 12 + M∗ 32 ) + ...

  37. [46]

    (A.20) 27

    + D2M∗ 24 + ( D4 + √ 3 ) M∗ 22 + D6 (√ 3M∗ 14 + M∗ 34 ) + D8 (√ 3M∗ 12 + M∗ 32 )) , G[9] 33 = 9 𝓁 ( D3 ( M∗ 23 + √ 3M∗ 33 ) + D7 (√ 3M∗ 13 + M∗ 33 ) + M∗ 33 ) , G[9] 34 = 9 2𝓁 ( D3M∗ 24 + √ 3D3M∗ 34 + ( D4 + √ 3 ) M∗ 23 + √ 3D4M∗ 33 + D7 (√ 3M∗ 14 + M∗ 34 ) + D8 (√ 3M∗ 13 + M∗...

    2019 doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.