Pith. sign in

REVIEW 5 major objections 6 minor 36 references

Identification of second-gradient elastic materials from planar hexagonal lattices. Part II: Mechanical characteristics and model validation

T0 review · 5 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper establishes that a periodic hexagonal lattice of axially deformable bars with three stiffnesses is, at scales large relative to its cell, energetically equivalent to a homogeneous second-gradient elastic continuum given by…

desk verdict Useful closed-form SGE identification from a hexagonal lattice with a transparent but underdetermined continuation rule; the validation is decent but stops short of 'excellent'. read the letter →

arxiv 1908.01640 v1 pith:IIQITBRA 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
keywords straingradientelasticitynon-centrosymmetricmaterialinternallengthhomogenizationhexagonallatticesecond-gradientenergyequivalencesimpleshearvalidation
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 completes a two-part identification programme by turning a discrete hexagonal lattice, built from hinged bars with three different axial stiffnesses, into a homogeneous second-gradient elastic continuum written in closed form through three constitutive matrices. At first order the lattice is just an isotropic Cauchy solid, but at second order the equivalent material acquires an internal length, a strain-curvature coupling that makes it non-centrosymmetric, and higher-order anisotropy, even though the local response remains isotropic. The authors validate the model by solving simple shear and uniaxial strain problems exactly for the lattice and for the continuum, showing that the second-gradient material tracks the exact lattice energy far more closely than the classical Cauchy model. They also map the bar-stiffness ratios for which the equivalent energy is positive definite, and show that even an indefinite equivalent can have positive stored energy once the body is large enough compared with the cell size.

What carries the argument

The central object is the trio of constitutive matrices $\mathbf{C}(k,\hat{k},\tilde{k})$, $\mathbf{M}(k,\hat{k},\tilde{k})$, and $\mathbf{A}(k,\hat{k},\tilde{k})$ of the standard second-gradient material, obtained from the condensed tensors $\mathbf{M}^*$ and $\mathbf{A}^*$ of Part I through the transformation $\mathbf{q}^* = \mathbf{Q}\,\mathbf{q}_{\mathrm{SGE}}$, $\mathbf{M} = \mathbf{M}^*\mathbf{Q} + \Delta\mathbf{M}$, $\mathbf{A} = \mathbf{Q}^T \mathbf{A}^*\mathbf{Q} + \Delta\mathbf{A}$, where the equilibrium constraints fix only part of $\mathbf{Q}$, $\Delta\mathbf{M}$, and $\Delta\mathbf{A}$. The identification leaves four coefficients undetermined ($Q_{35}$, $\Delta A_{11}$, $\Delta M_{15}$, $\Delta M_{16}$); the paper fixes them with the relaxation rule of Eq. (51), which corresponds to dropping the equilibrium constraint at the first identification step and yields closed-form rational expressions in the bar stiffnesses while preserving the positive-definiteness domain. The cell side $\ell$ enters explicitly, linearly in $\mathbf{M}$ and quadratically in $\mathbf{A}$, which is what gives the continuum its internal length. Symmetry analysis assigns $\mathbf{C}$ to $O(2)$, $\mathbf{M}$ to $Z_3$, and $\mathbf{A}$ to $D_6$; these symmetry restrictions together with the relaxation rule determine the remaining entries.

What would settle it

Subject a finite hexagonal lattice strip and its equivalent second-gradient continuum to three-point bending or a concentrated edge load, where the displacement field is not the quadratic equilibrium family used in the identification, and compare cell-averaged displacements and stored energies; if the SGE mismatch is no better than the Cauchy mismatch, the sufficiency assumption that the relaxed model works for arbitrary fields is refuted.

Watch

Extended reading notes

Core claim

The paper claims that the 'condensed' second-gradient material identified in Part I—by matching the lattice strain energy under quadratic displacement fields whose stress fields satisfy equilibrium—can be relaxed into a standard second-gradient elastic material with constitutive matrices $\mathbf{C}$, $\mathbf{M}$, and $\mathbf{A}$ written in closed form in Eq. (1). The Cauchy matrix $\mathbf{C}$ is isotropic with Lamé constants in Eq. (2); the coupling matrix $\mathbf{M}$ is non-centrosymmetric with $Z_3$ symmetry and vanishes exactly when the two non-perimeter bar stiffnesses are equal or when $k = \hat{k}\tilde{k}/(2(\hat{k}+\tilde{k}))$; the curvature matrix $\mathbf{A}$ has $D_6$ symmetry. The resulting material carries an internal length $\ell$, making it non-local and anisotropic in its higher-order response while locally isotropic. The paper validates this identification by solving simple shear and uniaxial strain boundary value problems exactly for both the lattice and the continuum, reporting that the second-gradient model tracks the exact lattice energy far more closely than the Cauchy model and captures a transverse displacement in uniaxial strain that a Cauchy solid cannot produce.

Load-bearing premise

The argument assumes that matching the lattice energy on the restricted family of equilibrium quadratic displacement fields, together with the specific rule chosen for the undetermined coefficients in Eq. (51), fixes a standard second-gradient material that also matches the lattice under arbitrary displacement fields; if that sufficiency step fails, the four validation problems would not be a genuine test.

Editorial extensions

If this is right

  • A finite hexagonal truss can be replaced by the closed-form continuum of Eq. (1) at scales a few times the cell size, cutting computational cost while retaining size-dependent behaviour through the internal length $\ell$.
  • The model predicts a measurable non-centrosymmetric effect: under uniaxial strain along one axis, the lattice develops a transverse displacement whenever $\hat{k}\ne\tilde{k}$ and $k\ne\hat{k}\tilde{k}/(2(\hat{k}+\tilde{k}))$, an effect invisible to a Cauchy or centrosymmetric continuum.
  • Strain-energy comparisons in Tables 2 and 3 show the equivalent SGE material reduces the error from roughly 8-27% for the Cauchy model to about 1-9% in shear and below about 2% for uniaxial strain along the cell direction.
  • Because the higher-order energy scales as $(\ell/\rho)^2$ relative to the classical energy, larger specimens should behave like the Cauchy solid while small specimens should show non-classical effects; the crossover ratio $\rho/\ell$ depends on the bar stiffness ratios.
  • Positive definiteness of the equivalent material is not guaranteed: some stiffness ratios, including the regular equal-stiffness honeycomb, give an indefinite energy, but the energy stored in a finite disk becomes positive once the disk radius is large enough, so finite-size stability is the physically relevant check.

Reading between the lines

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

  • Inference: the same matching procedure could be applied to other periodic truss topologies, such as triangular, kagome, square, or chiral lattices, with the symmetry classes of $\mathbf{M}$ and $\mathbf{A}$ encoding the lattice point group, so lattice geometry would translate directly into second-gradient constitutive data.
  • Inference: the relaxation rule of Eq. (51) is one admissible choice among several, since $Q_{35}$ and $\Delta A_{11}$ remain undetermined by symmetry; alternative standard materials with identical condensed behaviour exist, and comparing them against the same benchmarks would isolate how much of the reported accuracy comes from the identification rather than from the specific rule.
  • Inference: if the internal-length picture is right, elastic wave dispersion in a finite hexagonal lattice should deviate from the Cauchy prediction at wavenumbers near $1/\ell$, with the $\mathbf{M}$ term coupling shear and dilatational branches; measuring dispersion curves would validate the model in dynamics.
  • Inference: because swapping $\hat{k}$ and $\tilde{k}$ flips the sign of $m$ while leaving the other constitutive matrices unchanged, the same lattice geometry could act as a mechanically switchable non-centrosymmetric medium whose bending direction under uniaxial strain is set by the stiffness ordering.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. This paper completes the identification of a second-gradient elastic (SGE) continuum equivalent to a planar hexagonal lattice of axially deformable bars. Starting from the 'condensed' constitutive matrices C, M*, A* derived in Part I under equilibrium-satisfying quadratic displacement fields, the authors construct a 'standard' SGE material with constitutive matrices C, M, A given in closed form in Eq. (1). They analyze the symmetry classes of these tensors (isotropic C, Z3-symmetric M, D6-symmetric A), study positive definiteness of the condensed energy and its dependence on the bar stiffness ratios, and validate the model by comparing the lattice response with the SGE and Cauchy continuum responses under simple shear and uniaxial strain in four aligned loading configurations. The validation is reported as a reduction of strain-energy mismatch when the SGE model replaces the classical Cauchy model.

Significance. If the identification is accepted, the paper provides a closed-form, parameter-free second-gradient continuum model of a hexagonal lattice, with an internal length and non-centrosymmetric coupling terms, which is of genuine interest for micromechanics and metamaterial design. The authors give exact analytical lattice solutions (obtained with Mathematica) and explicit formulas for all constitutive coefficients, so the paper is checkable and reproducible in principle. The validation is not circular: the constitutive parameters are obtained from energy matching, not from fitting the validation responses, and the paper demonstrates a clear non-centrosymmetric effect (transverse displacement under uniaxial strain) that a Cauchy model cannot capture. The main weaknesses are the underdetermined extension from the condensed to the standard form and the mixed quality of the validation results, which fall short of the claimed 'excellent agreement'.

major comments (5)
  1. [§2.4, Eq. (51)] The extension from the condensed to the standard SGE form is underdetermined. Equations (14)-(16) leave four coefficients (Q35, ΔA11, ΔM15, ΔM16) unspecified by energy matching, and Eq. (51) fixes them by a rule described in words ('imposing an equivalence ... in which the equilibrium constraint on the coefficients βijk is neglected') rather than derived from an explicit energy-matching or minimization criterion. Since the matrices M and A in Eq. (1) depend on this choice, the claim that the resulting SGE material is energetically equivalent to the lattice for arbitrary displacement fields is not established. A concrete test would be an off-axis boundary value problem (e.g., simple shear or bending with a loading direction not aligned with the x1/x2 lattice axes) that activates the non-condensed curvature modes; the current validation only uses four aligned cases (Tables 2 and 3).
  2. [§3.1.3, Table 2] The reported errors undercut the 'excellent agreement' language in the Abstract and Section 1. For Ex2, simple shear parallel to the x2-axis, the SGE energy error is 19.3%, compared with 26.74% for Cauchy, so the improvement is modest and the absolute error remains large. For the same loading, Ex3 shows an SGE error of only 1.14%, indicating strong configuration dependence. The paper should either soften the global validation claim or explain why the x2-aligned shear case is an outlier.
  3. [§3.2.3, Table 3] Negative discrepancies (e.g., -6.83%, -7.47%, -8.23% for the uniaxial parallel to x1 cases) show that the SGE model is stiffer than the lattice in these configurations. This contradicts the impression of a consistently improved match and means the energy comparison is not one-sided. The statement that the advantage 'becomes clear' for uniaxial strain parallel to x2 is not supported by the last column of Table 3, where the SGE errors are -1.24%, 0.57%, and -0.72%. The authors should address why the equivalent SGE solid can be stiffer than the lattice and what this implies for the claimed energetic equivalence.
  4. [§3.2.1, Eq. (71)] The expression for τ211(x2) contains the coefficient a45, which is not defined in Eq. (1) or elsewhere in the paper; the A matrix in Eq. (1) uses only a11, a12, a16, a26, and a34. As written, the closed-form solution for uniaxial strain parallel to the x2-axis is not reproducible. Please define a45 or correct the typo (possibly it should be a26 or a16).
  5. [§2.4, after Eq. (51)] The assertion that the chosen values of Q35, ΔA11, ΔM15, and ΔM16 provide a positive-definiteness domain identical for the condensed and standard forms is not proved and is not obvious, because the standard form operates on a larger curvature space (six curvature components versus four in the condensed form). A symbolic or numerical verification of this equality should be provided, or the statement should be qualified.
minor comments (6)
  1. [§2.1, Eq. (33)] The word 'verifed' should be 'verified'.
  2. [§2.2] The word 'isoropic' in the sentence about the C matrix should be 'isotropic'.
  3. [§2.4] The phrase 'first indentification step' contains a typo: 'indentification' should be 'identification'.
  4. [Table 1 header] The header lists the ratios as 'ˆk/k and ˆk/k'; the second ratio should be '~k/k'.
  5. [§3.1.3, footnote 2] The statement that the dimensionless energy difference coincides with the dimensionless difference of the resultant tractions is non-obvious and should be derived or referenced, since it is used to interpret Tables 2 and 3.
  6. [Eq. (20)] In the last row of R[q](θ), there is a misplaced period after '( c(θ) +c(3θ))/2.'; the notation should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SGE parameters come from energy matching, not from fitting the validation responses, and the lattice comparisons are independent exact solutions.

full rationale

The constitutive matrices in Eq. (1) are not fitted to the validation boundary value problems. They are obtained from the closed-form energy matching established in Part I (Eqs. (6)-(7)) and from the stated extension rule in Section 2.4, Eq. (51), which removes the equilibrium constraint by imposing energy equivalence with the lattice for general quadratic displacement fields. The four remaining coefficients Q35, Delta-A11, Delta-M15, and Delta-M16 are fixed by that rule, not by the simple shear or uniaxial strain comparisons. The validation in Section 3 solves the lattice exactly with Mathematica and compares the lattice response with the continuum response, so it is an independent check of the chosen extension rather than a restatement of the identification inputs. The paper's reliance on Part I is a self-citation, but Part I is a separately published analytical derivation with stated assumptions, and Part II provides direct numerical lattice comparisons that do not depend on the cited closed forms for the validation data. The symmetry classes of M and A are imposed as constraints (Section 2.1.2) rather than independently predicted, but this is an explicit modeling choice, not a circular reduction of the central equivalence claim. The underdetermination of the condensed-to-standard extension and the residual, sometimes negative, energy mismatches in Tables 2-3 are robustness or correctness concerns, not circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the discrete lattice model, the energy-equivalence identification from Part I, and the particular relaxation that selects one standard SGE representation from an infinite family. The only effectively free choices are the four transformation coefficients fixed by the practical rule; all constitutive coefficients are derived functions of the three spring stiffnesses and the cell size. No new physical entities are postulated.

free parameters (3)
  • Q35 (condensed-to-standard transformation coefficient) = Expression in k, ^k, ~k given in Eq. (51), not a numeric fit
    One of the four coefficients left undetermined by symmetry requirements; fixed by the 'practical rule' that removes the equilibrium constraint. The final constitutive matrices depend on this choice, so the standard SGE representation is not unique.
  • Delta A11 (curvature stiffness contribution) = Expression in k, ^k, ~k given in Eq. (51), not a numeric fit
    Similarly chosen by the practical rule; affects the curvature energy and the validation responses.
  • Delta M15 and Delta M16 (coupling contributions) = Set to 0 by the practical rule in Eq. (51)
    Part of the non-uniqueness in building the standard SGE; setting them to zero is a modeling choice, not a consequence of the energy matching alone.
assumptions (5)
  • domain assumption The hexagonal lattice is made of axially deformable bars with hinged joints, so bars carry only axial force.
    Stated in Section 1 and Fig. 1; the entire energy matching depends on this discrete model.
  • domain assumption The equivalent SGE solid is defined by strain-energy matching under remote quadratic displacement fields whose generated stresses are in equilibrium, the 'condensed form'.
    Introduced in Part I [28] and used in Eq. (6); limits the identification to equilibrated fields.
  • domain assumption The 'standard' SGE energy in Eq. (11) with matrices C, M, A is admissible and complete for the homogenized material.
    Eq. (11) is posed as the continuum energy; the validity of second-gradient elasticity for this lattice is the central modeling assumption.
  • domain assumption Plane strain or plane stress reduction with the stated relation for lambda'.
    Eq. (2) and the note 'lambda' = lambda applies when plane strain prevails'.
  • domain assumption Positive definiteness of the constitutive matrix is the relevant stability criterion.
    Section 2.3 uses positive definiteness as the a-priori stability condition.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Identification of second-gradient elastic materials from planar hexagonal lattices. Part II: Mechanical characteristics and model validation." pith.science (2026). https://pith.science/paper/IIQITBRA

@misc{pith2026190801640,
  author       = {Pith},
  title        = {Pith review of: Identification of second-gradient elastic materials from planar hexagonal lattices. Part II: Mechanical characteristics and model validation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IIQITBRA}},
  note         = {Machine review of arXiv:1908.01640}
}
read the original abstract

Positive definiteness and symmetry of the constitutive tensors describing a second-gradient elastic (SGE) material, which is energetically equivalent to a hexagonal planar lattice made up of axially deformable bars, are analyzed by exploiting the closed form-expressions obtained in part I of the present study in the \lq condensed' form. It is shown that, while the first-order approximation leads to an isotropic Cauchy material, a second-order identification procedure provides an equivalent model exhibiting non-locality, non-centrosymmetry, and anisotropy. The derivation of the constitutive properties for the SGE from those of the \lq condensed' one (obtained by considering a quadratic remote displacement which generates stress states satisfying equilibrium) is presented. Comparisons between the mechanical responses of the periodic lattice and of the equivalent SGE material under simple shear and uniaxial strain show the efficacy of the proposed identification procedure and therefore validate the proposed constitutive model. This model reveals that, at higher-order, a lattice material can be made equivalent to a second-gradient elastic material exhibiting an internal length, a finding which is now open for applications in micromechanics.

Figures

Figures reproduced from arXiv: 1908.01640 by the authors.

Figure 1
Figure 1. The ‘condensed’ second-gradient elastic material (center), characterized by the constitutive matrices [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Polar diagrams of the curvature energy density [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. As for Fig. 2, but for the total energy density [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Regions in the bars’ stiffness (bk/k–ek/k) space corresponding to positive definite (green colour) or indefinite (red colour) equivalent second-gradient elastic material. Note the symmetry with respect to a line inclined at π/4 and that the case of equal stiffnessess, …
Figure 5
Figure 5. Figure 5: Parameter R, Eq. (47), as a function of the ratio ρ/`, for β222 = −β112 = −β211 = 1. Different pairs of bars’ stiffness are reported {bk/k, ek/k}, given by {4.5, 3.5} (green), {5, 0.2} (purple) and {10, 0.1} (orange). Note that the ratio is always positive at sufficien…
Figure 6
Figure 6. Figure 6: Upper row from left to right: Undeformed lattice and equivalent [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: A simple shear strain is applied aligned parallel to the [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: A simple shear strain is applied aligned parallel to the [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: A uniaxial strain is applied aligned parallel to the [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: A uniaxial strain aligned parallel to the [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: Deformed configurations (superimposed to the undeformed configuration sketched gray) and dimensionless [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 35 canonical work pages

  1. [1]

    Auffray, R

    N. Auffray, R. Bouchet, and Y. Br´ echet. Strain gradient elastic homogenization of bidimensional cellular media. International Journal of Solids and Structures , 47(13):1698–1710, 2010

    work page 2010

  2. [3]

    Bacca, D

    M. Bacca, D. Bigoni, F. Dal Corso, and D. Veber. Mindlin second-gradient elastic properties from dilute two- phase cauchy-elastic composites. part I: Closed form expression for the effective higher-order constitutive tensor. International Journal of Solids and Structures , 50(24):4010–4019, 2013

    work page 2013

  3. [4]

    Bacca, D

    M. Bacca, D. Bigoni, F. Dal Corso, and D. Veber. Mindlin second-gradient elastic properties from dilute two- phase cauchy-elastic composites part II: Higher-order constitutive properties and application cases. International Journal of Solids and Structures , 2013

    work page 2013

  4. [5]

    Bacigalupo

    A. Bacigalupo. Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits. Meccanica, 49(6):1407–1425, 2014

    work page 2014

  5. [6]

    Bacigalupo and L

    A. Bacigalupo and L. Gambarotta. Second-order computational homogenization of heterogeneous materials with periodic microstructure. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f¨ ur Angewandte Mathematik und Mechanik , 90(10-11):796–811, 2010

    work page 2010

  6. [7]

    Bacigalupo and L

    A. Bacigalupo and L. Gambarotta. Simplified modelling of chiral lattice materials with local resonators. Inter- national Journal of Solids and Structures , 83:126–141, 2016

    work page 2016

  7. [8]

    Bacigalupo and L

    A. Bacigalupo and L. Gambarotta. Wave propagation in non-centrosymmetric beam-lattices with lumped masses: Discrete and micropolar modeling. International Journal of Solids and Structures , 118:128–145, 2017

    work page 2017

  8. [9]

    Bigoni and W.J

    D. Bigoni and W.J. Drugan. Analytical derivation of cosserat moduli via homogenization of heterogeneous elastic materials. Journal of Applied Mechanics , 74(4):741–753, 2007

    work page 2007

Show all 36 references
  1. [10]

    Cosserat and F

    E. Cosserat and F. Cosserat. Sur la theorie des corps deformables . Herman, Paris, 1909

    1909

  2. [11]

    Dal Corso and L

    F. Dal Corso and L. Deseri. Residual stresses in random elastic composites: Nonlocal micromechanics-based models and first estimates of the representative volume element size. Meccanica, 48(8):1901–1923, 2013

    1901

  3. [12]

    Day, K.A

    A.R. Day, K.A. Snyder, E.J. Garboczi, and M.F. Thorpe. The elastic moduli of a sheet containing circular holes. Journal of the Mechanics and Physics of Solids , 40(5):1031–1051, 1992

    1992

  4. [13]

    W. J. Drugan and J. R. Willis. A micromechanics-based nonlocal constitutive equation and estimates of represen- tative volume element size for elastic composites. Journal of the Mechanics and Physics of Solids , 44(4):497–524, 1996

    1996

  5. [14]

    N. A. Fleck and J. W. Hutchinson. A reformulation of strain gradient plasticity. Journal of the Mechanics and Physics of Solids , 49(10):2245–2271, 2001

    2001

  6. [15]

    N. A. Fleck and J. R. Willis. A mathematical basis for strain-gradient plasticity theory-part I: Scalar plastic multiplier. Journal of the Mechanics and Physics of Solids , 57(1):161–177, 2009

    2009

  7. [16]

    S. Forest. Homogenization methods and the mechanics of generalized continua-part 2. Theoretical and applied mechanics, 28(29):113–144, 2002

    2002

  8. [17]

    Mechanics of generalized continua: construction by homogenizaton

    Samuel Forest. Mechanics of generalized continua: construction by homogenizaton. Le Journal de Physique IV , 8(PR4):Pr4–39, 1998

    1998

  9. [18]

    Cosserat overall modeling of heterogeneous materials

    Samuel Forest and Karam Sab. Cosserat overall modeling of heterogeneous materials. Mechanics Research Communications, 25(4):449–454, 1998

    1998

  10. [19]

    Gourgiotis and D

    P.A. Gourgiotis and D. Bigoni. Folding and faulting of an elastic continuum. Proc.R.Soc.A, 472:20160018, 2016

    2016

  11. [20]

    Gourgiotis and D

    P.A. Gourgiotis and D. Bigoni. Stress channelling in extreme couple-stress materials part II: Localized folding vs faulting of a continuum in single and cross geometries. Journal of the Mechanics and Physics of Solids , 88:169 – 185, 2016

    2016

  12. [21]

    Gourgiotis, T

    P.A. Gourgiotis, T. Zisis, and H. G. Georgiadis. On concentrated surface loads and green’s functions in the toupin-mindlin theory of strain-gradient elasticity. International Journal of Solids and Structures , 130-131:153– 171, 2018

    2018

  13. [22]

    W.T. Koiter. Couple-stresses in the theory of elasticity, parts I and II. Proc. K. Ned. Akad. Wet., Ser. B: Phys. Sci., 67:17–44, 1964

    1964

  14. [23]

    Lekhnitskii, P

    S.G. Lekhnitskii, P. Fern, J.J. Brandstatter, and E.H. Dill. Theory of elasticity of an anisotropic elastic body. Physics Today, 17:84, 1964

    1964

  15. [24]

    R.D. Mindlin. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis , 16(1):51–78, 1964. 24 Published in International Journal of Solids and Structures (2019) 176-177, 19-35 doi: https://doi.org/10.1016/j.ijsolstr.2019.07.009

    2019 doi

  16. [25]

    Mindlin and N.N

    R.D. Mindlin and N.N. Eshel. On first strain-gradient theories in linear elasticity. International Journal of Solids and Structures, 4(1):109 – 124, 1968

    1968

  17. [26]

    Ostoja-Starzewski, S.D

    M. Ostoja-Starzewski, S.D. Boccara, and I. Jasiuk. Couple-stress moduli and characteristic length of a two-phase composite. Mechanics Research Communications, 26(4):387–396, 1999

    1999

  18. [27]

    Papathanasiou, P.A

    T.K. Papathanasiou, P.A. Gourgiotis, and F. Dal Corso. Finite element simulation of a gradient elastic half-space subjected to thermal shock on the boundary. Applied Mathematical Modelling , 40(23-24):10181–10198, 2016

    2016

  19. [28]

    Rizzi, D

    G. Rizzi, D. Veber, F. Dal Corso, and D. Bigoni. Identification of second-gradient elastic materials from planar hexagonal lattices. part I: Analytical derivation of equivalent constitutive tensors. International Journal of Solids and Structures, 176–177:1–18, 2019

    2019

  20. [29]

    Smyshlyaev

    V.P. Smyshlyaev. Propagation and localization of elastic waves in highly anisotropic periodic composites via two-scale homogenization. Mechanics of Materials , 41(4):434–447, 2009

    2009

  21. [30]

    Smyshlyaev and K.D

    V.P. Smyshlyaev and K.D. Cherednichenko. On rigorous derivation of strain gradient effects in the overall behaviour of periodic heterogeneous media. Journal of the Mechanics and Physics of Solids , 48(6-7):1325–1357, 2000

    2000

  22. [31]

    Snyder, E.J

    K.A. Snyder, E.J. Garboczi, and A. R. Day. The elastic moduli of simple two-dimensional isotropic composites: Computer simulation and effective medium theory. Journal of applied physics , 72(12):5948–5955, 1992

    1992

  23. [32]

    T.-H. Tran, V. Monchiet, and G. Bonnet. A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media. International Journal of Solids and Structures , 49(5):783–792, 2012

    2012

  24. [33]

    Trovalusci, M

    P. Trovalusci, M. Ostoja-Starzewski, M.L. De Bellis, and A. Murrali. Scale-dependent homogenization of random composites as micropolar continua. European Journal of Mechanics-A/Solids , 49:396–407, 2015

    2015

  25. [34]

    Wheel, J.C

    M.A. Wheel, J.C. Frame, and P.E. Riches. Is smaller always stiffer? on size effects in supposedly generalised continua. International Journal of Solids and Structures , 67-68:84 – 92, 2015

    2015

  26. [35]

    J.R. Willis. Variational and related methods for the overall properties of composites. Advances in applied mechanics, 21:1–78, 1981

    1981

  27. [36]

    Zisis, P

    Th. Zisis, P. A. Gourgiotis, and H. G. Georgiadis. Contact mechanics in the framework of couple stress elasticity , volume 90. Advanced Structured Materials, 2018

    2018

  28. [37]

    Zisis, P.A

    Th. Zisis, P.A. Gourgiotis, Konstantinos P. Baxevanakis, and H.G. Georgiadis. Some basic contact problems in couple stress elasticity. International Journal of Solids and Structures , 51(11-12):2084–2095, 2014. 25

    2014

Pith tools

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