Pith. sign in

REVIEW 2 major objections 4 minor 58 references

On the synthesis of complete two-dimensional second-gradient continua: Tri-pantographic fabrics

T0 review · 2 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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

Pith's one-line read The paper defines completeness as pointwise positive definiteness of the second-gradient Hessian and proves that a tri-pantographic stored energy satisfies it.

desk verdict A clean completeness criterion for 2D second-gradient continua, proved for a postulated tri-pantographic energy; the microstructure-to-continuum step is openly conjectural, so the synthesis claim is conditional. read the letter →

arxiv 2608.05524 v1 pith:CRREQ4N3 submitted 2026-08-06 math-ph cond-mat.mtrl-scimath.MPphysics.class-ph

Casey Rodriguez , Emilio Barchiesi , Simon R. Eugster , Ivan Giorgio , Francesco dell'Isola This is my paper · ORCID
classification math-phcond-mat.mtrl-scimath.MPphysics.class-ph MSC 74A3574B2074Q05
keywords second-gradientcontinuapantographicfabricscompletenesspositivedefinitenessvirtualworkprinciplehomogenizationmechanicalmetamaterialsfiber-reinforcedelasticity
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 introduces a criterion called completeness for two-dimensional second-gradient elastic continua: at each configuration, the stored-energy Hessian with respect to the second gradient of the placement must be positive definite, so every nonzero second-gradient increment is quadratically controlled by the energy. Drawing on the virtual-work formulation of second-gradient continua, it derives constitutive relations, equilibrium equations, and boundary interactions, including line forces, line double-forces, and corner forces, for fibrous energies that depend on fiber stretch, stretch gradient, and curvature. Applying this criterion to pantographic microstructures, it shows that the classical pantographic sheet and the bi-pantographic fabric are incomplete, because certain mixed second-gradient components are not detected. It then adds a third oblique fiber family and proves that the resulting tri-pantographic continuum is complete, with the mixed derivative recovered through the oblique direction. The result matters for design because completeness is a local strong-convexity property of the highest-gradient part of the energy, a natural prerequisite for well-posed boundary-value problems, and the paper shows a concrete three-family architecture can achieve it.

What carries the argument

The central object is the Hessian $D^2_{FF}W$ of the stored energy with respect to the second-gradient variable $F=\nabla\nabla\chi$ at fixed first gradient, evaluated on admissible third-order tensors symmetric in the last two indices. Completeness is exactly the local positive definiteness of this Hessian. For the pantographic energies, the Hessian decomposes into rank-one dyadic squares indexed by fiber families: terms $2g(\rho_{\alpha})(A\cdot(\tau_{\alpha}\otimes D_{\alpha}\otimes D_{\alpha}))^2$ plus $2f(\rho_{\alpha})\rho_{\alpha}^{-2}(A\cdot(\nu_{\alpha}\otimes D_{\alpha}\otimes D_{\alpha}))^2$. This reduces the positivity question to whether the three symmetric projections $A[D_{\alpha}\otimes D_{\alpha}]$ determine $A$, and the oblique third family closes the algebra by coupling the two pure symmetric projections to the mixed projection. The virtual-work framework turns the same tensor expression into explicit line forces, line double-forces, and corner forces.

What would settle it

If one can exhibit a nonzero admissible third-order tensor $A$ with $A[D_{\alpha}\otimes D_{\alpha}]=0$ for $\alpha=1,2,3$ while $A[D_1\otimes D_2+D_2\otimes D_1]\ne 0$, the positive-definiteness proof fails; for $\eta\ne\pi/2$ the paper's algebra rules this out, so such a tensor would refute completeness. Separately, deriving the homogenized energy of the proposed discrete three-family lattice and finding cross-terms with negative coefficients, or finding $f$ or $g$ nonpositive in an admissible regime, would show that the continuum-level claim does not transfer to the discrete architecture.

Watch

Extended reading notes

Core claim

The central discovery is that a second-gradient continuum built from three fiber families, two orthogonal plus one oblique, has a stored energy $W=\sum_{\alpha=1}^3 [f(\rho_{\alpha})(\vartheta_{\alpha,\alpha})^2 + g(\rho_{\alpha})(\rho_{\alpha,\alpha})^2 + h(\rho_{\alpha})]$ (equation (4.29)) whose Hessian with respect to the second-gradient variable is pointwise positive definite, hence complete in the paper's sense. The proof is algebraic: for this energy the quadratic form $A\cdot D^2_{FF}W[A]$ is a sum of six squared directional contractions with strictly positive coefficients. Vanishing of the whole sum forces $A[D_{\alpha}\otimes D_{\alpha}]=0$ for $\alpha=1,2,3$; since $D_3=\cos\eta\,D_1+\sin\eta\,D_2$ with $\eta\in(0,\pi)\setminus\{\pi/2\}$, this forces $A[D_1\otimes D_2+D_2\otimes D_1]=0$ as well, so $A$ annihilates every symmetric second-order tensor and, by the minor symmetry of $A$, $A=0$. The same criterion exposes why two-family fabrics fail: bi-pantographic energies control only double derivatives along each fiber family and leave mixed derivatives $A[D_1\otimes D_2+D_2\otimes D_1]$ invisible.

Load-bearing premise

The stored energy (4.29) of the tri-pantographic continuum is postulated, not derived: the paper explicitly states in Section 4.3 that establishing a homogenization result for the proposed discrete three-family architecture lies beyond the scope of the present work, so the 'synthesis' claim rests on the unproven conjecture that such a discrete lattice has this effective energy.

Editorial extensions

If this is right

  • In a classical pantographic sheet the line double-force on an aligned rectangular edge is necessarily orthogonal to the current fiber direction, and the corner force vanishes; the paper derives these restrictions from the missing second-gradient components.
  • A bi-pantographic fabric detects only directional second derivatives taken twice along each of its two fiber families, so mixed second-gradient increments with equal transverse components cost no energy, for any positive moduli.
  • The tri-pantographic continuum is complete: every nonzero admissible second-gradient increment is quadratically controlled, pointwise, by the highest-gradient part of the stored energy; a uniform control constant requires the deformation gradient to stay in a compact admissible set bounded away from the singular stretch limit.
  • In the tri-pantographic fabric, the third oblique family makes the pointwise line-force and line-double-force expressions acquire both tangential and normal components, and it activates corner forces that can point in any planar direction as the stretch gradient and curvature vary.

Reading between the lines

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

  • Beyond the paper, the algebraic mechanism suggests a general design rule for higher-gradient metamaterials: choose fiber directions whose symmetric tensor products span the full space of symmetric second-order tensors, so the squared directional contractions determine every admissible second-gradient increment; three directions with one oblique are the minimal two-dimensional completion.
  • Beyond the paper, completeness should not be read as a guarantee of well-posedness: the paper's closing discussion notes that coercivity on H^2 would require uniform lower bounds, bounded coefficients, control of first-gradient coupling, and normalization conditions that eliminate the affine kernel of the Hessian seminorm.
  • A direct experimental extension would compare two-family and three-family pantographic lattices on deformation modes dominated by mixed second gradients; the paper's claim predicts that the three-family specimen shows positive, scale-sensitive energy for these modes while the two-family specimen shows near-zero resistance.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper introduces a notion of completeness for two-dimensional second-gradient elastic continua, defined as local positive definiteness of the stored-energy Hessian with respect to the second-gradient variable. It develops a variational framework based on the Principle of Virtual Work for fibrous continua whose energies depend on fiber stretch, stretch gradient, and curvature, and derives the associated equilibrium equations and boundary interactions. The framework is applied to pantographic-type models: the classical pantographic sheet is shown to be incomplete, the bi-pantographic fabric remains incomplete because mixed second derivatives are undetected, and a tri-pantographic continuum with three fiber families is shown to be complete via a direct kernel argument on the Hessian quadratic form. The paper also derives explicit boundary force and double-force expressions for the tri-pantographic case, including nonzero corner forces generated by the third oblique family, and proposes a discrete three-family architecture while explicitly stating that its homogenization is left open.

Significance. If the results hold, the paper contributes a clean algebraic criterion for when a second-gradient energy provides full pointwise control of all admissible second-gradient increments, and it identifies the third oblique fiber family as the key architectural ingredient. The Hessian computation (3.35) and the completeness proof for the postulated energy (4.29) are correct and easy to verify. The boundary-interaction formulas are explicit and physically informative. However, the paper's central 'synthesis' claim is conditional: the tri-pantographic energy is postulated, not derived from a discrete microstructure, and the paper states that the homogenization is beyond its scope. Within the paper's own definition of synthesis, this leaves the main constructive claim incomplete.

major comments (2)
  1. [§4.3, Eq. (4.29) and following paragraph] The tri-pantographic stored energy (4.29) is postulated, not derived from the proposed discrete three-family architecture; the paper states that establishing the homogenization 'lies beyond the scope of the present work.' Under the paper's own definition of synthesis in §1.1, a synthesis requires exhibiting a family of mechanical systems whose effective limit is the prescribed continuum. This is not supplied. If the actual homogenized energy of the proposed lattice contained cross-family couplings, the Hessian would not have the block-diagonal form (4.30)–(4.31), and the argument that a zero quadratic form forces A[D_α⊗D_α]=0 for each α separately would fail. Because the discrete cell with three obliquely intersecting beam families has a different hinge geometry than the orthogonal bi-pantographic cell, such couplings are a genuine possibility. I request either a homogenization analysis (or at least a formal asymptotic derivation) of the proposed cell, or a clear reframing of the contribution as a candidate complete continuum, with title and abstract adjusted accordingly.
  2. [§4.3, Eqs. (4.29) and (4.12)] The stored energy uses the same functions f(ρ_α), g(ρ_α), h(ρ_α) for all three families, with no dependence on the obliquity angle η. These functions were derived in [8] for a bi-pantographic fabric with two orthogonal families. For an oblique third family, the hinge geometry and the interaction between crossing families will generally change the effective coefficients; at minimum, f, g, h should depend on η and on the relative angles between families. Without a derivation, the completeness theorem characterizes a postulated model, not a class of physical fabrics. The authors should state this assumption explicitly and discuss whether the completeness property is robust to plausible η-dependent modifications of the moduli.
minor comments (4)
  1. [§1.1] The phrase 'a concrete candidate pantographic architecture that yields a complete two-dimensional continuum' in the introduction could be read as claiming that the discrete architecture yields the complete continuum; since the homogenization is left open, I suggest rewording to 'a continuum-level construction motivated by a candidate architecture.'
  2. [§4.1, Eq. (4.7)] The notation D^2_FF W[τ_γ⊗D_γ⊗D_γ]=0 is correct because the contraction yields the zero tensor, but the point is that the quadratic form vanishes; consider stating it as (τ_γ⊗D_γ⊗D_γ)·D^2_FF W[τ_γ⊗D_γ⊗D_γ]=0 for clarity.
  3. [§4.2, Eq. (4.19)] The mixed second-gradient A = a⊗(D_1⊗D_2+D_2⊗D_1) is symmetric in the last two indices and is therefore an admissible tangent-space element; the paper could make this explicit to avoid any apparent conflict with the symmetry of actual placement second gradients.
  4. [Title header] The typeset header contains stray spaces in 'TRI-P ANTOGRAPHIC F ABRICS'; this should be corrected in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the completeness proof for the postulated tri-pantographic energy is self-contained algebra, and the unproven discrete homogenization is an explicitly stated open limitation rather than a circular reduction.

full rationale

The paper's central mathematical claim is that the tri-pantographic stored energy (4.29) is complete. The proof is self-contained: Eq. (4.32)-(4.33) writes the Hessian quadratic form as a sum of strictly positive squares; Eq. (4.35) concludes each contraction A[D_alpha (x) D_alpha] vanishes; and Eq. (4.36)-(4.38) uses the obliquity condition eta in (0,pi)\ {pi/2} to obtain the mixed symmetric contraction, yielding A=0 on all symmetric tensors. This argument depends only on positivity of f and g and on the chosen directions D1=E1, D2=E2, D3=cos(eta)E1+sin(eta)E2, not on any fitted parameter or on the validity of the cited homogenization results. The self-citations to [8] and [9] provide the motivating forms of f,g,h, but the theorem uses only f,g>0, so those citations are not load-bearing for the completeness proof. The paper explicitly labels the tri-pantographic energy as postulated: 'we postulate the stored energy W=...' (Eq. (4.29)), and it explicitly flags the discrete-to-continuum step as open: 'Establishing such a homogenization result, however, lies beyond the scope of the present work.' That is an honest statement of an unproven synthesis conjecture, not a case where a prediction is equivalent to its inputs by construction. The boundary-interaction derivations follow from the Principle of Virtual Work and the constitutive definitions, and the incompleteness examples for the pantographic sheet and bi-pantographic fabric are direct computations from the stated energies. No step in the paper reduces by definition to its own inputs, and no fitted or predicted quantity is relabeled as a first-principles result. Therefore the appropriate finding is no significant circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 2 invented entities

The central completeness theorem is self-contained once the tri-pantographic energy form is accepted. The heavy self-citation in references does not affect the proof. The main uncharged input is the ad hoc postulation of the tri-pantographic energy and the corresponding discrete architecture.

free parameters (1)
  • Third fiber orientation η = η ∈ (0,π) \ {π/2}, arbitrary
    The completeness proof requires the third family not be parallel to D1 or D2 (η≠0,π) and not orthogonal (η≠π/2); any such angle works, so it is a design parameter chosen by hand rather than fitted to data.
assumptions (4)
  • domain assumption Principle of Virtual Work holds for the internal virtual work functional (3.1) and external work (3.5)
    This is the foundational postulate from which equilibrium equations and boundary interactions are derived; it is standard in generalized continuum mechanics.
  • domain assumption Stored energy W is twice continuously differentiable and depends only on fiber stretch, stretch gradient, and curvature (3.23)
    The constitutive relations and Hessian formulas require differentiability and this specific dependence.
  • domain assumption The pantographic sheet energy (4.4) and bi-pantographic energy (4.11) are the correct effective energies for those microstructures, as derived heuristically in [20] and [8]
    The paper adopts these energies from prior literature without re-deriving them.
  • ad hoc to paper The tri-pantographic energy (4.29) uses the same functions f,g,h as the bi-pantographic model and is a valid continuum energy for a three-family fabric
    Postulated directly at the continuum level; no homogenization of a discrete three-family lattice is provided.
invented entities (2)
  • Tri-pantographic continuum (three-family second-gradient continuum)
    purpose: A continuum model whose stored energy is complete, i.e., positive definite in the second-gradient variable.
    The energy (4.29) is postulated; no experimental or discrete homogenization evidence is given.
  • Discrete three-family architecture
    purpose: Proposed concrete microstructure that would realize the tri-pantographic continuum.
    The paper states that rigorous homogenization remains open, so the realizability is conjectural.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the synthesis of complete two-dimensional second-gradient continua: Tri-pantographic fabrics." pith.science (2026). https://pith.science/paper/CRREQ4N3

@misc{pith2026260805524,
  author       = {Pith},
  title        = {Pith review of: On the synthesis of complete two-dimensional second-gradient continua: Tri-pantographic fabrics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CRREQ4N3}},
  note         = {Machine review of arXiv:2608.05524}
}
read the original abstract

We introduce a notion of completeness for two-dimensional second-gradient elastic continua and propose a microstructural route toward its synthesis. A continuum is said to be complete if the Hessian of the stored energy with respect to the second-gradient variable is locally positive definite, so that every nonzero admissible increment of the placement second gradient is quadratically controlled in the highest-order part of the energy about each configuration. Starting from the Principle of Virtual Work, we derive the constitutive relations, equilibrium equations, and admissible boundary interactions for a broad class of fibrous second-gradient continua whose stored energies depend on fiber stretch, stretch gradients, and curvature. The theory is then applied to several continua motivated by pantographic microstructures. Classical pantographic sheets are shown to be incomplete, while bi-pantographic fabrics enlarge the class of components of the second gradient detected by the energy but remain incomplete. Finally, we formulate a tri-pantographic continuum associated with a proposed three-family architecture and prove that it is complete. The examples illustrate how microstructural architecture influences both the completeness properties of an effective continuum and the pointwise form of its higher-order boundary interactions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

58 extracted references · 57 canonical work pages

  1. [8]

    Barchiesi, S

    E. Barchiesi, S. R. Eugster, F. dell’Isola, and F. Hild. Large in-plane elastic deformations of bi-pantographic fabrics: asymptotic homogenization and experimental validation.Mathematics and Mechanics of Solids, 25(3):739–767, 2020

    work page 2020

  2. [1]

    B. E. Abali and E. Barchiesi. Additive manufacturing introduced substructure and computational determination of metamaterials parameters by means of the asymptotic homogenization.Continuum Mechanics and Thermodynamics, 33(4):993–1009, 2021

    work page 2021

  3. [2]

    B. E. Abali, B. Vazic, and P. Newell. Influence of microstructure on size effect for metamaterials applied in composite structures.Mechanics Research Communications, 122:103877, 2022

    work page 2022

  4. [3]

    Afshar, S

    R. Afshar, S. Jeanne, and B. E. Abali. Nonlinear material modeling for mechanical characterization of 3-D printed PLA polymer with different infill densities.Applied Composite Materials, 30(3):987–1001, 2023

    work page 2023

  5. [4]

    Alibert, P

    J.-J. Alibert, P. Seppecher, and F. dell’Isola. Truss modular beams with deformation energy depending on higher displacement gradients.Mathematics and Mechanics of Solids, 8(1):51–73, 2003

    work page 2003

  6. [5]

    Aydin, B

    G. Aydin, B. C. Sarar, M. E. Yildizdag, and B. E. Abali. Investigating infill density and pattern effects in additive manufacturing by characterizing metamaterials along the strain-gradient theory.Mathematics and Mechanics of Solids, 27(10):2002–2016, 2022

    work page 2002

  7. [6]

    Aydin, M

    G. Aydin, M. E. Yildizdag, and B. E. Abali. Strain-gradient modeling and computation of 3-D printed metamaterials for verifying constitutive parameters determined by asymptotic homogenization. InTheoretical Analyses, Computations, and Experiments of Multiscale Materials: A Tribute to Francesco dell’Isola, pages 343–357. Springer International Publishing, ...

    work page 2022

  8. [7]

    Barchiesi, F

    E. Barchiesi, F. dell’Isola, F. Hild, and P. Seppecher. Two-dimensional continua capable of large elastic extension in two independent directions: Asymptotic homogenization, numerical simulations and experimental evidence.Mechanics Research Communications, 103:103466, 2020

    work page 2020

Show all 58 references
  1. [9]

    Barchiesi, S

    E. Barchiesi, S. R. Eugster, L. Placidi, and F. dell’Isola. Pantographic beam: a complete second gradient 1d-continuum in plane.Zeitschrift f¨ ur angewandte Mathematik und Physik, 70(5):135, Aug 2019

    2019

  2. [10]

    Barchiesi, J

    E. Barchiesi, J. Harsch, G. Ganzosch, and S. R. Eugster. Discrete versus homogenized continuum modeling in finite deformation bias extension test of bi-pantographic fabrics.Continuum Mechanics and Thermodynamics, 35(3):863–876, 2023

    2023

  3. [11]

    Barchiesi, M

    E. Barchiesi, M. Spagnuolo, and L. Placidi. Mechanical metamaterials: a state of the art.Mathematics and Mechanics of Solids, 24(1):212–234, 2019

    2019

  4. [12]

    Boutin, F

    C. Boutin, F. dell’Isola, I. Giorgio, and L. Placidi. Linear pantographic sheets: asymptotic micro-macro models identification. Mathematics and Mechanics of Complex Systems, 5(2):127–162, 2017

    2017

  5. [13]

    Camar-Eddine and P

    M. Camar-Eddine and P. Seppecher. Determination of the closure of the set of elasticity functionals.Archive for Rational Mechanics and Analysis, 170(3):211–245, 2003

    2003

  6. [14]

    Carcaterra and I

    A. Carcaterra and I. Giorgio. Exoskeletons by microelectromechanical textures attached to an elastic structure: Higher-order gradients metamaterials and soft robotics.Mathematics and Mechanics of Complex Systems, 14(3):397–426, 2026. 16

    2026

  7. [15]

    Chekeres, V

    O. Chekeres, V. Salnikov, and F. D’Annibale. From approximation of dissipative systems to representative space-time volume elements for metamaterials.Continuum Mechanics and Thermodynamics, 36(6):1597–1605, 2024

    2024

  8. [16]

    Spira mirabilis

    A. Ciallella, F. D’Annibale, D. Del Vescovo, and I. Giorgio. Deformation patterns in a second-gradient lattice annular plate composed of “Spira mirabilis” fibers.Continuum Mechanics and Thermodynamics, 35(4):1561–1580, 2023

    2023

  9. [17]

    De Angelo, M

    M. De Angelo, M. Spagnuolo, F. D’Annibale, A. Pfaff, K. Hoschke, A. Misra, C. Dupuy, P. Peyre, J. Dirrenberger, and M. Pawlikowski. The macroscopic behavior of pantographic sheets depends mainly on their microstructure: experimental evidence and qualitative analysis of damage ...

    2019

  10. [18]

    dell’Isola, A

    F. dell’Isola, A. Della Corte, and I. Giorgio. Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives.Mathematics and Mechanics of Solids, 22(4):852–872, 2017

    2017

  11. [19]

    dell’Isola, A

    F. dell’Isola, A. Della Corte, I. Giorgio, and D. Scerrato. Pantographic 2d sheets: Discussion of some numerical investigations and potential applications.International Journal of Non-Linear Mechanics, 80:200–208, 2016. Dynamics, Stability, and Control of Flexible Structures

    2016

  12. [20]

    dell’Isola, I

    F. dell’Isola, I. Giorgio, M. Pawlikowski, and N. L. Rizzi. Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium.Proceedings of the Royal Society of London A: Mathematical, Physica...

    2016

  13. [21]

    dell’Isola, A

    F. dell’Isola, A. Misra, L. Murcia Terranova, and E. Turco. Exotic deformation modes of curvature gradient beams and their potential realization through truss motif.Journal of Applied Mechanics, pages 1–18, 2026

    2026

  14. [22]

    dell’Isola, S

    F. dell’Isola, S. Moschini, and E. Turco. Towards the synthesis of planar beams whose deformation energy depends on the third gradient of displacement.Mathematics and Mechanics of Complex Systems, 12(4):573–597, 2024

    2024

  15. [23]

    dell’Isola, P

    F. dell’Isola, P. Seppecher, J.-J. Alibert, T. Lekszycki, R. Grygoruk, M. Pawlikowski, D. Steigmann, I. Giorgio, U. Andreaus, E. Turco, M. Golaszewski, N. Rizzi, C. Boutin, V. A. Eremeyev, A. Misra, L. Placidi, E. Barchiesi, L. Greco, M. Cuomo, A. Cazzani, A. Della Corte, A. B...

    2019

  16. [24]

    dell’Isola, P

    F. dell’Isola, P. Seppecher, and A. Della Corte. The postulations ` a la D’Alembert and ` a la Cauchy for higher gradient continuum theories are equivalent: a review of existing results.Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Scienc...

    2015

  17. [25]

    dell’Isola, P

    F. dell’Isola, P. Seppecher, L. Placidi, E. Barchiesi, and A. Misra. Least action and virtual work principles for the formulation of generalized continuum models. In F. dell’Isola and D. Steigmann, editors,Discrete and Continuum Models for Complex Metamaterials, pages 327–394....

    2020

  18. [26]

    dell’Isola, P

    F. dell’Isola, P. Seppecher, M. Spagnuolo, E. Barchiesi, F. Hild, T. Lekszycki, I. Giorgio, L. Placidi, U. Andreaus, M. Cuomo, S. R. Eugster, A. Pfaff, K. Hoschke, R. Langkemper, E. Turco, R. Sarikaya, A. Misra, M. De Angelo, F. D’Annibale, A. Bouterf, X. Pinelli, A. Misra, B....

    2019

  19. [27]

    Desmorat

    B. Desmorat. Energy theorems for micromorphic media.Mathematics and Mechanics of Complex Systems, 14(1):1–10, 2026

    2026

  20. [28]

    Desmorat, M

    B. Desmorat, M. Spagnuolo, and E. Turco. Stiffness optimization in nonlinear pantographic structures.Mathematics and Mechanics of Solids, 25(12):2252–2262, 2020

    2020

  21. [29]

    the method of virtual power in the mechanics of continuous media, I: Second-gradient theory

    M. Epstein and R. Smelser. An appreciation and discussion of Paul Germain’s “the method of virtual power in the mechanics of continuous media, I: Second-gradient theory”.Mathematics and Mechanics of Complex Systems, 8(2):191–199, 2020

    2020

  22. [30]

    V. A. Eremeyev and E. Turco. Enriched buckling for beam-lattice metamaterials.Mechanics Research Communications, 103:103458, 2020

    2020

  23. [31]

    L. C. Evans.Partial Differential Equations, volume 19 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2 edition, 2010

    2010

  24. [32]

    R. Fedele. Deformation-induced coupling of the generalized external actions in third-gradient materials.Zeitschrift f¨ ur angewandte Mathematik und Physik, 73(5):218, 2022

    2022

  25. [33]

    R. Fedele. Third-gradient continua: nonstandard equilibrium equations and selection of work conjugate variables.Mathematics and Mechanics of Solids, 27(10):2046–2072, 2022

    2022

  26. [34]

    Fedele, L

    R. Fedele, L. Placidi, and F. Fabbrocino. A review of inverse problems for generalized elastic media: formulations, experiments, synthesis.Continuum Mechanics and Thermodynamics, 36(6):1413–1453, 2024

    2024

  27. [35]

    Fedele and D

    R. Fedele and D. J. Steigmann. Lagrangian and eulerian formulations of second-grade elasticity via convected coordinates. Mathematics and Mechanics of Complex Systems, 13(3):377–389, 2025

    2025

  28. [36]

    P. Germain. La m´ ethode des puissances virtuelles en m´ ecanique des milieux continus - 1` ere partie, th´ eorie du second gradient.Journal de M´ ecanique, 12(2):235–274, 1973

    1973

  29. [37]

    P. Germain. The method of virtual power in the mechanics of continuous media, I: Second-gradient theory.Mathematics and Mechanics of Complex Systems, 8(2):153–190, 2020. Translated by M. Epstein and R. E. Smelser

    2020

  30. [38]

    Giorgio, M

    I. Giorgio, M. Spagnuolo, U. Andreaus, D. Scerrato, and A. M. Bersani. In-depth gaze at the astonishing mechanical behavior of bone: A review for designing bio-inspired hierarchical metamaterials.Mathematics and Mechanics of Solids, 26(7):1074–1103, Dec. 2021. 17

    2021

  31. [39]

    Maurin, F

    F. Maurin, F. Greco, and W. Desmet. Isogeometric analysis for nonlinear planar pantographic lattice: discrete and continuum models.Continuum Mechanics and Thermodynamics, 31(4):1051–1064, Jul 2019

    2019

  32. [40]

    Misra, N

    A. Misra, N. Nejadsadeghi, M. De Angelo, and L. Placidi. Chiral metamaterial predicted by granular micromechanics: verified with 1D example synthesized using additive manufacturing.Continuum Mechanics and Thermodynamics, 32(5):1497–1513, 2020

    2020

  33. [41]

    Moschini, L

    S. Moschini, L. Murcia Terranova, and F. D’Annibale. Kinematics of zigzagged articulated parallelograms with articulated braces (ZAPAB) mechanisms.Mathematics and Mechanics of Complex Systems, 13(4):471–500, 2025

    2025

  34. [42]

    Murcia Terranova

    L. Murcia Terranova. Planar one-dimensional continua whose energy depends on the gradient of curvature: an overview of their applications in metamaterials design.Mathematics and Mechanics of Complex Systems, 13(2):127–166, 2025

    2025

  35. [43]

    Murcia Terranova, E

    L. Murcia Terranova, E. Turco, A. Misra, and F. dell’Isola. Computational identification of double-bending stiffness: from zigzagged articulated parallelograms with articulated braces (ZAPAB) structures to pure-curvature gradient planar inextensible 1D continua.Comptes Rendus....

    2025

  36. [44]

    Nejadsadeghi, L

    N. Nejadsadeghi, L. Placidi, M. Romeo, and A. Misra. Frequency band gaps in dielectric granular metamaterials modulated by electric field.Mechanics Research Communications, 95:96–103, 2019

    2019

  37. [45]

    Pancella and F

    C. Pancella and F. D’Annibale. A Timoshenko-like equivalent beam model for the static analysis of a chiral metamaterial. Continuum Mechanics and Thermodynamics, 37(4):59, 2025

    2025

  38. [46]

    Placidi, U

    L. Placidi, U. Andreaus, and I. Giorgio. Identification of two-dimensional pantographic structure via a linear d4 orthotropic second gradient elastic model.Journal of Engineering Mathematics, 103(1):1–21, 2017

    2017

  39. [47]

    Placidi, F

    L. Placidi, F. Di Girolamo, and R. Fedele. Variational study of a Maxwell–Rayleigh-type finite length model for the preliminary design of a tensegrity chain with a tunable band gap.Mechanics Research Communications, 136:104255, 2024

    2024

  40. [48]

    Placidi, A

    L. Placidi, A. Misra, E. Barchiesi, R. Luciano, and F. Fabbrocino. Novel material anisotropy optimization based upon granular micromechanics.Comptes Rendus. M´ ecanique, 354(G1):165–182, 2026

    2026

  41. [49]

    Rodriguez and F

    C. Rodriguez and F. dell’Isola. Balance laws versus the Principle of Virtual Work and the limited scope of Noll’s theorem. Mathematics and Mechanics of Solids, Apr. 2026. OnlineFirst

    2026

  42. [50]

    Seppecher

    P. Seppecher. Microscopic interpretation of strain-gradient and generalized continuum models. In A. Bertram and S. Forest, editors,Mechanics of Strain Gradient Materials, volume 600 ofCISM International Centre for Mechanical Sciences, pages 71–99. Springer Cham, 2020

    2020

  43. [51]

    Spagnuolo

    M. Spagnuolo. Symmetrization of mechanical response in fibrous metamaterials through micro-shear deformability.Symmetry, 14(12):2660, 2022

    2022

  44. [52]

    Spagnuolo, V

    M. Spagnuolo, V. A. Eremeyev, F. D’Annibale, and F. Hild. Rescaling and mesoshear deformations of lattice metamaterials. International Journal of Solids and Structures, 315:113349, 2025

    2025

  45. [53]

    E. Turco. How the properties of pantographic elementary lattices determine the properties of pantographic metamaterials. InNew Achievements in Continuum Mechanics and Thermodynamics: A Tribute to Wolfgang H. M¨ uller, pages 489–506. Springer International Publishing, Cham, 2019

    2019

  46. [54]

    E. Turco. Heavy changes in pantograph mechanics with slight repositioning of nodes.Comptes Rendus. M´ ecanique, 354(G1):183–201, 2026

    2026

  47. [55]

    Turco and A

    E. Turco and A. Bilotta. Designing multistable metamaterials inspired by Kresling tubes.Mechanics Research Communications, 142:104348, 2024

    2024

  48. [56]

    Turco, F

    E. Turco, F. dell’Isola, A. Cazzani, and N. L. Rizzi. Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models.Zeitschrift f¨ ur angewandte Mathematik und Physik, 67(4):85, Jun 2016

    2016

  49. [57]

    M. E. Yildizdag, B. C. Sarar, A. Salvatori, G. D’Ovidio, and E. Turco. Analysis of transmission and reflection characteristics of linear plane waves in pantographic lattices.Zeitschrift f¨ ur angewandte Mathematik und Physik, 74(5):178, 2023

    2023

  50. [58]

    M. E. Yildizdag, C. A. Tran, E. Barchiesi, M. Spagnuolo, F. dell’Isola, and F. Hild. A multi-disciplinary approach for mechanical metamaterial synthesis: a hierarchical modular multiscale cellular structure paradigm. InState of the Art and Future Trends in Material Modeling, p...

    2019

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.