Pith. sign in

REVIEW 3 major objections 5 minor 53 references

Curvature-Guided Mechanics and Design of Spinodal and Shell-Based Architected Materials

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read The directional stiffness and strength of shell-based architected materials can be predicted from surface curvature alone, through a mesh-based stretch-to-bend energy ratio.

desk verdict Geometric proxies for anisotropy in shell-based architected materials are useful and broadly validated, but the E* ∝ Γ² scaling is a fitted bridge rather than a derived result—worth peer review with that issue addressed. read the letter →

arxiv 2505.21509 v1 pith:XCFQIIHX submitted 2025-05-15 cond-mat.mtrl-sci cond-mat.softphysics.app-ph

Somayajulu Dhulipala , Carlos M. Portela This is my paper · ORCID
classification cond-mat.mtrl-scicond-mat.softphysics.app-ph
keywords spinodalarchitectedmaterialsshell-basedmetamaterialscurvaturemechanicalanisotropystretching-to-bendingenergyCahn-Hilliardmorphologiesgeometricdesignmetrics
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 sets out to show that the mechanical anisotropy of thin-shell architected materials—aperiodic spinodals, triply periodic minimal surfaces, and even hollow truss lattices—is governed by how surface curvature distributes strain energy between stretching and bending under load. It derives, from affine shell kinematics on a mesh, two geometry-only metrics: a stretch-to-bend energy ratio Γ that predicts directional stiffness, and a dimensionless strength proxy Γ_s that predicts directional yield strength. The metrics are validated against finite-element homogenization and microscale compression experiments on additively manufactured spinodal samples. If the claim holds, designers can tune a microstructure's directional stiffness and strength by computing simple curvature statistics from a mesh, without a per-design finite-element solve.

What carries the argument

The carrying mechanism is an affine-deformation Kirchhoff-Love shell calculation on a triangle mesh. Every element is locally a paraboloid, the displacement field is prescribed as the macroscopic uniaxial gradient projected onto the element, and the curvilinear strain-displacement relations are integrated through the thickness to separate areal stretching energy W_s, which is linear in thickness h and proportional to sin²α, from areal bending energy W_b, which is cubic in h and controlled by projected directional, net, and Gaussian curvature terms. Taking the ratio removes the material constants and produces the geometric proxies Γ and Γ_p. The named objects are the stretch-to-bend ratio Γ, the strength proxy Γ_s, and the local bending proxy β_p = κ²_{d,p} + D²_p − K_p.

What would settle it

Hold the per-element principal curvatures and surface normals fixed but rearrange their spatial layout on the mesh—for example, cluster all high-curvature elements in one region instead of scattering them. If finite-element homogenization then changes the directional stiffness while Γ and Γ_p remain unchanged, the geometric proxy does not carry all the information needed to predict anisotropy.

Watch

Extended reading notes

Core claim

The central discovery is that the directional mechanics reduce to a ratio of two energy totals computed by pure geometry. Each mesh element is treated as a shallow paraboloid with principal curvatures κ1 and κ2; imposing the affine displacement field u0 = U_g(r·e_d)e_d and integrating the plane-stress strain energy through the shell thickness yields closed-form areal stretching and bending energies. Summed over the mesh, these give Γ = W_t,s/W_t,b, with directional stiffness E* ∝ Γ², and Γ_s = (W_t,$s^{3}$/W_t,$b^{2}$)^{1/2}, with directional yield strength σ_y* ∝ Γ_s. The same reduction yields local proxies—sin²α for stretching and β_p = κ²_{d,p} + D²_p − K_p for bending—that depend only on normals and curvature, so the anisotropy ranking of a morphology can be read off its mesh without homogenization.

Load-bearing premise

The load-bearing premise is that every shell element deforms with exactly the same displacement gradient as the macroscopic loading direction, so the proxies ignore non-affine deformation and stress redistribution; the paper explicitly states that its predictions are dominated by this affine-deformation assumption.

Editorial extensions

If this is right

  • Directional stiffness of any thin shell-based mesh can be ranked by computing Γ or Γ_p along candidate loading directions, no finite-element solve required.
  • The same geometry-only calculation gives a directional yield surface through Γ_s, so strength-aware design can be done before material selection or fabrication.
  • Since the derivation uses only shell kinematics, it transfers from spinodal morphologies to TPMS structures and hollow truss lattices whose members behave as shells.
  • A single mesh precomputation of normals, curvatures, and areas serves any modulus and, to a weak degree, any Poisson ratio, because the material constants factor out of the proxies.
  • The anisotropy ratio predicted from Γ is thickness-independent, which confines the claim to the thin-shell regime 0.01 < κh < 0.1 stated in the paper.

Reading between the lines

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

  • The affine assumption is the natural first correction: scaling each element's effective displacement gradient by its local stiffness could extend the proxies to thicker shells and finite strains without abandoning the geometry-only idea.
  • Because Γ_p depends only on per-element normal orientation and curvature, inverse design could target the surface-normal distribution rather than the full morphology, which may be easier to realize in phase-separation synthesis.
  • The mesh-based metric could be run on tomographic or image-segmented meshes of fabricated samples, offering a fast quality-control screen for anisotropy before mechanical testing.
  • It is plausible the exponents survive dynamics: if rate dependence enters only through the constituent modulus and yield strength, the geometric ranking of directions would remain valid under impact loading.
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

3 major / 5 minor

Summary. The manuscript develops a shell-theory-based kinematic framework for predicting the anisotropic stiffness and strength of shell-based architected materials, including aperiodic spinodal morphologies, TPMS structures, and hollow truss lattices. Starting from a paraboloid representation of each mesh element and an affine-displacement assumption (Eq. 22), the authors derive per-element stretching and bending energy densities (Eqs. 34-35) and total energies W_t,s and W_t,b (Eqs. 36-37). They introduce the ratio Γ = W_t,s/W_t,b as a proxy for stiffness anisotropy, assert the relation E* ∝ Γ², and define a strength proxy Γ_s = sqrt(W_t,s³/W_t,b²) (Eq. 57), together with geometric variants Γ_p and Γ_s,p that can be computed directly from mesh attributes. The framework is compared with FEA homogenization under periodic and affine boundary conditions and with microscale uniaxial compression experiments on columnar, lamellar, and isotropic spinodal morphologies.

Significance. If the central E* ∝ Γ² relation and the strength proxy can be substantiated, the paper would be a useful contribution: it provides closed-form geometric metrics computable from a mesh without solving a boundary-value problem, with linear computational cost and fast convergence. The kinematic derivation of W_s and W_b is internally consistent under the stated thin-shell, affine assumptions, and the experimental results (Fig. 7c,d) show encouraging trend-level correlations. The paper is also candid about several limitations, including the affine-deformation assumption and the thin-shell regime. However, the mapping from energy ratio to stiffness is currently an empirical plot-matching step rather than a derived consequence of the mechanics, and the strength model inherits this mapping, so the central claim is not yet fully supported.

major comments (3)
  1. [Section 4, Eqs. (34)-(37) and Fig. 4e-h] The claim that directional stiffness satisfies E* ∝ Γ² is asserted rather than derived. From the total affine energy W_t = W_t,s + W_t,b and the prescribed macroscopic strain ε = U_g, the affine upper-bound stiffness is C_aff = 2(W_t,s + W_t,b)/(U_g²V) = [2W_t,s/(U_g²V)](1 + 1/Γ), which contains a 1/Γ correction, not a Γ² dependence; the squared proportionality would require the directional variation of W_t,s itself to scale as Γ², which is not shown. The paper's support for E* ∝ Γ² consists of visually comparing (Γ/Γ_max)² with normalized stiffness surfaces in Fig. 4e-h, which is a calibration rather than a derivation. Because Eq. (55) in Section 5.2 imports E* ∝ Γ², and the strength proxy Γ_s is then obtained by substitution, any error in this exponent propagates directly into the strength predictions. I ask the authors either to derive the exponent from the kinematics or to provide an out-of-sample test that does not use the same FEA data for calibration and validation, for example a scatter plot of E*/E*_max versus Γ/Γ_max with the fitted exponent reported.
  2. [Section 5.2, Eqs. (52)-(57)] The strength prediction assumes that at the macroscopic yield point every element experiences the same affine displacement gradient U_g = U_g,lim, determined solely by the condition that at least 3% of the element volume is perpendicular to the loading direction. This ignores non-affine deformation and stress redistribution, both of which can be substantial in cellular solids. Moreover, the FEA strength validation in Section 5.1 uses the same 3%-of-volume yield definition, so part of the agreement between theory and FEA reflects a consistency of yield definition rather than independent confirmation. The authors should test the equal-displacement assumption directly, for example by comparing element-level strain energies from FEA at the 3% yield point with the affine prediction, and should report whether the strength proxy is sensitive to the choice of yield volume fraction.
  3. [Section 4, Fig. 4h and accompanying text] The hollow-octet case is a concrete counterexample to the claimed predictive power of the framework. The framework matches homogenization under affine boundary conditions but not under periodic boundary conditions, and the text concedes that the predictions are 'dominated by the affine-deformation assumption.' Since spinodal and TPMS samples deform under less restrictive, periodic-compatible conditions, matching the affine reference is not a validation. This case demonstrates that at least one tested geometry has non-affine deformation significant enough to break the proxy, which undermines the statement that the framework applies to any shell-based morphology. I recommend either restricting the claim to geometries where non-affine effects are shown to be small, or adding an out-of-sample validation against periodic-boundary-condition homogenization for several shell-based morphologies.
minor comments (5)
  1. [Eq. (58)] The left-hand side of Eq. (58) writes Γ_s,p = sqrt(W_t,s³/W_t,b³), but the definition in Eq. (57) and the right-hand side of Eq. (58) correspond to sqrt(W_t,s³/W_t,b²); this appears to be a typo that should be corrected to avoid confusion.
  2. [Section 4, near Fig. 4d] The text defines Γ = W_t,s/W_b,k, but the denominator should be W_t,b; please correct this notational error.
  3. [Eq. (55)] The symbol U_C is used without definition; the authors should define it explicitly (presumably the unit-cell volume) before first use.
  4. [Fig. 4e caption] The caption for Fig. 4e(iv) says 'normalized stiffness plotted against bending to stretching energy ratio,' but the text discusses (Γ/Γ_max)²; please label the axes and state whether the comparison involves a fixed power law or a fitted exponent.
  5. [Throughout] There are several typographical slips, including 'Massachusetts Insitute' in the affiliation and 'strenghten' in Section 6; a careful proofread is recommended.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the core energy framework is derived from shell kinematics, while the empirical E* ∝ Γ² relation used in the strength proxy is a calibrated approximation with independent experimental support, not a definitional equivalence.

full rationale

The paper's central stiffness derivation is self-contained: Eqs. (22)–(35) define an affine Kirchhoff–Love shell kinematic model from which W_s and W_b are computed per element and summed to give Γ = W_t,s/W_t,b. This is a first-principles calculation independent of the FEA used for validation. The later assertion E* ∝ Γ² is not derived from those equations but is established empirically by comparing (Γ/Γmax)² to homogenized elastic surfaces (Section 4, Fig. 4e–h); Γ and E* are separately defined quantities, so this is a calibrated empirical relation rather than a definitional reduction. The strength derivation in Section 5.2 imports that relation to obtain Γ_s = (W_t,s³/W_t,b²)^(1/2), and the resulting proxy is then checked against independent directional-strength FEA (Fig. 6f) and microscale experiments (Fig. 7d). Although the affine-deformation assumption limits accuracy, as the paper itself acknowledges for the hollow-octet case, an acknowledged modeling limitation is not circularity. The consistent use of the 3% volume yield criterion in both theory and FEA is a shared operational definition, not a concealed input. No self-citation chain or uniqueness theorem forces the claimed results, and the experimental correlation provides external grounding. Therefore no circular step can be exhibited by construction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard shell theory plus two problem-specific choices: the affine deformation ansatz and the 3% volume yield criterion. The empirical E*∝Γ^2 mapping enters as a calibrated relation rather than a derived one. No new physical entities are introduced.

free parameters (2)
  • Yield volume fraction threshold = 3%
    Strength in both FEA and theory is defined as the effective stress at which 3% of the element volume has reached the constituent yield stress. This number is chosen by the authors (following Bonatti and Mohr) and is not derived; it directly affects both the theoretical and FEA strength values and is shared by both, making part of the validation circular.
  • Stiffness-energy mapping exponent = 2 (E* proportional to (W_t,s/W_t,b)^2)
    The relation E* ∝ Γ^2 is inferred by comparing the kinematic energy ratio Γ against FEA homogenization results for the same morphologies (Fig. 4e-h). It is an empirical calibration rather than a derivation, and it is subsequently used in the derivation of the strength proxy Γ_s.
assumptions (5)
  • standard math Kirchhoff-Love thin-shell kinematics and small-strain displacement field
    Invoked at Section 4 before Eq. (17); neglects transverse shear and requires κh < 0.1 (Appendix B).
  • standard math Paraboloid approximation of each surface element
    Appendix A expands a smooth surface to second order about a point, giving Eq. (9) with principal curvatures.
  • domain assumption Affine mid-surface deformation
    Eq. (22) sets u0 = U_g(r·e_d)e_d for every element; the paper states predictions are dominated by this assumption (Fig. 4e).
  • ad hoc to paper Yield onset at 3% volume fraction
    Section 5.2 uses the same 3% volume criterion as the FEA strength method, so the theoretical and numerical yield definitions are not independent.
  • domain assumption Strength governed by stretching alone
    Section 5.2 evaluates yield from in-plane stretching stresses and discards bending terms, valid only in the thin-shell, small-curvature limit.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Curvature-Guided Mechanics and Design of Spinodal and Shell-Based Architected Materials." pith.science (2026). https://pith.science/paper/XCFQIIHX

@misc{pith2026250521509,
  author       = {Pith},
  title        = {Pith review of: Curvature-Guided Mechanics and Design of Spinodal and Shell-Based Architected Materials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XCFQIIHX}},
  note         = {Machine review of arXiv:2505.21509}
}
read the original abstract

Additively manufactured (AM) architected materials have enabled unprecedented control over mechanical properties of engineered materials. While lattice architectures have played a key role in these advances, they suffer from stress concentrations at sharp joints and bending-dominated behavior at high relative densities, limiting their mechanical efficiency. Additionally, high-resolution AM techniques often result in low-throughput or costly fabrication, restricting manufacturing scalability of these materials. Aperiodic spinodal architected materials offer a promising alternative by leveraging low-curvature architectures that can be fabricated through techniques beyond AM. Enabled by phase separation processes, these architectures exhibit tunable mechanical properties and enhanced defect tolerance by tailoring their curvature distributions. However, the relation between curvature and their anisotropic mechanical behavior remains poorly understood. In this work, we develop a theoretical framework to quantify the role of curvature in governing the anisotropic stiffness and strength of shell-based spinodal architected materials. We introduce geometric metrics that predict the distribution of stretching and bending energies under different loading conditions, bridging the gap between curvature in doubly curved shell-based morphologies and their mechanical anisotropy. We validate our framework through finite element simulations and microscale experiments, demonstrating its utility in designing mechanically robust spinodal architectures. This study provides fundamental insights into curvature-driven mechanics, guiding the optimization of next-generation architected materials for engineering applications.

Figures

Figures reproduced from arXiv: 2505.21509 by the authors.

Figure 1
Figure 1. (a) Renders of shell-based architected materials exhibiting a spectrum from periodic to aperiodic structures. (b) [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. (a) Normalized Gaussian curvature vs normalized mean curvature plotted for various shell-based architected [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. (a) Schematic of a general paraboloid with an inset showing a magnified view near the origin. (b) Mid-surface [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: (a) Schematic of mesh and important parameters of the mesh shown (inset). Theoretical (kinematic) prediction [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: (i) Stretching proxy, (ii) bending proxy, decomposed into (iii) projected net curvature, (iv) projected Gaussian [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Directional strength of a columnar spinodal morphology with stress-strain plots obtained from implicit FEA [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: (a) SEM image of a columnar morphology. (b) Stress-strain plots along di [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

53 extracted references · 21 canonical work pages

  1. [1]

    Kadic, G

    M. Kadic, G. W. Milton, M. van Hecke, M. Wegener, 3D metamaterials, Nature Reviews Physics 1 (3) (2019) 198–210. doi:10.1038/s42254-018-0018-y. URLhttps://www.nature.com/articles/s42254-018-0018-y

    doi:10.1038/s42254-018-0018-y 2019

  2. [2]

    J. U. Surjadi, C. M. Portela, Enabling three-dimensional architected materials across length scales and timescales, Nature Materials 24 (4) (2025) 493–505. doi:10.1038/s41563-025-02119-8. URLhttps://www.nature.com/articles/s41563-025-02119-8

    doi:10.1038/s41563-025-02119-8 2025

  3. [3]

    Zheng, H

    X. Zheng, H. Lee, T. H. Weisgraber, M. Shusteff, J. DeOtte, E. B. Duoss, J. D. Kuntz, M. M. Biener, Q. Ge, J. A. Jackson, S. O. Kucheyev, N. X. Fang, C. M. Spadaccini, Ultralight, ultrastiffmechanical metamaterials, Science 344 (6190) (2014) 1373–1377. doi:10.1126/science.1252291. URLhttps://www.science.org/doi/10.1126/science.1252291

    doi:10.1126/science.1252291 2014

  4. [4]

    L. R. Meza, S. Das, J. R. Greer, Strong, lightweight, and recoverable three-dimensional ceramic nanolattices, Science 345 (6202) (2014) 1322–1326. doi:10.1126/science.1255908. URLhttps://www.science.org/doi/10.1126/science.1255908

    doi:10.1126/science.1255908 2014

  5. [5]

    Bauer, A

    J. Bauer, A. Schroer, R. Schwaiger, O. Kraft, Approaching theoretical strength in glassy carbon nanolattices, Nature Materials 15 (4) (2016) 438–443. doi:10.1038/nmat4561. URLhttps://www.nature.com/articles/nmat4561

    doi:10.1038/nmat4561 2016

  6. [6]

    C. M. Portela, A. Vidyasagar, S. Kr ¨odel, T. Weissenbach, D. W. Yee, J. R. Greer, D. M. Kochmann, Extreme me- chanical resilience of self-assembled nanolabyrinthine materials, Proceedings of the National Academy of Sciences 117 (11) (2020) 5686–5693. doi:10.1073/pnas.1916817117. URLhttps://www.pnas.org/content/117/11/5686

    doi:10.1073/pnas.1916817117 2020

  7. [7]

    Bauer, L

    J. Bauer, L. R. Meza, T. A. Schaedler, R. Schwaiger, X. Zheng, L. Valdevit, Nanolattices: An Emerging Class of Mechanical Metamaterials, Advanced Materials 29 (40) (2017) 1701850. doi:10.1002/adma.201701850. URLhttps://onlinelibrary.wiley.com/doi/abs/10.1002/adma.201701850

    doi:10.1002/adma.201701850 2017

  8. [8]

    Crook, J

    C. Crook, J. Bauer, A. Guell Izard, C. Santos de Oliveira, J. Martins de Souza e Silva, J. B. Berger, L. Valdevit, Plate-nanolattices at the theoretical limit of stiffness and strength, Nature Communications 11 (1) (2020) 1579. doi:10.1038/s41467-020-15434-2. URLhttps://www.nature.com/articles/s41467-020-15434-2

    doi:10.1038/s41467-020-15434-2 2020

Show all 53 references
  1. [9]

    J. B. Berger, H. N. G. Wadley, R. M. McMeeking, Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness, Nature 543 (7646) (2017) 533–537. doi:10.1038/nature21075. URLhttp://www.nature.com/articles/nature21075

    2017 doi

  2. [10]

    L. J. Gibson, M. F. Ashby, Cellular Solids: Structure and Properties, 2nd Edition, Cambridge Solid State Science Series, Cambridge University Press, Cambridge, 1997. doi:10.1017/CBO9781139878326. URLhttps://www.cambridge.org/core/books/cellular-solids/BC25789552BAA8E3CAD5E1D105612AB5

    1997 doi

  3. [11]

    Glaesener, S

    R. Glaesener, S. Kumar, C. Lestringant, T. Butruille, C. Portela, D. Kochmann, Predicting the influence of geomet- ric imperfections on the mechanical response of 2d and 3d periodic trusses, Acta Materialia 254 (2023) 118918. doi:https://doi.org/10.1016/j.actamat.2023.118918. ...

    2023

  4. [12]

    L. Liu, P. Kamm, F. Garc ´ıa-Moreno, J. Banhart, D. Pasini, Elastic and failure response of imperfect three- dimensional metallic lattices: the role of geometric defects induced by selective laser melting, Journal of the Mechanics and Physics of Solids 107 (2017) 160–184. doi:...

    2017 doi

  5. [13]

    Gross, P

    A. Gross, P. Pantidis, K. Bertoldi, S. Gerasimidis, Correlation between topology and elastic properties of imperfect truss-lattice materials, Journal of the Mechanics and Physics of Solids 124 (2019) 577–598. doi:https://doi.org/10.1016/j.jmps.2018.11.007. URLhttps://www.scien...

    2019 doi

  6. [14]

    Pellegrino, C

    S. Pellegrino, C. R. Calladine, Matrix analysis of statically and kinematically indeterminate frameworks, Interna- tional Journal of Solids and Structures 22 (4) (1986) 409–428. doi:10.1016/0020-7683(86)90014-4. URLhttps://www.sciencedirect.com/science/article/pii/0020768386900144

    1986

  7. [15]

    Deshpande, N

    V . Deshpande, N. Fleck, M. Ashby, Effective properties of the octet-truss lattice material, Journal of the Mechanics and Physics of Solids 49 (8) (2001) 1747–1769. doi:https://doi.org/10.1016/S0022-5096(01)00010-2. URLhttps://www.sciencedirect.com/science/article/pii/S0022509...

    2001 doi

  8. [16]

    V . S. Deshpande, M. F. Ashby, N. A. Fleck, Foam topology: bending versus stretching dominated architectures, Acta Materialia 49 (6) (2001) 1035–1040. doi:10.1016/S1359-6454(00)00379-7. URLhttps://www.sciencedirect.com/science/article/pii/S1359645400003797

    2001 doi

  9. [17]

    C. M. Portela, J. R. Greer, D. M. Kochmann, Impact of node geometry on the effective stiffness of non-slender three-dimensional truss lattice architectures, Extreme Mechanics Letters 22 (2018) 138–148. 27 doi:10.1016/j.eml.2018.06.004. URLhttps://www.sciencedirect.com/science/...

    2018 doi

  10. [18]

    Hsieh, B

    M.-T. Hsieh, B. Endo, Y . Zhang, J. Bauer, L. Valdevit, The mechanical response of cellular materials with spinodal topologies, Journal of the Mechanics and Physics of Solids 125 (2019) 401–419. doi:10.1016/j.jmps.2019.01.002. URLhttps://www.sciencedirect.com/science/article/p...

    2019 doi

  11. [19]

    N. G. Dou, R. A. Jagt, C. M. Portela, J. R. Greer, A. J. Minnich, Ultralow thermal conductivity and me- chanical resilience of architected nanolattices, Nano Letters 18 (8) (2018) 4755–4761, pMID: 30022671. doi:10.1021/acs.nanolett.8b01191. URLhttps://doi.org/10.1021/acs.nanol...

    2018 doi

  12. [20]

    L. R. Meza, G. P. Phlipot, C. M. Portela, A. Maggi, L. C. Montemayor, A. Comella, D. M. Kochmann, J. R. Greer, Reexamining the mechanical property space of three-dimensional lattice architectures, Acta Materialia 140 (2017) 424–432. doi:https://doi.org/10.1016/j.actamat.2017.0...

    2017 doi

  13. [21]

    Y . Wang, X. Zhang, Z. Li, H. Gao, X. Li, Achieving the theoretical limit of strength in shell-based carbon nanolat- tices, Proceedings of the National Academy of Sciences 119 (34) (2023) e2119536119, publisher: Proceedings of the National Academy of Sciences. doi:10.1073/pnas...

    2023 doi

  14. [22]

    Tancogne-Dejean, M

    T. Tancogne-Dejean, M. Diamantopoulou, M. B. Gorji, C. Bonatti, D. Mohr, 3d plate-lattices: An emerging class of low-density metamaterial exhibiting optimal isotropic stiffness, Advanced Materials 30 (45) (2018) 1803334. doi:https://doi.org/10.1002/adma.201803334. URLhttps://o...

    2018 doi

  15. [23]

    Bonatti, D

    C. Bonatti, D. Mohr, Smooth-shell metamaterials of cubic symmetry: Anisotropic elasticity, yield strength and specific energy absorption, Acta Materialia 164 (2019) 301–321. doi:10.1016/j.actamat.2018.10.034. URLhttps://www.sciencedirect.com/science/article/pii/S1359645418308346

    2019 doi

  16. [24]

    S. C. Han, J. W. Lee, K. Kang, A new type of low density material: Shellular, Advanced Materials 27 (37) (2015) 5506–5511. doi:https://doi.org/10.1002/adma.201501546. URLhttps://advanced.onlinelibrary.wiley.com/doi/abs/10.1002/adma.201501546

    2015 doi

  17. [25]

    B. D. Nguyen, S. C. Han, Y . C. Jung, K. Kang, Design of the p-surfaced shellular, an ultra- low density material with micro-architecture, Computational Materials Science 139 (2017) 162–178. doi:https://doi.org/10.1016/j.commatsci.2017.07.025. URLhttps://www.sciencedirect.com/...

    2017 doi

  18. [26]

    Al-Ketan, R

    O. Al-Ketan, R. Rezgui, R. Rowshan, H. Du, N. X. Fang, R. K. Abu Al-Rub, Microarchitected stretching- dominated mechanical metamaterials with minimal surface topologies, Advanced Engineering Materials 20 (9) (2018) 1800029. doi:https://doi.org/10.1002/adem.201800029. URLhttps:...

    2018 doi

  19. [27]

    Zhang, S

    L. Zhang, S. Feih, S. Daynes, S. Chang, M. Y . Wang, J. Wei, W. F. Lu, Energy absorption characteristics of metallic triply periodic minimal surface sheet structures under compressive loading, Additive Manufacturing 23 (2018) 505–515. doi:https://doi.org/10.1016/j.addma.2018.0...

    2018 doi

  20. [28]

    M.-S. Pham, C. Liu, I. Todd, J. Lertthanasarn, Damage-tolerant architected materials inspired by crystal microstruc- ture, Nature 565 (7739) (2019) 305–311. doi:10.1038/s41586-018-0850-3. URLhttps://doi.org/10.1038/s41586-018-0850-3

    2019 doi

  21. [29]

    S. Shi, Y . Li, B.-N. Ngo-Dinh, J. Markmann, J. Weissm ¨uller, Scaling behavior of stiffness and strength of hierar- chical network nanomaterials, Science 371 (6533) (2021) 1026–1033. doi:10.1126/science.abd9391. URLhttps://www.science.org/doi/abs/10.1126/science.abd9391

    2021 doi

  22. [30]

    I. C. Cheng, A. M. Hodge, High temperature morphology and stability of nanoporous ag foams, Journal of Porous Materials 21 (4) (2014) 467–474. doi:10.1007/s10934-014-9793-8. URLhttps://doi.org/10.1007/s10934-014-9793-8

    2014 doi

  23. [31]

    I. C. Cheng, A. M. Hodge, Morphology, oxidation, and mechanical behavior of nanoporous cu foams, Advanced Engineering Materials 14 (4) (2012) 219–226. doi:https://doi.org/10.1002/adem.201100252. URLhttps://advanced.onlinelibrary.wiley.com/doi/abs/10.1002/adem.201100252

    2012 doi

  24. [32]

    Cheng, A

    I.-C. Cheng, A. M. Hodge, Strength scale behavior of nanoporous ag, pd and cu foams, Scripta Materialia 69 (4) (2013) 295–298. doi:https://doi.org/10.1016/j.scriptamat.2013.04.023. URLhttps://www.sciencedirect.com/science/article/pii/S1359646213002340

    2013 doi

  25. [33]

    Tsujioka, N

    N. Tsujioka, N. Ishizuka, N. Tanaka, T. Kubo, K. Hosoya, Well-controlled 3D skeletal epoxy-based monoliths obtained by polymerization induced phase separation, Journal of Polymer Science Part A: Polymer Chemistry 46 (10) (2008) 3272–3281. doi:https://doi.org/10.1002/pola.22665...

    2008 doi

  26. [34]

    M. N. Lee, A. Mohraz, Bicontinuous macroporous materials from bijel templates, Advanced Materials 22 (43) 28 (2010) 4836–4841. doi:https://doi.org/10.1002/adma.201001696. URLhttps://advanced.onlinelibrary.wiley.com/doi/abs/10.1002/adma.201001696

    2010 doi

  27. [35]

    Fern ´andez-Rico, S

    C. Fern ´andez-Rico, S. Schreiber, H. Oudich, C. Lorenz, A. Sicher, T. Sai, V . Bauernfeind, S. Heyden, P. Carrara, L. D. Lorenzis, R. W. Style, E. R. Dufresne, Elastic microphase separation produces robust bicontinuous materials, Nature Materials 23 (1) (2024) 124–130. doi:10...

    2024 doi

  28. [36]

    Vidyasagar, S

    A. Vidyasagar, S. Kr ¨odel, D. M. Kochmann, Microstructural patterns with tunable mechanical anisotropy obtained by simulating anisotropic spinodal decomposition, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474 (2218) (2018) 20180535. do...

    2018

  29. [37]

    Kumar, S

    S. Kumar, S. Tan, L. Zheng, D. M. Kochmann, Inverse-designed spinodoid metamaterials, npj Computational Materials 6 (1) (2020) 1–10, publisher: Nature Publishing Group. doi:10.1038/s41524-020-0341-6. URLhttps://www.nature.com/articles/s41524-020-0341-6

    2020 doi

  30. [38]

    Soyarslan, S

    C. Soyarslan, S. Bargmann, M. Pradas, J. Weissm ¨uller, 3d stochastic bicontinuous microstructures: Generation, topology and elasticity, Acta Materialia 149 (2018) 326–340. doi:https://doi.org/10.1016/j.actamat.2018.01.005. URLhttps://www.sciencedirect.com/science/article/pii/...

    2018 doi

  31. [39]

    Thakolkaran, M

    P. Thakolkaran, M. Espinal, S. Dhulipala, S. Kumar, C. M. Portela, Experiment-informed finite- strain inverse design of spinodal metamaterials, Extreme Mechanics Letters 74 (2025) 102274. doi:https://doi.org/10.1016/j.eml.2024.102274. URLhttps://www.sciencedirect.com/science/a...

    2025

  32. [40]

    Lazarus, H

    A. Lazarus, H. C. B. Florijn, P. M. Reis, Geometry-induced rigidity in nonspherical pressurized elastic shells, Phys. Rev. Lett. 109 (2012) 144301. doi:10.1103/PhysRevLett.109.144301. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.109.144301

    2012 doi

  33. [41]

    Taffetani, F

    M. Taffetani, F. Box, A. Neveu, D. Vella, Limitations of curvature-induced rigidity: How a curved strip buckles under gravity, EPL (Europhysics Letters) 127 (1) (2019) 14001. doi:10.1209/0295-5075/127/14001. URLhttps://doi.org/10.1209/0295-5075/127/14001

    2019 doi

  34. [42]

    R. Wu, K. W. Kwan, A. H. W. Ngan, Printed miniature robotic actuators with curvature-induced stiffness control inspired by the insect wing, Bioinspiration and Biomimetics 16 (4) (2021) 046018. doi:10.1088/1748-3190/abffec. URLhttps://doi.org/10.1088/1748-3190/abffec

    2021 doi

  35. [43]

    E. Ramm, W. A. Wall, Shell structures—a sensitive interrelation between physics and numerics, International Journal for Numerical Methods in Engineering 60 (1) (2004) 381–427. doi:10.1002/nme.967. URLhttps://onlinelibrary.wiley.com/doi/abs/10.1002/nme.967

    2004 doi

  36. [45]

    Zheng, S

    L. Zheng, S. Kumar, D. M. Kochmann, Data-driven topology optimization of spinodoid metamaterials with seamlessly tunable anisotropy, Computer Methods in Applied Mechanics and Engineering 383 (2021) 113894. doi:10.1016/j.cma.2021.113894. URLhttps://www.sciencedirect.com/science...

    2021

  37. [46]

    F. V . Senhora, E. D. Sanders, G. H. Paulino, Optimally-tailored spinodal architected materials for multiscale design and manufacturing, Advanced Materials 34 (26) (2022) 2109304. doi:https://doi.org/10.1002/adma.202109304. URLhttps://advanced.onlinelibrary.wiley.com/doi/abs/1...

    2022 doi

  38. [47]

    Al-Ketan, R

    O. Al-Ketan, R. K. Abu Al-Rub, MSLattice: A free software for generating uniform and graded lattices based on triply periodic minimal surfaces, Material Design & Processing Communications 3 (6) (2021) e205. doi:10.1002/mdp2.205. URLhttps://onlinelibrary.wiley.com/doi/abs/10.10...

    2021 doi

  39. [48]

    V . Pini, J. J. Ruz, P. M. Kosaka, O. Malvar, M. Calleja, J. Tamayo, How two-dimensional bending can extraordi- narily stiffen thin sheets, Scientific Reports 6 (1) (2016) 29627. doi:10.1038/srep29627. URLhttps://www.nature.com/articles/srep29627

    2016 doi

  40. [49]

    T. A. Witten, Stress focusing in elastic sheets, Reviews of Modern Physics 79 (2) (2007) 643–675. doi:10.1103/RevModPhys.79.643. URLhttps://link.aps.org/doi/10.1103/RevModPhys.79.643

    2007 doi

  41. [50]

    Y . Guo, S. Sharma, S. Kumar, Inverse designing surface curvatures by deep learning, Advanced Intelligent Systems 6 (6) (2024) 2300789. arXiv:https://advanced.onlinelibrary.wiley.com/doi/pdf/10.1002/aisy.202300789, doi:https://doi.org/10.1002/aisy.202300789. URLhttps://advance...

    2024 doi

  42. [51]

    Rusinkiewicz, Estimating curvatures and their derivatives on triangle meshes, in: Proceedings

    S. Rusinkiewicz, Estimating curvatures and their derivatives on triangle meshes, in: Proceedings. 2nd International Symposium on 3D Data Processing, Visualization and Transmission, 2004. 3DPVT 2004., 2004, pp. 486–493. doi:10.1109/TDPVT.2004.1335277. 29 URLhttps://ieeexplore.i...

    2004 arXiv

  43. [52]

    J. N. Reddy, Theory and Analysis of Elastic Plates and Shells, 2nd Edition, CRC Press, Boca Raton, 2006, Ch. 11, pp. 403–478. doi:10.1201/9780849384165

    2006 doi

  44. [53]

    URLhttp://archive.org/details/dli.ernet.449352

    SokolnikoffI.s., Mathematical Theory Of Elasticity, Mcgraw-hill Book Company New York and London, 1956. URLhttp://archive.org/details/dli.ernet.449352

    1956

  45. [54]

    A. N. Palazotto, Nonlinear analysis of shell structures, Washington, DC : American Institute of Aeronautics and Astronautics, 1992. URLhttp://archive.org/details/nonlinearanalysi0000pala 30 Figure C.9: Histogram of stretching-to-bending energies plotted for elements against th...

    1992

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.