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This paper develops a hybrid material point method framework that reproduces the large-deformation three-point bending response of pressurized tubes and tape springs, including cross-sectional ovalization and contact, in good to excellent a

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2026-08-02 01:37 UTC pith:QCNST46J

load-bearing objection A useful but over-claimed validation study: the new part is applying an existing hybrid-MPM method to three-point bending of tubes and tape springs, and it deserves review, but the advertised 'excellent agreement' partly rests on unmeasured E and a fitted adapter correction. the 5 major comments →

arxiv 2607.14594 v1 pith:QCNST46J submitted 2026-07-16 cond-mat.soft cond-mat.mtrl-scicond-mat.stat-mech

Numerical and experimental framework for bending elasticity of highly flexible slender structures

Shunsuke Nomura , Satsuki Shibuya , Isamu Hashiguchi , Ryuichi Tarumi , Tomohiko G. Sano This is my paper
classification cond-mat.soft cond-mat.mtrl-scicond-mat.stat-mech
keywords tubespipestape springsBrazier instabilitymaterial point methodhybrid finite-element/meshfreecontact mechanicsthree-point bending
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to establish that a hybrid finite-element/material-point method (hybrid-MPM) can simulate the three-point bending test of highly flexible slender structures — specifically pressurized cylindrical tubes and tape springs — with enough fidelity to capture the characteristic coupling between lengthwise bending and cross-sectional ovalization (Brazier instability), including contact with supports and indenter. The authors validate the framework against desktop experiments and classical analytical predictions, finding good agreement for tubes (up to the onset of instability) and excellent agreement for tape springs (even past snap-through). If this holds, the framework offers a practical computational tool for designing and analyzing large-deformation, contact-rich slender structures such as soft robots and deployable structures, where conventional FEM struggles with contact and remeshing.

Core claim

The central claim is that the hybrid-MPM approach, which alternately uses Lagrangian finite elements for shell elasticity and an Eulerian grid for contact resolution, can replicate the force-displacement curves of three-point bending tests for elastic tubes and tape springs. For pressurized tubes, the simulated maximum bending force matches both experiments and Brazier's formula F* = F0* sqrt(1 + p/p*), and the force-displacement response agrees up to the peak; discrepancies after the peak are attributed to unmodeled fluid-structure interaction and adapter boundary conditions. For tape springs, the simulated force response agrees with experiments in detail, including the linear-to-nonlinear

What carries the argument

The central mechanism is the hybrid-MPM time integration: after each explicit finite-element step updates nodal velocities from internal and external forces (including internal pressure), the velocities are mapped onto an Eulerian grid (F2G), where contact between bodies is detected via the normal relative velocity and corrected by enforcing non-penetration and momentum conservation (neglecting tangential friction), then mapped back to FE nodes (G2F). This combines the accurate shell elasticity of FEM with the robust contact handling of MPM. The analytical counterpart is Brazier's energy minimization over the cross-sectional ovalization amplitude zeta, yielding closed-form force-displacement

Load-bearing premise

The correspondence between simulation and experiment assumes that tangential (friction) forces at contacts and fluid-structure interaction inside the pressurized tube are negligible; if either materially changes the deformed cross-section, the predicted force-displacement curves, especially after the peak, will not match.

What would settle it

Measure the force-displacement curve of a pressurized tube in three-point bending with the tube's ends sealed while keeping the internal fluid volume fixed (so fluid cannot be ejected), and compare with the open-reservoir case: if the two curves differ significantly after the peak, fluid-structure interaction is important and the current constant-pressure simulation is missing a crucial effect.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The hybrid-MPM framework can simulate three-point bending of flexible tubes and tape springs without explicit contact algorithms or remeshing.
  • The simulated maximum bending force for pressurized tubes follows the classical prediction F* = F0* sqrt(1 + p/p*), confirming that internal pressure stiffens the cross-section against ovalization.
  • For tape springs, the simulation reproduces the force response beyond kink formation, something the infinitely-long-shell theory fails to capture.
  • The framework provides a foundation for simulating contact-rich large deformations in soft robots and deployable structures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The neglect of tangential friction at contacts may explain residual discrepancies in post-peak tube response; adding a Coulomb-friction contact model would be a direct test.
  • The saturation of the simulated tube force after the peak versus the experimental decrease suggests that fluid-structure interaction (water being ejected from the tube) contributes to the unloading, so a coupled FSI simulation could improve predictions.
  • The same hybrid-MPM formalism could be extended to other cross-section geometries (e.g., rectangular or corrugated tubes) and to dynamic loading, retaining the same contact machinery.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. The paper presents a hybrid finite-element/material-point-method (hybrid-MPM) framework for simulating three-point bending of highly flexible slender structures, using elastic tubes and tape springs as canonical examples. The authors derive classical Brazier-type moment–curvature relations, run hybrid-MPM simulations with contact, perform desktop experiments, and compare force–displacement curves and peak forces. They report good agreement for pressurized tubes up to the peak force and excellent agreement for tape springs across several geometries, and conclude that the framework is robust for predicting large deformation of structures involving complex contact.

Significance. If the validation were independent and the claims properly scoped, this would be a useful contribution: it combines a robust contact-handling numerical method with canonical experiments on two technologically relevant structures, and it provides explicit analytical formulas for the force–displacement response. The paper is also commendably candid about several discrepancies, including the tube post-peak behavior and the neglect of friction and fluid–structure interaction. However, the central validation loop is not yet closed because key material parameters are asserted rather than measured and one experimental correction is fitted to the very data being compared. These issues must be addressed before the paper can support the strong claim in the abstract.

major comments (5)
  1. [§4.1 and §5.1] The Young's moduli used in the simulations, E=0.47 MPa for the VPS tube and E=3.8 GPa for the polyester tape, are stated without any independent measurement, datasheet reference, or calibration protocol. Since the initial stiffness K0 (Eq. (6)) and the force scale F0 (Eqs. (4) and (17)) are both linear in E, selecting E can force the linear-regime slope and the overall force scale to match. The abstract's inference that agreement implies robustness is therefore not yet a parameter-free validation. Please provide independent tensile-test data or a cited datasheet, and include a sensitivity study (e.g., ±10% in E) to show that the agreement is not imposed by parameter choice.
  2. [§4.2.2 and §4.3] The adapter tension f=0.22 tanh(0.05Δ) N is fitted to the experimentally measured force-displacement data 'when the tube elasticity is negligible' and then subtracted from the load-cell output before comparison with simulation. This is a fitted correction, not an independent mechanical characterization of the adapters, and it directly shapes the peak and post-peak force curve. The acknowledged post-peak discrepancy (simulation saturates, experiment decreases) could be partly an artifact of this subtraction. Please provide raw load-cell curves, a separately measured adapter-only force response, and error bounds on f. Also, Sec. 4.3 states that the adapters are 'accounted for by applying axial forces in the simulations,' but Sec. 4.1 does not specify these axial forces; clarify what was actually applied.
  3. [§3.1 and §4.1] The Neo-Hookean model in Eq. (22) uses Lamé parameter λ=Eν/{(1+ν)(1-2ν)}. For the tube, Sec. 4.1 sets ν=0.5, for which λ is singular and the stored energy is ill-defined. If the authors intend a nearly incompressible material, they must state the actual value used (e.g., ν=0.49 or 0.495) and report its influence on the results. As written, the tube simulation input is internally inconsistent.
  4. [§2.1, Eq. (4), §4.3] The conversion κ=8Δ/L² is stated without derivation and is not the standard relation for a point-loaded simply supported beam, where the maximum curvature is κ=12Δ/L². The paper attributes the later theoretical peak displacement (Sec. 4.3) to kink localization and fluid effects, but a factor of 1.5 in the assumed curvature would itself shift the predicted peak to smaller Δ and alter the comparison in Fig. 3(a). Please justify the geometric relation or replace it with a more appropriate curvature–displacement relation for three-point bending, and discuss the sensitivity of the conclusions to this assumption.
  5. [Abstract and §6] The abstract claims that 'the excellent agreement between the simulation and the experiments implies that the hybrid-MPM framework provides a robust computational framework for predicting the large deformation of structures involving complex contact.' For tubes, however, Sec. 4.3 and Sec. 6 explicitly state that the post-peak force is not captured: the simulation saturates while the experiment decreases. The conclusion should be scoped to pre-peak and peak-force behavior, and the tape-spring results should be presented as the primary demonstration of post-kink agreement. A more cautious wording would prevent the main claim from overstating the demonstrated predictive capability.
minor comments (6)
  1. [Eq. (8)] The first term inside the parentheses appears to be κ̃, but consistency with Eq. (2) and the subsequent derivation requires κ̃². Please correct this typographical error.
  2. [Table 1 and §3] Several numerical parameters that are essential for reproducibility—grid spacing Δg, time step Δt, damping coefficient γ_d, and mesh density—are not listed in Table 1 or elsewhere. Please report these values or provide a reference where they are fixed.
  3. [Fig. 3 and Fig. 4] The captions do not fully define the line styles and symbols. For example, Fig. 3(b) states 'The same symbols represent the same thickness' but does not identify which symbol corresponds to which h value. Please make the legends self-contained.
  4. [§4.2.2] The fitting of f=0.22 tanh(0.05Δ) is described in one sentence. Please report the number of data points, the fitting range, and the uncertainty, so that the correction can be assessed.
  5. [References] Reference [62] contains the malformed DOI '10.1103/hv9t-3h5w' and appears incomplete. Please verify the bibliographic information.
  6. [General] The phrase 'post-buckled' in Sec. 6 is used where 'post-peak' or 'post-instability' may be more precise; the tube does not necessarily undergo buckling in the classical sense at the force maximum. Minor language tightening throughout would improve readability.

Circularity Check

0 steps flagged

No significant circularity: classical theory and independent simulations; E and adapter-fit concerns are validation gaps, not by-construction reductions.

full rationale

The claimed derivation chain is self-contained. The tube theory (Eqs. 1–10) and tape-spring theory (Eqs. 11–18) are classical Brazier-type energy minimizations with explicit displacement ansatze and boundary conditions; they are not defined in terms of the target force–displacement curves. The hybrid-MPM simulation uses Neo-Hookean elasticity with stated E, ν, and pressure, and the contact algorithm is taken from external literature (Bardenhagen et al., Ref. [58]). The experimental comparisons are newly reported, and the paper frankly discloses discrepancies (e.g., tube force saturates in simulation while experimental force decreases post-peak, Sec. 4.3), which is not the signature of an agreement imposed by construction. The E values in Secs. 4.1 and 5.1 are asserted without independent measurement or citation, so the initial stiffness K0 and force scale F0 are not demonstrated to be parameter-free; however, the paper nowhere states that E was calibrated to the target bending curves, so no specific reduction such as Eq. (6) being fitted to the measured F–Δ data can be exhibited. The adapter correction f = 0.22 tanh(0.05Δ) in Sec. 4.2.2 is a tare-type correction fitted in a regime where the tube elasticity is stated to be negligible and is subtracted before comparing with the simulation; it is not a parameter of the simulation and does not, by itself, force the simulated curve to match. Self-citations [40,41] for prior validation of the hybrid-MPM framework are not load-bearing because the present manuscript adds independent experimental validation and the underlying numerical formulation cites external method papers [35–38,42–44]. No uniqueness theorem or ansatz is imported from the authors' own prior work in a way that determines the central result. Thus no circular step meeting the required standard can be identified.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

No new particles, forces, or conserved quantities are postulated. The ledger instead captures material parameters and modeling choices that the central claim depends on; several are asserted without independent measurement or documentation.

free parameters (5)
  • Young's modulus of VPS tube, E = 0.47 MPa
    Assigned in Sec. 4.1 without citation or independent tensile measurement; if inferred from the same three-point bending curves, the simulated initial stiffness is fitted, not predicted.
  • Young's modulus of polyester tape, E = 3.8 GPa
    Assigned in Sec. 5.1 without citation or independent measurement.
  • Poisson ratio of tube, ν = 0.5
    Incompressible value; λ in Eq. (22) is singular at ν=0.5 and no mixed-formulation/locking treatment is described.
  • Adapter tension correction coefficients = 0.22 N, 0.05 1/mm in f=0.22 tanh(0.05Δ)
    Fitted to the same force-displacement data (Sec. 4.2.2) and subtracted from load-cell output; makes the tube comparison partially curve-fit.
  • Numerical integration parameters (Δg, Δt, γ_d, mesh density) = not reported
    Defined in Sec. 3 but no values given; chosen by hand and necessary to reproduce force curves.
axioms (6)
  • domain assumption Neo-Hookean hyperelastic constitutive law (Eq. 22) for tube and tape materials
    No material characterization shows VPS or polyester is neo-Hookean at the strains reached; Poisson ratio 0.5 creates near-incompressibility.
  • standard math Brazier ansatz w=-Rζ cos 2θ and inextensible circumferential centerline for tubes
    Classical approximation from Refs [2,21] used to derive Eqs. (1)-(4); assumes uniform ovalization and infinite length.
  • standard math Tape cross-section mode w=-Rζ cos(πθ/(2β)) with moment-free edges
    Fourier mode satisfying K_θ=0 at θ=±β, used for Eqs. (14)-(18).
  • domain assumption Frictionless contact with Bardenhagen et al. normal-contact correction
    Sec. 3.2.2 neglects tangential forces; authors claim sufficient but no friction-sensitivity study is provided.
  • domain assumption Tube internal pressure modeled as static surface force; fluid dynamics neglected
    Sec. 4.3 states fluid is not implemented and may be crucial; pressure is applied as f_pres on the inner surface.
  • domain assumption Uniform curvature relation κ=8Δ/L^2 for force-displacement conversion
    Used in Eq. (4) to map M-κ to F-Δ; valid for small sagitta of a circular arc, not exact for three-point beam bending; may explain theoretical peak at larger Δ.

pith-pipeline@v1.3.0-alltime-deepseek · 18027 in / 12252 out tokens · 127999 ms · 2026-08-02T01:37:54.237950+00:00 · methodology

0 comments
read the original abstract

Slender structures are highly flexible, spanning several orders of magnitude in length scale. Their deformation depends on the slenderness of their cross sections, highlighting that the elasticity and geometry of structures are intrinsically coupled. The deformation of the cross-section becomes significant, particularly when tubes and pipes are subjected to bending, known as the Brazier instability. Although the bending performance of slender structures is quantified experimentally using a canonical three-point bending test, their numerical counterparts remain under-explored because complex contact mechanics must be implemented in simulations. In this study, we develop a computational framework to simulate experimental three-point bending tests using a hybrid material point method (hybrid-MPM) approach, which integrates Lagrangian finite element and Eulerian finite difference frameworks. We adapt our framework to elastic tubes and tape springs as canonical examples that exhibit characteristic bending deformation in which the cross-sectional and lengthwise bending are coupled. The predictions of numerical simulations are validated against desktop experiments and classical theory. The excellent agreement between the simulation and the experiments implies that the hybrid-MPM framework provides a robust computational framework for predicting the large deformation of structures involving complex contact, such as soft robots and deployable structures.

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Reference graph

Works this paper leans on

62 extracted references · 28 canonical work pages

  1. [1]

    Pergamon Press, London (1980)

    Landau, L.D., Lifshitz, E.M.: Theory of Elasticity. Pergamon Press, London (1980)

    1980

  2. [2]

    Cambridge university press, Cam- bridge (1983)

    Calladine, C.R.: Theory of Shell Structures. Cambridge university press, Cam- bridge (1983)

    1983

  3. [3]

    Nature Reviews Physics1, 425–436 (2019) https://doi.org/10.1038/s42254-019-0063-1

    Vella, D.: Buffering by buckling as a route for elastic deformation. Nature Reviews Physics1, 425–436 (2019) https://doi.org/10.1038/s42254-019-0063-1

    work page doi:10.1038/s42254-019-0063-1 2019

  4. [4]

    Current Opin- ion in Colloid & Interface Science40(2019) https://doi.org/10.1016/j.cocis.2019

    Holmes, D.P.: Elasticity and stability of shape-shifting structures. Current Opin- ion in Colloid & Interface Science40(2019) https://doi.org/10.1016/j.cocis.2019. 02.008

    work page doi:10.1016/j.cocis.2019 2019

  5. [5]

    Oxford University Press, Oxford (2010)

    Audoly, B., Pomeau, Y.: Elasticity and Geometry: From Hair Curls to the Non- linear Response of Shells. Oxford University Press, Oxford (2010)

    2010

  6. [6]

    World Scientific, Singapore (2010)

    Baˇ zant, Z.P., Cedolin, L.: Stability of Structures. World Scientific, Singapore (2010). https://doi.org/10.1142/7828

    work page doi:10.1142/7828 2010

  7. [7]

    Standard Test Methods for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials

    ASTM Standard D790-17. Standard Test Methods for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials. ASTM International, West Conshohocken, PA (2017). https://doi.org/10.1520/ D0790-17

    2017

  8. [8]

    Physical Review Letters128, 058101 (2022) https://doi.org/10.1103/ PhysRevLett.128.058101

    Qiu, L., Hutchinson, J.W., Amir, A.: Bending instability of rod-shaped bac- teria. Physical Review Letters128, 058101 (2022) https://doi.org/10.1103/ PhysRevLett.128.058101

    2022

  9. [9]

    Journal of Agricultural Meteorology16, 149–152 (1961) https://doi.org/10.2480/agrmet.16.149

    Tani, N.: The lodging of rice plant caused by wind force. Journal of Agricultural Meteorology16, 149–152 (1961) https://doi.org/10.2480/agrmet.16.149

    work page doi:10.2480/agrmet.16.149 1961

  10. [10]

    Annual Review of Fluid Mechanics44, 453–478 (2011) https://doi.org/ 10.1146/annurev-fluid-120710-101200

    Dumais, J., Forterre, Y.: Vegetable dynamicks: The role of water in plant move- ments. Annual Review of Fluid Mechanics44, 453–478 (2011) https://doi.org/ 10.1146/annurev-fluid-120710-101200

    work page doi:10.1146/annurev-fluid-120710-101200 2011

  11. [11]

    PLoS Computational Biology11(2015) https: //doi.org/10.1371/journal.pcbi.1004037

    Bastien, R., Douady, S., Moulia, B.: A unified model of shoot tropism in plants: Photo-, gravi- and propio-ception. PLoS Computational Biology11(2015) https: //doi.org/10.1371/journal.pcbi.1004037

    work page doi:10.1371/journal.pcbi.1004037 2015

  12. [12]

    Annals of Botany101, 1379–1389 (2008) https://doi.org/10.1093/aob/mcn047

    Grubb, P.J., Jackson, R.V., Barberis, I.M., Bee, J.N., Coomes, D.A., Dominy, N.J., Fuente, M.A.S.D.L., Lucas, P.W., Metcalfe, D.J., Svenning, J.-C., Turner, 21 I.M., Vargas, O.: Monocot leaves are eaten less than dicot leaves in tropical low- land rain forests: Correlations with toughness and leaf presentation. Annals of Botany101, 1379–1389 (2008) https:...

    work page doi:10.1093/aob/mcn047 2008

  13. [13]

    Soft Matter14, 8636–8642 (2018) https: //doi.org/10.1039/C8SM01355A

    Bende, N.P., Yu, T., Corbin, N.A., Dias, M.A., Santangelo, C.D., Hanna, J.A., Hayward, R.C.: Overcurvature induced multistability of linked conical frusta: how a bendy straw holds its shape. Soft Matter14, 8636–8642 (2018) https: //doi.org/10.1039/C8SM01355A

    work page doi:10.1039/c8sm01355a 2018

  14. [14]

    Proceedings of the National Academy of Sciences 114, 5624–5628 (2017) https://doi.org/10.1073/pnas.1620612114

    Saito, K., Nomura, S., Yamamoto, S., Niiyama, R., Okabe, Y.: Investigation of hindwing folding in ladybird beetles by artificial elytron transplantation and microcomputed tomography. Proceedings of the National Academy of Sciences 114, 5624–5628 (2017) https://doi.org/10.1073/pnas.1620612114

    work page doi:10.1073/pnas.1620612114 2017

  15. [15]

    Proceedings of the Royal Society of London

    Seffen, K.A., Pellegrino, S.: Deployment dynamics of tape springs. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences455, 1003–1048 (1999) https://doi.org/10.1098/rspa.1999.0347

    arXiv 1999

  16. [16]

    application to the folding of tape springs

    Guinot, F., Bourgeois, S., Cochelin, B., Blanchard, L.: A planar rod model with flexible thin-walled cross-sections. application to the folding of tape springs. Inter- national Journal of Solids and Structures49, 73–86 (2012) https://doi.org/10. 1016/j.ijsolstr.2011.09.011

    2012

  17. [17]

    EPL (Europhysics Letters)123, 14001 (2018) https://doi.org/10.1209/0295-5075/123/14001

    Matsumoto, D., Sano, T.G., Wada, H.: Pinching an open cylindrical shell: Extended deformation and its persistence. EPL (Europhysics Letters)123, 14001 (2018) https://doi.org/10.1209/0295-5075/123/14001

    work page doi:10.1209/0295-5075/123/14001 2018

  18. [18]

    EPL77 (2007) https://doi.org/10.1209/0295-5075/77/40003

    Mahadevan, L., Vaziri, A., Das, M.: Persistence of a pinch in a pipe. EPL77 (2007) https://doi.org/10.1209/0295-5075/77/40003

    work page doi:10.1209/0295-5075/77/40003 2007

  19. [19]

    Nature599, 229–233 (2021) https://doi.org/10.1038/ s41586-021-04029-6

    Jones, T.J., Jambon-Puillet, E., Marthelot, J., Brun, P.-T.: Bubble cast- ing soft robotics. Nature599, 229–233 (2021) https://doi.org/10.1038/ s41586-021-04029-6

    2021

  20. [20]

    Penguin, London (2009)

    Gordon, J.E.: Structures: or Why Things Don’t Fall Down. Penguin, London (2009)

    2009

  21. [21]

    Proceedings of the Royal Society of London

    Brazier, L.G.: On the flexure of thin cylindrical shells and other ”thin” sections. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character116, 104–114 (1927) https://doi.org/10. 1098/rspa.1927.0125

    arXiv 1927

  22. [22]

    Journal of Applied Mechanics28, 112–116 (1961) https://doi.org/ 10.1115/1.3640420 22

    Seide, P., Weingarten, V.I.: On the buckling of circular cylindrical shells under pure bending. Journal of Applied Mechanics28, 112–116 (1961) https://doi.org/ 10.1115/1.3640420 22

    work page doi:10.1115/1.3640420 1961

  23. [23]

    Physical Review Letters99, 076101 (2007) https://doi.org/10.1103/ PhysRevLett.99.076101

    Ghatak, A., Das, A.L.: Kink instability of a highly deformable elastic cylin- der. Physical Review Letters99, 076101 (2007) https://doi.org/10.1103/ PhysRevLett.99.076101

    2007

  24. [24]

    Journal of the Mechanics and Physics of Solids123, 138–148 (2019) https://doi.org/10

    Seffen, K.A., Wang, B., Guest, S.D.: Folded orthotropic tape-springs. Journal of the Mechanics and Physics of Solids123, 138–148 (2019) https://doi.org/10. 1016/j.jmps.2018.09.017

    2019

  25. [25]

    EPL127(2019) https://doi

    Taffetani, M., Box, F., Neveu, A., Vella, D.: Limitations of curvature-induced rigidity: How a curved strip buckles under gravity. EPL127(2019) https://doi. org/10.1209/0295-5075/127/14001

    work page doi:10.1209/0295-5075/127/14001 2019

  26. [26]

    Thin-Walled Structures124, 538–545 (2018) https://doi.org/10.1016/j.tws.2017.12.033

    Walker, M.G., Seffen, K.A.: On the shape of bistable creased strips. Thin-Walled Structures124, 538–545 (2018) https://doi.org/10.1016/j.tws.2017.12.033

    work page doi:10.1016/j.tws.2017.12.033 2018

  27. [27]

    Theoretical and Applied Mechanics Japan55, 31–40 (2006) https://doi.org/10.11345/nctam.55.31

    TANAKA, M., NOGUCHI, H.: Brazier instability analysis by nonlinear finite element method. Theoretical and Applied Mechanics Japan55, 31–40 (2006) https://doi.org/10.11345/nctam.55.31

    work page doi:10.11345/nctam.55.31 2006

  28. [28]

    Nature521, 467–475 (2015) https://doi.org/10.1038/nature14543

    Rus, D., Tolley, M.T.: Design, fabrication and control of soft robots. Nature521, 467–475 (2015) https://doi.org/10.1038/nature14543

    doi:10.1038/nature14543 2015

  29. [29]

    Proceedings of the National Academy of Sciences of the United States of America112, 10863–10868 (2015) https://doi.org/10.1073/pnas.1504947112

    Overvelde, J.T.B., Kloek, T., D’Haen, J.J.A., Bertoldi, K.: Amplifying the response of soft actuators by harnessing snap-through instabilities. Proceedings of the National Academy of Sciences of the United States of America112, 10863–10868 (2015) https://doi.org/10.1073/pnas.1504947112

    work page doi:10.1073/pnas.1504947112 2015

  30. [30]

    International Journal of Control, Automation and Systems15, 3–15 (2017) https://doi.org/10.1007/s12555-016-0462-3

    Lee, C., Kim, M., Kim, Y.J., Hong, N., Ryu, S., Kim, H.J., Kim, S.: Soft robot review. International Journal of Control, Automation and Systems15, 3–15 (2017) https://doi.org/10.1007/s12555-016-0462-3

    work page doi:10.1007/s12555-016-0462-3 2017

  31. [31]

    Science Robotics5(2020) https://doi.org/10

    Gorissen, B., Melancon, D., Vasios, N., Torbati, M., Bertoldi, K.: Inflatable soft jumper inspired by shell snapping. Science Robotics5(2020) https://doi.org/10. 1126/scirobotics.abb1967

    2020

  32. [32]

    Advanced Intelligent Systems2(2020) https://doi.org/10.1002/aisy

    El-Atab, N., Mishra, R.B., Al-Modaf, F., Joharji, L., Alsharif, A.A., Alamoudi, H., Diaz, M., Qaiser, N., Hussain, M.M.: Soft actuators for soft robotic applications: A review. Advanced Intelligent Systems2(2020) https://doi.org/10.1002/aisy. 202000128

    doi:10.1002/aisy 2020

  33. [33]

    Science Advances8(2022) https:// doi.org/10.1126/sciadv.add3788

    Chi, Y., Hong, Y., Zhao, Y., Li, Y., Yin, J.: Snapping for high-speed and high- efficient butterfly stroke-like soft swimmer. Science Advances8(2022) https:// doi.org/10.1126/sciadv.add3788

    work page doi:10.1126/sciadv.add3788 2022

  34. [34]

    Micromachines13, 651 (2022) https://doi.org/10.3390/mi13050651 23

    Liu, T., Hao, G.: Design of deployable structures by using bistable compliant mechanisms. Micromachines13, 651 (2022) https://doi.org/10.3390/mi13050651 23

    work page doi:10.3390/mi13050651 2022

  35. [35]

    Computer Physics Communications87, 236–252 (1995) https: //doi.org/10.1016/0010-4655(94)00170-7

    Sulsky, D., Zhou, S.-J., Schreyer, H.L.: Application of a particle-in-cell method to solid mechanics. Computer Physics Communications87, 236–252 (1995) https: //doi.org/10.1016/0010-4655(94)00170-7

    doi:10.1016/0010-4655(94)00170-7 1995

  36. [36]

    ACM Trans

    Jiang, C., Schroeder, C., Selle, A., Teran, J., Stomakhin, A.: The affine particle-in- cell method. ACM Trans. Graph.34(4) (2015) https://doi.org/10.1145/2766996

    doi:10.1145/2766996 2015

  37. [37]

    ACM Trans

    Hu, Y., Li, T.-M., Anderson, L., Ragan-Kelley, J., Durand, F.: Taichi: a lan- guage for high-performance computation on spatially sparse data structures. ACM Trans. Graph.38(6) (2019) https://doi.org/10.1145/3355089.3356506

    arXiv 2019

  38. [38]

    Advances in Applied Mechanics, vol

    de Vaucorbeil, A., Nguyen, V.P., Sinaie, S., Wu, J.Y.: Chapter Two - Material point method after 25 years: Theory, implementation, and applications. Advances in Applied Mechanics, vol. 53, pp. 185–398. Elsevier (2020). https://doi.org/10. 1016/bs.aams.2019.11.001

    2020

  39. [39]

    Current Robotics Reports (2023) https://doi.org/10.1007/s43154-023-00105-z

    Qin, L., Peng, H., Huang, X., Liu, M., Huang, W.: Modeling and simulation of dynamics in soft robotics: a review of numerical approaches. Current Robotics Reports (2023) https://doi.org/10.1007/s43154-023-00105-z

    doi:10.1007/s43154-023-00105-z 2023

  40. [40]

    Advanced Robotics Research1(2025) https://doi.org/10.1002/adrr.202500041

    Abe, T., Hashiguchi, I., Nakahara, Y., Kobayashi, S., Tarumi, R., Takahashi, H., Ishigami, G., Sano, T.G.: Snap and jump: How elastic shells pop out. Advanced Robotics Research1(2025) https://doi.org/10.1002/adrr.202500041

    work page doi:10.1002/adrr.202500041 2025

  41. [41]

    Transactions of the JSME (in Japanese)92, 25–00202 (2026) https://doi.org/10.1299/transjsme

    Hashiguchi, I., Yoshimura, K., Kobayashi, S., Tarumi, R.: Numerical analysis of pneumatically driven soft robots with contact using fem-mpm. Transactions of the JSME (in Japanese)92, 25–00202 (2026) https://doi.org/10.1299/transjsme. 25-00202

    work page doi:10.1299/transjsme 2026

  42. [42]

    Earthquake Engineering & Structural Dynamics4(3), 245–249 (1976) https://doi.org/10.1002/eqe.4290040305

    Hinton, E., Rock, T., Zienkiewicz, O.C.: A note on mass lumping and related processes in the finite element method. Earthquake Engineering & Structural Dynamics4(3), 245–249 (1976) https://doi.org/10.1002/eqe.4290040305

    work page doi:10.1002/eqe.4290040305 1976

  43. [43]

    Computer Methods in Applied Mechanics and Engineering 195(44), 5983–5994 (2006) https://doi.org/10.1016/j.cma.2005.10.008

    Wu, S.R.: Lumped mass matrix in explicit finite element method for transient dynamics of elasticity. Computer Methods in Applied Mechanics and Engineering 195(44), 5983–5994 (2006) https://doi.org/10.1016/j.cma.2005.10.008

    work page doi:10.1016/j.cma.2005.10.008 2006

  44. [44]

    ACM Trans

    Hu, Y., Fang, Y., Ge, Z., Qu, Z., Zhu, Y., Pradhana, A., Jiang, C.: A moving least squares material point method with displacement discontinuity and two-way rigid body coupling. ACM Trans. Graph.37(4) (2018) https://doi.org/10.1145/ 3197517.3201293

    arXiv 2018

  45. [45]

    Butterworth-Heinemann, Oxford (2013)

    Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z.: The Finite Element Method: Its Basis and Fundamentals, 7th edn. Butterworth-Heinemann, Oxford (2013)

    2013

  46. [46]

    Extreme Mechanics Letters55, 101788 24 (2022) https://doi.org/10.1016/j.eml.2022.101788

    Sano, T.G., Johanns, P., Grandgeorge, P., Baek, C., Reis, P.M.: Exploring the inner workings of the clove hitch knot. Extreme Mechanics Letters55, 101788 24 (2022) https://doi.org/10.1016/j.eml.2022.101788

    arXiv 2022

  47. [47]

    Journal of Applied Mechanics88(2), 024501 (2021) https://doi.org/10.1115/1.4049023

    Baek, C., Johanns, P., Sano, T.G., Grandgeorge, P., Reis, P.M.: Finite element modeling of tight elastic knots. Journal of Applied Mechanics88(2), 024501 (2021) https://doi.org/10.1115/1.4049023

    work page doi:10.1115/1.4049023 2021

  48. [48]

    ACM SIGGRAPH 2016 Courses (2016) https://doi.org/10.1145/2897826.2927348

    Jiang, C., Schroeder, C., Selle, A., Teran, J., Stomakhin, A.: The material point method for simulating continuum materials. ACM SIGGRAPH 2016 Courses (2016) https://doi.org/10.1145/2897826.2927348

    arXiv 2016

  49. [49]

    Biopolymers34(3), 415 (1994) https://doi.org/10.1002/bip

    Chirico, G., Langowski, J.: Kinetics of dna supercoiling studied by brownian dynamics simulation. Biopolymers34(3), 415 (1994) https://doi.org/10.1002/bip. 360340313

    doi:10.1002/bip 1994

  50. [50]

    In: ACM Siggraph 2008 Papers, pp

    Bergou, M., Wardetzky, M., Robinson, S., Audoly, B., Grinspun, E.: Discrete elastic rods. In: ACM Siggraph 2008 Papers, pp. 1–12 (2008)

    2008

  51. [51]

    Journal of the Mechanics and Physics of Solids160(2022) https://doi.org/10.1016/j.jmps

    Sano, T.G., Pezzulla, M., Reis, P.M.: A Kirchhoff-like theory for hard magnetic rods under geometrically nonlinear deformation in three dimensions. Journal of the Mechanics and Physics of Solids160(2022) https://doi.org/10.1016/j.jmps. 2021.104739

    arXiv 2022

  52. [52]

    ACM Transactions on Graphics (Proc

    Bertails, F., Audoly, B., Cani, M.-P., Querleux, B., Leroy, F., L´ evˆ eque, J.-L.: Super-helices for predicting the dynamics of natural hair. ACM Transactions on Graphics (Proc. ACM SIGGRAPH’06)25, 1180–1187 (2006) https://doi.org/10. 1145/1141911.1142012 . ACM

    Pith/arXiv arXiv 2006

  53. [53]

    Applied Mechanics Reviews, 1–88 (2025)

    Huang, W., Hao, Z., Li, J., Tong, D., Guo, K., Zhang, Y., Gao, H., Hsia, K.J., Liu, M.: A tutorial on simulating nonlinear behaviors of flexible structures with the discrete differential geometry (DDG) method. Applied Mechanics Reviews, 1–88 (2025)

    2025

  54. [54]

    Communications Materials4(1), 59 (2023)

    Sano, T.G., Hohnadel, E., Kawata, T., M´ etivet, T., Bertails-Descoubes, F.: Ran- domly stacked open cylindrical shells as functional mechanical energy absorber. Communications Materials4(1), 59 (2023)

    2023

  55. [55]

    Physical Review E107(2), 025003 (2023)

    Crassous, J.: Discrete-element-method model for frictional fibers. Physical Review E107(2), 025003 (2023)

    2023

  56. [56]

    Geophysical Journal International139(3), 806–822 (1999)

    Komatitsch, D., Tromp, J.: Introduction to the spectral element method for three- dimensional seismic wave propagation. Geophysical Journal International139(3), 806–822 (1999)

    1999

  57. [57]

    formulation

    Oakley, D.R., Knight, N.F.: Adaptive dynamic relaxation algorithm for non- linear hyperelastic structures part i. formulation. Computer Methods in Applied Mechanics and Engineering126(1), 67–89 (1995) https://doi.org/10.1016/ 0045-7825(95)00805-B 25

    1995

  58. [58]

    Computer Methods in Applied Mechanics and Engineering 187(3), 529–541 (2000) https://doi.org/10.1016/S0045-7825(99)00338-2

    Bardenhagen, S.G., Brackbill, J.U., Sulsky, D.: The material-point method for granular materials. Computer Methods in Applied Mechanics and Engineering 187(3), 529–541 (2000) https://doi.org/10.1016/S0045-7825(99)00338-2

    work page doi:10.1016/s0045-7825(99)00338-2 2000

  59. [59]

    International Journal for Numeri- cal Methods in Engineering76(6), 922–948 (2008) https://doi.org/10.1002/nme

    Steffen, M., Kirby, R.M., Berzins, M.: Analysis and reduction of quadrature errors in the material point method (MPM). International Journal for Numeri- cal Methods in Engineering76(6), 922–948 (2008) https://doi.org/10.1002/nme. 2360

    doi:10.1002/nme 2008

  60. [60]

    Yoshida, K., Wada, H.: Mechanics of a snap fit. Phys. Rev. Lett.125, 194301 (2020) https://doi.org/10.1103/PhysRevLett.125.194301

    work page doi:10.1103/physrevlett.125.194301 2020

  61. [61]

    Yoshida, K., Wada, H.: Indentation of an elastic arch on a frictional substrate: Pinning, unfolding, and snapping. Phys. Rev. E109, 045001 (2024) https://doi. org/10.1103/PhysRevE.109.045001

    work page doi:10.1103/physreve.109.045001 2024

  62. [62]

    Matsumoto, Y., Yoshida, K., Sano, T.G.: Sliding of a cylindrical shell into a rigid hole. Phys. Rev. E113, 035501 (2026) https://doi.org/10.1103/hv9t-3h5w 26

    work page doi:10.1103/hv9t-3h5w 2026