Finite-element workflow · reduced mechanics
Separate what is publicly executable from what still needs solver exports: inspect the open geometry/YACS workflow, then calculate an independent undeformed elastic cell using the paper’s published homogenization equations.
Calculated reduced cell · not FEM
Evaluate the initial, undeformed sinusoidal flute with the compliance rotation and A/D integration published in equations 1–11. Every geometry and material change recomputes the cell in the browser.
Input provenance. E₁=3509.2 MPa, E₂=1330.4 MPa and E₃=0.0509 MPa are the paper’s Table 2 FCT estimates. Geometry, shear moduli (G₁₂=600 MPa; G₁₃=G₂₃=5 MPa) and Poisson ratios (0.3) are explicit demonstration assumptions because the paper does not provide a reusable calibrated input package. They are not claimed as experimental values.
Effective thickness
4.2655 mm
equation 10
Effective modulus
4.0165 MPa
equation 11 · reduced cell
Flute take-up factor
1.3902
arc length / pitch
Maximum local angle
54.73°
|atan(dH/dx)|
| row | 1 | 2 | 3 |
|---|---|---|---|
| 1 | 1.465e+1 | 1.679e+0 | 0.000e+0 |
| 2 | 1.679e+0 | 3.702e+2 | 0.000e+0 |
| 3 | 0.000e+0 | 0.000e+0 | 9.447e+0 |
| row | 1 | 2 | 3 |
|---|---|---|---|
| 1 | 4.735e+1 | 5.419e+0 | 0.000e+0 |
| 2 | 5.419e+0 | 5.245e+2 | 0.000e+0 |
| 3 | 0.000e+0 | 0.000e+0 | 2.598e+1 |
The implementation builds the paper’s 6×6 compliance matrix, rotates it about the global y-axis, inverts the reduced 3×3 block, and integrates A and D over one period with 512 trapezoidal intervals. Units remain MPa and mm throughout.
Validator acceptance: a flat cell must recover the input paper thickness; 512 intervals must agree with 4096 intervals to within 0.05% for both effective quantities; matrices must remain symmetric; increasing flute height must increase the take-up factor.
Boundary: this is an independent calculator for the undeformed elastic cell. It is not a solver execution, FEM mesh, FCT replay, calibrated compression curve, contact model, Weibull softening model or reproduction of the paper’s validation specimens.
Compression mechanics · complete public-source audit
The CC BY 4.0 paper links SCT, ECT and FCT experiments to inverse calibration, deformed-cell homogenization, Weibull softening and a final single-solid FEM proof of concept. The calculator above executes only its undeformed elastic equations 1–11; the remaining stages below are cited evidence, not browser-generated results.
Coverage state
1 executed subset · 3 source-only stages
No copied figures · no inferred curves
Equations 1–11 transform an orthotropic compliance matrix over a sinusoidal flute and integrate A/D to estimate effective thickness and through-thickness modulus.
ProjectHub: The browser calculator executes this undeformed elastic subset with explicit geometry, material inputs and units.
Replay gap: Deformed overlapping functions Hᵢ(x,t), arc-length optimization inputs and the full compression sequence required by equations 12–14.
SCT strength and ECT response calibrate paper E_MD and E_CD using a buckling analysis and Tsai-Wu criterion.
ProjectHub: Eight published material/flute summaries are shown as cited evidence; no ECT solver is executed.
Replay gap: Raw SCT/ECT observations, specimen dimensions and conditioning, Tsai-Wu strengths, buckling model, optimizer settings and per-run predictions.
FCT response calibrates E_ZD and feeds the homogenized effective stiffness and thickness; the paper reports one analyzed sample.
ProjectHub: The four reported modulus values are cited below; the experimental load–deformation curve is not reconstructed.
Replay gap: Machine-readable load–deformation samples, geometry, force/displacement units and boundary conditions, optimizer objective/tolerance and correction-factor provenance.
Two Weibull distributions weight the homogenized global response for early contact degradation and stochastic flute buckling; a second specimen tests the fitted parameters before a single-solid linear-elastic FEM proof of concept.
ProjectHub: The two parameter sets and statistical comparisons are cited; no curve, field or solver replay is presented.
Replay gap: Both raw compression histories, the exact softening composition and normalization, specimen geometry, load/support steps, custom-material input, mesh, solver/version and nodal/element outputs.
| Material / flute | E_MD | E_CD |
|---|---|---|
| Mixed B-flute | 1.68 ± 0.54 | 1.10 ± 0.37 |
| Mixed C-flute | 1.69 ± 0.76 | 1.15 ± 0.57 |
| Mixed E-flute | 1.46 ± 0.35 | 1.14 ± 0.23 |
| Recycled B-flute | 1.59 ± 0.85 | 1.12 ± 0.36 |
| Recycled C-flute | 1.47 ± 0.53 | 1.20 ± 0.60 |
| Recycled E-flute | 1.77 ± 0.58 | 1.13 ± 0.62 |
| Non-recycled B-flute | 1.75 ± 0.44 | 1.08 ± 0.21 |
| Non-recycled C-flute | 1.89 ± 0.90 | 1.11 ± 0.51 |
| Variable | Value |
|---|---|
| Experimental effective E | 5.588 |
| Optimized E_MD,FCT | 3509.2 |
| Optimized E_CD,FCT | 1330.4 |
| Optimized E_Z,FCT | 0.0509 |
| Specimen | Contact β / η | Buckling β / η |
|---|---|---|
| Calibration E1 | 0.14 / 4.82e-7 | 1.31 / 0.96 |
| Independent E2 | 0.11 / 4.22e-9 | 1.22 / 0.95 |
| Test | Contact p | Buckling p |
|---|---|---|
| Likelihood-ratio | 1.00 | 1.00 |
| Kolmogorov–Smirnov | <0.05 | 0.76 |
| Bootstrap | 0.74 | 0.53 |
Public implementation boundary: The public repository supplies geometry generation and YACS workflow definitions. It does not supply a portable mesh, solver result archive, the paper's experimental tables as machine-readable data, a compression history or the final proof-of-concept FEM field export.
Decision: Keep the live undeformed homogenization calculator as a separate reduced model. Add a replay only after the complete experimental histories, deformed geometry contract and solver/export provenance are public and reusable.
Ricardo Fitas, Heinz Joachim Schaffrath and Samuel Schabel. arXiv:2507.02189v1, submitted 2 July 2025. CC BY 4.0. Repository commit 6264c13dd01c. Verified 2026-10-05.
Flat-crush FEM · paper and public-code audit
The CC BY 4.0 study documents a two-dimensional Salome-Meca/Code_Aster workflow for three corrugated cells. The public MIT repository contains real geometry, mesh, contact and nonlinear-solver orchestration; it does not archive the generated fields needed to replay the paper in this browser.
Evidence state
3 cases · 5 reported displacement states
Source-audited · not browser-executed FEM
STEP 1
Python/NURBS construction of fluting, liners and rigid compression blocks, followed by Salome mesh generation and MED export.
STEP 2
Orthotropic paper layers and an isotropic aluminium compression block use the published elastic constants listed below.
STEP 3
Penalty contact, fixed lower block and imposed vertical displacement of the upper block are assembled for a nonlinear Code_Aster solve.
STEP 4
The workflow requests displacement and stress/contact-pressure fields and contains MED/result export plus ParaView post-processing steps.
| Case | Wavelength | Amplitude | Paper thickness | Reported displacement states |
|---|---|---|---|---|
| A / Geometry 1 | 10 mm | 3 mm | 0.2 mm | 0.625, 1.25, 2.5 mm |
| C / Geometry 2 | 8 mm | 2.5 mm | 0.1 mm | 1.25 mm |
| E / Geometry 3 | 4 mm | 1.5 mm | 0.1 mm | 0.6 mm |
| Symbol | Value | Unit |
|---|---|---|
| E11 | 1.7090e+9 | Pa |
| E22 | 9.1800e+8 | Pa |
| E33 | 8.9947e+6 | Pa |
| G12 | 4.8473e+8 | Pa |
| G13 | 5.0000e+8 | Pa |
| G23 | 2.6229e+7 | Pa |
| nu12 | 0.3998 | — |
| nu13 | 0.001 | — |
| nu23 | 0.001 | — |
Aluminium block: E = 1.250e+11 Pa; ν = 0.25. No paper figure or field was copied.
general_geometry.pySalome 2022 script for the two-dimensional NURBS geometry.blob 393d1eb101caYACS Schemes/Board simulation 2.xmlYACS study loop, geometry/mesh preparation and simulation orchestration.blob 4ebc4e6c1903YACS Schemes/Corrugated_Board_FEA_Simulation.xmlCode_Aster contact workflow with displacement loading, nonlinear solve and MED/result export commands.blob 52512890d9d7Available: Public geometry and YACS source, solver commands, material constants, three geometric cases and paper-reported output states.
Missing for replay: Committed generated MED mesh, result MED files, machine-readable displacement/pressure fields, load-deflection histories, exact solver build/runtime and regression reference outputs.
Decision: Do not interpolate paper figures or call the browser homogenization model a flat-crush FEM replay. Add an interactive replay only after exact reusable exports and solver provenance are public.
Symmetry 2025, 17(2), 257; peer-reviewed version of Preprints 2024, 202412.2302.v1. CC BY 4.0. Repository commit 6264c13dd01c. Verified 2026-10-05.
The CC BY 4.0 paper publishes the initial sinusoidal geometry, compliance transformation, A/D integration and effective-property equations. This page implements only that auditable elastic subset. The public MIT repository remains the source for geometry generation and YACS orchestration.