2D Honeycomb Cell Mechanics: Bending versus Stretching, Stiffness-to-Weight and Wall Buckling

Presets

Live readouts

Graphs

Stress against strain of the selected lattice in its plane (elastic line, buckling or yield plateau, densification)

Modulus against relative density (log-log): stretching-dominated lattices have slope 1, the hexagonal honeycomb slope 3

Strength against wall thickness ratio t/l: wall buckling, yield, in-plane buckling and plastic collapse

Sandwich panel: how many times stiffer than a solid plate of equal mass, against core height

Numerical checks

running...

How to use

Pick a preset (aerospace aluminium core, natural beeswax comb, plastic or paper cores) and press play: the lattice is compressed and released and every wall is coloured by its bending moment. Try: (1) switch the cell shape from hexagonal to triangular at the same t/l and see how much stiffer the stretching lattice is; (2) lower t/l to 0.01 and watch the strength graph, where the buckling line drops below the yield line; (3) raise t/l to 0.2: the yield line takes over; (4) raise the strain slider past the buckling strain and read the warning; (5) in the sandwich graph move the core height and watch the stiffness advantage over an equal-mass solid plate. Space pauses, R restarts. The white dots in all graphs mark the current design.

The science

A honeycomb is a cellular solid: a thin-walled lattice that is mostly air. Two numbers describe it: the wall thickness ratio τ = t/l (wall thickness over cell edge length) and the relative density, the fraction of the volume that is material. For a regular hexagonal cell with mitred walls the area of the wall ring divided by the area of the cell (measured on the wall centrelines) is exactly

ρ*/ρ_s = 1 − (1 − τ/√3)² ≈ 2τ/√3 (thin walls)

Why hexagons: wall per area. Among regular tilings of the plane, cells of equal area need a total wall length of 2.28 √A (triangles), 2.00 √A (squares) or 1.86 √A (hexagons) per unit area. Hexagons use the least wall, which is the “honeycomb conjecture” (proved by Hales, 2001) and the reason bees can save wax. The page checks this number from the lattice geometry.

Bending versus stretching. When a lattice is loaded in its plane, its joints decide how it deforms. Triangular cells are rigid: every wall is simply stretched or compressed, and E*/E_s = ρ*/(3ρ_s) (pin-jointed limit). Square cells loaded along their walls also stretch: E*/E_s = ρ*/(2ρ_s). A hexagonal cell is a mechanism that is only held by the joint stiffness, so the inclined walls bend like beams, and the stiffness drops with the cube of the wall thickness:

E*/E_s = (4/√3) (t/l)³ ≈ 2.3 (t/l)³ = 1.5 (ρ*/ρ_s)³, ν* → 1

The in-plane graph of the 2D page plots exactly this: slope 1 for stretching, slope 3 for bending. The hexagon is far softer in its plane at the same density, but it is very light and it can deform without breaking, which is why it is also used as a crushable energy absorber. This page does not use the formula alone: it assembles the periodic beam-frame finite-element model of the lattice unit cell (Euler-Bernoulli beams, three degrees of freedom per node, periodic fluctuation fields plus free macro strains so that the load is uniaxial stress) and solves it live. The dots in the first graph are that solution.

Out of plane. Loaded along the cell axis the walls are simply compressed, so the core behaves like a solid reduced by the density: E₃ = (ρ*/ρ_s) E_s and, for a regular hexagon, the shear modulus G₁₃* ≥ 0.577 G_s (t/l) (a lower bound, Gibson and Ashby). The wall is a thin plate of width l; it buckles elastically at

σ_wall = k π² E_s / (12 (1 − ν_s²)) · (t/l)², σ_core = (ρ*/ρ_s) σ_wall vs. σ_yield = (ρ*/ρ_s) σ_ys

where k = 4 for hinged edges and 6.97 for clamped edges (the slider). The smaller of the two is the flatwise strength: thin walls buckle, thick walls crush. In the plane a regular honeycomb buckles elastically at σ_el* ≈ 0.22 E_s (t/l)³ or collapses plastically at σ_pl* ≈ 0.5 σ_ys (t/l)², and densifies at a strain of about 1 − 1.4 ρ*/ρ_s (Gibson and Ashby, Cellular Solids).

Sandwich panels. Two thin face sheets with a honeycomb core between them put the material far from the neutral axis. In three-point bending with span L, load F and width b:

D ≈ E_f b t_f (h + t_f)² / 2, S = G₁₃ b (h + t_f)² / h, δ = F L³ / (48 D) + F L / (4 S)

The page compares the panel with a solid plate of the same mass, whose stiffness D = E b t³/12 grows with the cube of its (small) thickness. Doubling the core height roughly quadruples the panel stiffness at almost no extra mass: that is stiffness-to-weight scaling.

Honest simplifications. (1) The finite-element solution is linear and small-strain: the picture exaggerates displacements by the slider factor, and above the buckling strain it no longer predicts the real collapse. (2) Beams are Euler-Bernoulli (no shear deformation) with rigid joints and ignore the double wall thickness of bonded foil cells; the exact coefficient of the beam formula (4/√3 = 2.31) is within a few percent of the FE result for t/l < 0.1 and falls below it for thick walls. (3) The buckling stress k is an edge-restraint estimate, not a computed eigenvalue; real cores also have imperfections that lower the peak. (4) The post-buckling plateau, the fold sawtooth and the fold wavelength (∝ (t l²)^1/3, coefficient empirical) of the 3D crush test are empirical shapes, not a solution of the folding mechanics. (5) Material values are representative (beeswax is an order of magnitude). (6) The sandwich estimate neglects face wrinkling, core indentation and the contribution of the core to bending. (7) Bonding, adhesives, node double walls and cell expansion are not modelled.

Checks run in the page. The FE modulus is compared with the beam formula at t/l = 0.02, the FE moduli of the square and triangular lattices with the exact stretching values 1/2 and 1/3, the isotropy of the hexagonal result, the density formula with the shoelace area of the mitred ring, the wall-per-area value with the closed form, and the sandwich deflection with a numerical double integration.