Gel–Sol Elastomer Bench

Elastomer sheet 64 × 64 cells · 20 mm sheet · electroded where painted · clamped frame

Paint

Building sheet model…

DOF — Ritz vectors — Peak |u| — Drawn ×1

Region	Area strain at f	at 2f	Lag at f	at 2f
Gel	—	—	—	—
Viscoelastic	—	—	—	—
Sol	—	—	—	—

Voltage V(t) = V_DC + V_AC sin 2πft; Maxwell stress c·V² (c = ε₀ε_r/d²) has a mean, a part at f (only if V_DC ≠ 0) and a part at 2f.
Both oscillating parts drive the sheet; the mean prestrain is not drawn.
All three material laws are linear viscoelastic, so on their own they only answer at f and 2f. Higher harmonics come from two sheet nonlinearities on gel cells: the film thins as it stretches (Maxwell stress ∝ V²J²) and neo-Hookean stiffening. A harmonic-balance solve finds the periodic state.
Each cell uses its own viscoelastic G*(ω, T): gel = fractional Kelvin–Voigt, viscoelastic fluid = Winter–Chambon S(iω)ⁿ (colour runs gel → sol as n goes 0 → 1), sol = Newtonian dashpot.
Inertia ρ = 1000 kg/m³ gives the sheet damped in-plane modes; drive 2f near a peak in the sheet spectrum to excite one.
Repainting or changing the grid re-bakes a reduced model (168 Ritz vectors); sliders then update instantly.
Brightness = instantaneous area strain (lighter = stretched, darker = compressed).
Drawn true to scale for small motion; beyond that the drawn stretch grows logarithmically and is capped so no cell folds. The table shows true values.
Sol enclosed by gel follows the gel; sol at a free edge flows ~100× more and sets the drawing scale.
Blank cells are unelectroded passive membrane (soft limit): they carry no stress but stretch with the painted cells out to the clamped frame.
Slowmo runs at the real drive frequency × the playback factor (Canvas group), so motion speeds up with f; “Cap 10 Hz” limits the on-screen rate.
Sound (top bar) plays the harmonic lines of the driven spectrum. Raise f to ≳ 30 Hz to hear it; at the default 1 Hz the tones are below hearing.

Gel · fractional Kelvin–Voigt G*(ω) = G_e(iω)^α + V(iω)^β

G′(f) — G″(f) — tan δ f / 2f — γ at f — γ at 2f — Loss per voltage cycle —

Viscoelastic fluid · Winter–Chambon G*(ω) = S(iω)ⁿ · n = 0 spring (gel) → n = 1 dashpot (sol)

δ = nπ/2 (all f) — G′(f) — G″(f) — γ at f — γ at 2f — Loss per voltage cycle —

Sol · Newtonian dashpot G*(ω) = iωη

η(T) = η·a_T — G′(f) — G″(f) — tan δ f / 2f — γ at f — γ at 2f — Loss per voltage cycle —

Sheet spectrum · driven response integrated over the grid harmonic balance, h·f for h = 1…12 · u_rms = √(Σ m_i|U_i|² / Σ m_i)

u_rms at f — u_rms at 2f — 3f…12f — Nonlinear solve — Sound off

Resonance peaks —