The 3D model plots strain from a single closed-form formula evaluated instantly at whatever field is applied. This 2D companion instead simulates a genuine lattice of coupled magnetic domains: each cell in a 34×10 grid carries its own tilt angle θ, its own slightly disordered local anisotropy field Hk (grain-to-grain dispersion), and relaxes toward its energy minimum at a finite rate rather than jumping there instantly — exactly how a real polycrystalline actuator's domain structure behaves.
Each cell minimises the same coherent-rotation energy that produces the 3D model's formula:
E(θ) = −H_k·cosθ − H·sinθ − J·Σ cos(θ − θ_neighbor)
∂E/∂θ = 0 ⇒ tanθ = H/H_k (the 3D model's exact equilibrium)
dθ/dt = −(1/τ)·∂E/∂θ (finite-rate relaxation, τ ≈ 150 ms)
λ(t) = λs · mean(sin²θ) over the lattice
Because relaxation is not instantaneous, the strain lattice cannot keep up with a fast-changing drive field — it lags behind H(t). Plotted against the field, that lag opens a genuine hysteresis loop in the strain-vs-field phase portrait. This is textbook Debye-relaxation behaviour: the phase lag φ = atan(2πfτ) grows monotonically toward 90° as frequency rises, but the strain *amplitude* the lattice can reach shrinks at the same time (it simply runs out of time to respond), so the loop's area — phase lag times shrinking amplitude — actually peaks near f ≈ 1/(2πτ) (≈1 Hz here) and narrows again above it. Sweep the frequency slider slowly and watch the loop open, peak, then close back down — a real signature of finite domain-wall mobility that a single instantaneous formula cannot produce.
- Hbias / Hac — DC operating point and AC swing of the drive field H(t) = Hbias + Hacsin(2πft), identical to the 3D model.
- Drive frequency — sweeps past the domain lattice's own relaxation rate (~1 Hz): the hysteresis loop widens on the way up to that rate and narrows again past it as the lattice's response amplitude collapses.
- Preload / Hk — raises the mean local anisotropy field across the lattice, flattening the strain response the same way it does in the 3D model.
Real-world relevance: real Terfenol-D actuators do show frequency-dependent hysteresis and eddy-current-like lag at high drive rates — this is the microscopic mechanism (finite domain-wall mobility) behind that macroscopic effect.