Wolff Remodelling as a Jamming Yield Threshold
Bone remodelling is a jamming yield transition. Bone density is the R19 state, mechanical load is the drive, and the spinodal s*=0.5324 IS the yield threshold. Sustained supra-threshold load flips density to the dense basin (jump 0.82); the load loop shows hysteresis (area 0.99) — bone memory. Threshold and memory are FORCED by the bistable substrate [F].
A quasi-static load sweep crosses to the dense basin only above the spinodal (crossing load 0.54 > 0): half-threshold load leaves density at 0.06 while 1.5× threshold drives it to 1. The up and down branches enclose a hysteresis loop (area 0.99), the mechanical memory behind Wolff's law. Setpoint cited (Frost mechanostat) [L]; absolute density [O].
Density as a bistable state
Wolff's law — bone adapts its density to the loads it bears — is, in the jamming framework, a yield/unjamming transition. Bone density is the settled R19 state: a resorbed (low) basin and a dense (high) basin separated by a barrier. Mechanical load is the drive h that tilts the double well.
The spinodal s* = 2(γ/3)3/2 = 0.5324 (with the bone master RUNX2, γ=1.2414) is exactly the yield threshold: below it the resorbed basin survives; above it that basin vanishes and density jumps to the dense state. The transition is near-discontinuous, with a density jump of 0.82 at the crossing.
Hysteresis = bone memory
Loading then unloading does not retrace the same path: the up and down branches enclose a hysteresis loop of area 0.99. The dense state, once formed, persists as the load is removed — the mechanical memory that lets trained bone stay strong. Half-threshold load leaves density at 0.06 (no spurious formation); 1.5× threshold drives it to full mineralization (1).
Grades
The existence of a threshold, the near-discontinuous jump and the hysteresis loop are FORCED by the bistable substrate [F]. The numerical setpoint is anchored to the Frost mechanostat [L]. Absolute bone mineral density in g·cm⁻³ is not fixed by the substrate and is graded [O].