Structural-State Closure of the Full Stick–Slip Cycle: Reproducing the Inter-Slip Locked Phase and Delineating Its Boundary from Rate-and-State Friction

Statistical studies of stick–slip (the published PES companion [1]) describe the avalanche endpoint: event-size distributions, precursory signals, and critical slowdown. What remains unresolved is the full time course: during the long inter-slip (locked) loading period, how does the contact network of the frictional interface evolve continuously, and can a closed model reproduce the complete sequence and delimit its boundary against the continuous state-evolution equations of rate-and-state friction (RSF)? This paper develops a structural-state closure in which two unresolved interfacial coordinates carry the dynamics: a sliding-generated structure s (relaxing at a loading-rate-independent time \(τN=1 / b\) ), and a contact maturation m (Dieterich ageing). A high-speed liquid-lubrication floor closes the slip phase and renders the free block–spring selfsustaining. Three results follow. (i) \(T 1\) exact reduction: on the PES-validated speed branch the steady-state of the closure is identical to the published two-component ice-friction law of Pan [1] to a maximum deviation of \(3.0 ×10⁻⁸\) , and the free slider produces repeated stick– slip (153 events; mean recurrence 25.4; mean stress drop 0.302). (ii) Locked-phase two-channel divergence: during loading the structure softens (s relaxes) while maturation hardens (m ages), so the static strength is non-monotone (strength dip 0.071 at \(t ~eq 19.9\) ), with slip nucleating near \(t ~eq 688\) . A single-state RSF law predicts monotone ageing only and cannot reproduce this dip. (iii) Testable predictions: (P2) the non-monotone dip of Dillavou and Rubinstein [5] is reproduced, but the experimental memory scaling \(Tₘᵢₙ ∝ TW\) requires upgrading the structure channel to a power-law relaxation spectrum with logarithmic maturation — then the closure reproduces \(Tₘᵢₙ=0.59 TW\) against the AOI value \(0.30 TW\) ; (P1) the precursory time span shortens with load-point velocity, consistent with the loading-rate series of Wang and Dresen [7]. We delineate the boundary: RSF performs well in monotone-aging and velocity-strengthening regimes, whereas the closure compensates for non-monotone recovery and memory scaling. All model results are reproducible from the enclosed reduced equations; experimental numbers are distinguished from model demonstrations throughout.