Archwire-mediated torque transfer between adjacent brackets: an analytical model with finite element verification

Torque expression in orthodontic treatment is influenced by bracket play and the mechanical behaviour of the archwire. In a continuous archwire, the free wire segment between adjacent brackets behaves as a torsional spring whose stiffness depends on its length. This study developed a simple analytical framework to predict how interbracket distance and an assumed bracket-play angle together determine the model-predicted moment from a twisted wire segment under an assumed dead-zone model, and stated explicitly which parts of the framework were numerically verified and which were assumed. A one-dimensional lumped-parameter model treated the free wire span between two brackets as a linear torsional spring (Saint-Venant torsion of a rectangular cross-section) in series with an idealized backlash (dead-zone) element representing bracket play. Bracket play was not simulated; its values were taken from previously reported in vitro measurements for 0.019 × 0.025-inch stainless steel (SS) archwire (8.4°–18.7°). Only the elastic spring term was verified numerically, using four three-dimensional finite element analysis (FEA) models of an isolated, straight SS wire segment (0.019 × 0.025-inch; free lengths of 4, 8, 12 and 16 mm) solved in FreeCAD 1.0.2 with the CalculiX solver. Each model was fixed at one end, and a 5° rotation about the wire’s long axis was prescribed at the other end through a rigid-body coupling. No bracket slot, wire–slot contact, friction, ligation, teeth or multi-bracket assembly was modelled. Mesh convergence and peak von Mises stress were assessed. FEA reaction moments exceeded the analytical Saint-Venant predictions by 1.1%–1.9% across the four spans (isolated 0.019 × 0.025-inch wire, generic linear-elastic properties, 5° twist, end warping restrained). At the 5° twist used for the reported results, refining the mesh from 0.15 mm to 0.08 mm changed the reaction moment by less than 0.5% for both the shortest (4 mm) and longest (16 mm) spans. The peak von Mises stress in the 4 mm model at 5° was 1,135 MPa, below the reported yield strength of SS archwire. Combining the verified spring term with the assumed dead zone gives the model-predicted moment T_model(L, θ_applied) = (G·K/L) × (π/180) × max(0, θ_applied − θ_play). The stiffness ratios between short-span and long-span configurations (4/12 mm and 8/16 mm; about 3.0 and 2.0) follow directly from k ∝ 1/L. Predictions at excess twists beyond about 7° (4 mm span) are estimated to exceed the elastic range. Within the assumptions of the model, the torsional stiffness of the interbracket wire segment increases as span length decreases, so that a given activation beyond the assumed bracket-play angle would give a larger model-predicted moment on a shorter span.