A formula for the displacement of an arch wire when subjected to a second-order couple.

A new formula, expressing the local angular displacement of an orthodontic beam when subjected to a second-order couple applied at midspan, has been developed and analyzed. The computed displacements were compared with the results of ex vivo testing. There was good agreement between the results from the theoretical evaluation and the bench testing. Second-order activation of an orthodontic beam can be described in four sequential phases. The initial displacement is influenced by the second-order clearance between bracket-slot and wire as well as the relationship between the bracket-slot width and interbracket distance. During phase II there is a nonlinear relationship between applied couple and rotational displacement. Within phases three and four displacement is linearly related to the interbracket distance, provided the relationship between the bracket-slot width and interbracket distance remains constant. For a given tooth size, the second-order beam stiffness is exponentially related to bracket width. The experiments also show that even small deflections of thin stainless steel wires can lead to second-order couples of large magnitudes when using a clinically relevant interbracket distance. Consequently, it is important that the orthodontist evaluates his or her choice of bracket width and arch wire stiffness in each clinical case in order to avoid supra-physiologic force levels.