Analytical modelling of Kirschner wires in Ilizarov circular external fixators using a tensile model

Mechanical characteristics of orthopaedic fixators, such as the stiffness and stability, directly influence the mechanobiological environment in which the bone is healed. In circular external fixators, the transfixing Kirschner wires are the major contributors to the biomechanics involved. A comprehensive understanding of the mechanical behaviour of the wires is therefore the key to biomechanical analysis of the Ilizarov fixator. In this study, to model the behaviour of the wires, a purely theoretical approach has been adopted to obtain explicit equations for a solely tension-based model formulation. Mathematical modelling leads to new algebraic polynomials whose solutions are the exact maximum deflection, angle of deflection, and total (final) tension in the wire. The predictions are compared with published experimental and computational findings on the deflection and stiffness of the wires, and analytical explanations are provided for the previously observed behaviours. Parametric (practical) implications of this type of abstraction include the fact that the angle of deflection as well as the tension in the wire are independent of its length. The inverse proportionality of wire stiffness to its length is also deduced. The findings are applicable to tensile elements (ropes, chains, etc.), provided that the tensile deformation (elongation) can be deemed to be the dominant mode of deformation.