Buckling of adaptive elastic bone-plate: theoretical and numerical investigation

During day-to-day activities, many bones in the axial and appendicular skeleton are subjected to repetitive, cyclic loading that often results directly in an increased risk of bone fracture. In clinical orthopedics, trabecular fatigue fractures are observed as compressive stress fractures in the proximal femur, vertebrae, calcaneus and tibia, that are often preceded by buckling and bending of microstructural elements (Müller et al. in J Biomechanics 31:150 1998; Gibson in J Biomechanics 18:317-328 1985; Gibson and Ashby in Cellular solids 1997; Lotz et al. in Osteoporos Int 5:252-261 1995; Carter and Hayes in Science 194:1174-1176 1976). However, the relative importance of bone density and architecture in the etiology of these fractures are poorly understood and consequently not investigated from a biomechanical point of view. In the present contribution, an attempt is made to formulate a bone-plate buckling theory using Cowin's concepts of adaptive elasticity (Cowin and Hegedus in J Elast 6:313-325 1976; Hegedus and Cowin J Elast 6:337-352 1976). In particular, the buckling problem of a Kirchhoff-Love bone plate is investigated numerically by using the finite difference method and an iterative solving approach (Chen in Comput Methods Appl Mech Eng 167:91-99 1998; Hildebland in Introduction to numerical analysis 1974; Richtmyer and Morton in Difference methods for initial-value problems 1967).