On the minimization and maximization of the strain energy density in cortical bone tissue

Universal minimization and maximization of the strain energy density, while possible in materials with cubic symmetry, is not possible for cortical long bone with its orthotropic material symmetry. Illustrating this point, it is shown that the stress state obtained when an axial load is applied along the long axis of a long bone at the midshaft is a minimizer of the strain energy density, while the stress state obtained when a load is applied perpendicular to the long axis, and perpendicular to the surface, of the mid-diaphysis of a long bone is a maximizer of the strain energy density. Thus, the bone tissue at the midshaft of a long bone is designed by nature so that it has the greatest stiffness in the direction of its long axis and its greatest impact loading resistance in the transverse direction.