The elastic anisotropy of bone

In modeling the anisotropic properties of hydroxyapatite (HAp), Katz found that two kinds of phenomenological relationships held among the elastic stiffness coefficients. Firstly, there are three linear combinations--(c11 + c22 + c33), (c44 + c55 + c66), (c12 + c13 + c23)--which arise naturally when computing the isotropic averages of anisotropic crystal systems over all possible spatial orientations. Secondly, the degree of elastic anisotropy in such crystal systems is characterized by two specific factors: (a) the ratio of the linear compressibility along the unique axis to that perpendicular to it, (c11 + c12 - 2c23)(c33 - c13); and (b) the ratio of the two shear moduli, c44/c66. There have been a number of experiments in recent years which have used either mechanical methods or ultrasonic techniques to measure the anisotropic elastic properties of bovine and human cortical bone. Analyses of data from these experiments show that the above relationships also play a significant role in characterizing the elastic anisotropy in bone.