Answer:
Bone is a live tissue which is responsible for sustaining the human body. It can grow and self-repair. Bones are submitted to the action of the muscles loads and the gravity. Long bones, as femurs, for instance, provide stability and support for a person to remain standing or walking.
Many researches have been done in Biomechanics area. In order to position this paper along with the other bone anisotropy papers, a short overview of the Biomechanical works were provided, freely classifying them in different areas/approaches. Among the papers that deal with the bone anisotropy, there are those that describe the structural bone details. These papers are named here as micro/nano papers, as in (Carnelli et al. 2013) and in (Baumann et al. 2012). Others papers only consider the macroscopic effects and are named here as macro papers, as it is this manuscript. There are papers that use Finite Element software to model bone, named here as numerical papers, as in (Kenedi and Vignoli 2014), in (San Antonio et al. 2012) like this manuscript. Other papers use theoretical/analytical methodologies, as mechanics of solids, theory of elasticity, homogeneization theory and so on. These papers are named here as analytical papers, as in (Toridis 1969) like this manuscript as well. Experimental approaches can be also used, through the utilization of sensors/transducers to measure diverse mechanical characteristics of bones, as for instance, to obtain better elastic material constants to describe such a complex material as bone. These papers are named here as experimental papers, as in (Allena and Clusel 2014). Also there are papers that cover two or more areas; these papers are named here as multi-area papers.
2 MATERIAL ANISOTROPY
Bones, from a macroscopic point of view, can be classified as non-homogeneous, porous and anisotropic tissue, (Doblaré et al. 2004). At a human femur cortical and trabecular bone tissues can coexist, although for the medial cross section analyzed in this work only cortical bone is present. It is very difficult to obtain experimentally bone elastic mechanical properties. Some authors like (Taylor et al. 2002) have obtained orthotropic bone elastic properties indirectly, through the utilization of modal analysis and Finite Element Method approaches. To overcome this difficulty authors like (Jones 1998) and (Krone and Schuster 2006) present different constitutive relationships to model bone behavior, among them, there are three constitutive relationships that are especially important for this work: the isotropic, the transversally isotropic and the orthotropic.
The isotropic materials have only two independent mechanical elastic constants, the Young modulus E and the Poisson ratio ν. The transversally isotropic materials have five independent mechanical elastic constants, two Young modulli, one shear modulus and two Poisson ratios. The orthotropic materials have nine independent mechanical elastic constants, three Young modulli, three shear modulli and three Poisson ratios, (Jones 1998).
These mechanical elastic constants are placed at the stiffness matrixS, which relates stresses and strains. Hooke's law can also be written in a different form using a compliance matrix C as
where ejr are the strain components,Cjrlm are the compliance matrix components and τlm are the stress components. Note thate, C and τ are tensors.
The geometric compatibility and the equilibrium equations are represented, respectively, by equations (2) and (3)
where u are the displacements, x are the coordinates and f are the body forces. Also note that these equations can be expanded according to the coordinate system.
At next section the analytical model is described in details. The principal stresses and principal strains expressions are explicitly presented as well as the correspondent principal angles.