## 2 Materials and Methods  
In this section, the novel analytical solution of a piecewise homogeneous annulus subjected to applied displacement constraints is presented. The analytical solution was used as a simplified model of the behavior of a femur section during the hip prosthesis implant. The hypotheses of the model are plane strain and piecewise constant homogeneous elastic material that simulate heterogeneity of bone tissue. More accurate description is possible using Functional Graded Materials where graded transition can be obtained [25]. To study the interaction between the prosthesis and the bone tissue by means of an analytical approach that overcome the difficulties and the computational effort due to large computational approaches, the femoral diaphysis was modeled as a biphasic hollow cylinder (Figure 1 left). The internal cavity represented the diaphyseal canal, while the central and the external phases the trabecular and cortical bone, respectively. The only geometrical parameter of the model is the radius of the generic annulus, the bone regions were described by the geometric loci whose definition inequalities are:  
\[ \begin{array}{ll}\text{{\it Femoral\;Diaplyseal\;Canal:}}&\text{r}\leq\text{r }_{0},\quad 0\leq\vartheta\leq 2\pi\\ \text{{\it Trabecular\;Bone:}}&\text{r}_{0}<\text{r}\leq\text{r}_{T},\;0\leq \vartheta\leq 2\pi\text{{\it with\;r}}_{T}=\text{r}_{0}+t_{T}\\ \text{{\it Cortical\;Bone:}}&\text{r}_{T}<\text{r}\leq\text{r}_{C},\;0\leq \vartheta\leq 2\pi\text{{\it with\;r}}_{C}=\text{r}_{T}+t_{C}\end{array}  \tag{1}\]where \(r_{0}\) is the internal radius, \(r_{T}\) and \(t_{T}\) are respectively the trabecular radius and the trabecular thickness and \(r_{C}\) and \(t_{C}\) denote the radius and the thickness of the cortical region. The cancellous and the cortical bone tissues were supposed--at the macroscopic scale of interest--as overall isotropic and linear elastic materials [26]. However, the spatially inhomogeneous mechanical properties due to the different levels of bone mineral density were taken into account, averaging the actual stiffness and strength values over proper domains of the idealized hollow bi-phase cylinder, denoting trabecular and cortical bone tissue elastic moduli, densities, yield and ultimate strains, respectively as \(E_{T}\), \(\rho_{T}\), \(\varepsilon_{Y}^{T}\), \(\varepsilon_{ult}^{T}\) and \(E_{C}\), \(\rho_{C}\), \(\varepsilon_{Y}^{C}\), \(\varepsilon_{ult}^{C}\). Poisson coefficients were indicated as \(\
u_{T}\) and \(\
u_{C}\), assuming \(\
u_{T}=\
u_{C}\) due to the small influence of the Poisson coefficient variability to the mechanical response. In the literature one can find values varying at most in the interval [0.3-0.1], but this is often a result of the characteristic size of the specimen used for experimental tests and of the introduction of anisotropic elasticity. In the case in which the isotropy is instead assumed as a result of the small size of the ideal specimen (e.g., the voxel on which the average mineral content measured via Quantitative Computed Tomography (QCT) and the elastic coefficients are so evaluated), the porosity of the trabecular tissue can be considered to be stereological and the eventual inhomogeneity becomes responsible for the spatial variation of the elastic moduli. Under this assumption (isotropy at the scale of the voxel), in the transition from dense (cortical) to porous (trabecular/spongy) bone, the Poisson ratio can be considered to be constant, as the theory of homogenization and standard solutions in micromechanics ensure [27; 28; 29]. However, we performed some sensitivity analyses and verified that variabilities of the Poisson coefficient in the above-mentioned range does not affect the results significantly, as it is easy to verify. In the case of THA without the use of the cement, in order to reach the press-fit of the prosthesis into the femur, the stem component volume results generally 10% greater than the volume prepared by the surgeon in the diaphysis canal.