## 2 Materials and Methods
For this reason, due to the insertion of the prosthesis, the stem component is forced to occupy the undersized surgically prepared volume into the femoral  
Figure 1: Three dimensional model of the femoral diaphysis in cylindrical coordinates system (**left**). Cross-section of the model (**right**). diaphysis, by pressing on the surrounding bone tissue by a prescribed radial displacement, namely \(u_{0}\) (Figure 1 right). The value of prescribed displacement is derived by starting from the slight volume discrepancy due to the fact the surgical procedure suggest inserting a prosthesis whose volume is slightly greater than the volume of the diaphyseal canal to so guarantee the best fit and at early stage stability of the implant. Under the hypotheses of axial symmetry as well as in real geometry of the problem this volume difference is then reflected in a prescribed displacement, due to the pushing of the prosthesis toward the innermost walls of the diaphyseal canal [18]. With the aim of simulating this scenario, a linear elastic analysis was carried out in the hypothesis of small displacements through an analytical procedure implemented in the software Mathematica(r) (Wolfram Research, Champaign, IL, USA, 2019). By using a cylindrical reference system \((r,\delta,z)\), the kinematics of the problem was described by means of the following displacement field  
\[\mathbf{u}^{k}=(u^{k}_{r},u^{k}_{\delta},u^{k}_{z}) \tag{2}\]  
with  
\[ \begin{array}{l}u^{k}_{r}=u^{k}(r)\\ u^{k}_{\delta}=0\\ u^{k}_{z}=\varepsilon_{0}z\end{array}  \tag{3}\]  
\(u^{k}_{r},u^{k}_{\delta},u^{k}_{z}\) denoting respectively the radial, circumferential and longitudinal components of the displacement, and the apex \(k=\{T,C\}\) referring to the bone tissue phase type (\(T\): trabecular, \(C\): cortical). The strain tensor was expressed in the Voigt's notation  
\[\epsilon^{k}=(\epsilon^{k}_{rr},\epsilon^{k}_{\delta\delta},\epsilon^{k}_{zz},\epsilon^{k}_{gz},\epsilon^{k}_{rz},\epsilon^{k}_{r\delta})=\left(\frac{du^{k }_{r}}{dr},\frac{u^{k}_{r}}{r},\epsilon_{0},0,0,0\right) \tag{4}\]  
being \(\epsilon^{k}_{rr}=\frac{du^{k}_{r}}{dr}\) the radial strain, \(\epsilon^{k}_{\delta\delta}=\frac{u^{k}_{r}}{r}\) the circumferential strain and \(\epsilon^{k}_{zz}=\varepsilon_{0}\) the longitudinal strain. In particular, the axial strain, \(\varepsilon_{0}\), is considered constant and is calculated by means of equilibrium along the axis of the bone. At any point of the bone, it was assumed the linear elastic constitutive equation was valid and the cylindrical symmetry of the structure allows reducing the equilibrium equations to the sole one in the radial direction involving the radial and the circumferential stresses \(\sigma^{k}_{rr}\) and \(\sigma^{k}_{\delta\delta}\) in each material phases  
\[\frac{d\sigma^{k}_{rr}}{dr}+\frac{1}{r}(\sigma^{k}_{rr}-\sigma^{k}_{\delta \delta})=0 \tag{5}\]  
By considering the stress-strain constitutive equation and the kinematical relationships recalled above, the well-known equilibrium equation in term of displacements can be written down, whose solution has the form  
\[u^{k}_{r}=C^{k}_{1}r+C^{k}_{2}r^{-1} \tag{6}\]  
The integration constants \(C^{k}_{1}\) and \(C^{k}_{2}\) were determined by imposing the boundary and interface conditions, namely  
* Boundary conditions \[ \begin{array}{l}r=r_{0}\ :u^{T}_{r}(r_{0})=u_{0}\\ r=r_{C}\ :\sigma^{C}_{rr}(r_{C})=0\end{array} \] (7)
* Interface conditions \[ \begin{array}{l}r=r_{T}:\sigma^{T}_{rr}(r_{T})=\sigma^{C}_{rr}(r_{T})\\ u^{T}_{r}(r_{T})=u^{T}_{r}(r_{T})\end{array}