# FEM-Based Compression Fracture Risk Assessment in Osteoporotic Lumbar Vertebra L1
## 3 Numerical Results and Discussion
### Influence of Vertebra Cortical Shell Buckling  
_Mechanical properties._ Different properties are assigned for particular phases. The dense cortical shell phase is characterised by volume density \(\rho_{cor}=1850\,\mathrm{kg/m^{3}}\)[37; 38], while the density of fully degenerated cancellous bone is \(\rho_{can}=100\,\mathrm{kg/m^{3}}\). The cortical phase is modelled as an isotropic elastic-plastic continuum. The elastic properties are defined by elasticity modulus \(E_{cor}=8.0\,\mathrm{GPa}\), and the Poisson's ratio is \(\
u_{cor}=0.3\)[14; 39]. The plastic properties are defined on the basis of perfect plasticity and obey the [PERSON] yield criteria. The yield stress is \(\sigma_{Y}=64\,\mathrm{MPa}\)[14]. This value is further used as the ultimate strength constant. The trabecular phase is modelled as an elastic orthotropic continuum. The elastic modulus of the trabecular bone in the vertical (longitudinal) direction is calculated according to the formula given in Reference [40] as follows:  
\[E_{can,zz}=4.73\rho_{can}^{1.56} \tag{13}\]  
Thereby, the transverse elastic modulus is assumed to be the fraction of the longitudinal modulus, thus  
\[E_{can,xx}=E_{can,yy}=0.1E_{can,zz} \tag{14}\]  
The [PERSON]'s ratio is \(\
u_{can,xy}=0.3\), and \(\
u_{can,xz}=\
u_{can,yz}=0.2\). _Finite element method._ The thin-walled domain of the cortical shell was discretised by shell finite elements. The shell element applied is a four-node element with six degrees of freedom at each node. Such an element is associated with plasticity and larger strain and describes structure buckling. It is suitable for analysing thin to moderately thick shell structures. The finite element mesh of cortical shells contains 3094 nodes and 2976 shell elements. A cancellous bone, endplates, and posterior bone models were meshed with volumetric finite elements. This type of solid element is a higher-order 3-D 20-node solid element that allows quadratic displacement behaviour. The element supports plasticity, large deflection, and large strain capabilities. Finally, the solid phase was described by a 3-D mesh containing 348,138 nodes and 147,814 solid elements. The meshed model is presented in Figure 10a. Shell and volumetric domains may be connected in a different manner, and two computational finite element models were finally generated. The first model reflects a healthy cortex state. A connection between the cortical wall and trabecular phase is implemented as a contact of two solids (Figure 10b). In the perfect case, the connection is modelled as a bonded contact, where no sliding or separation between faces or edges is allowed. For the shell-solid constraint option, an internal set of force-distributed constraints between nodes on the shell edges and nodes on the solid surface is created. In the model, each shell node acts as the master node and associated solid nodes act as slave nodes. In the case of osteopororotic degradation, the bond is weakened. The second model reflects the limit case with a fully degenerated interface. The degradation effect could be evaluated by removing the connecting bond, and the gap was imposed between the shell and the solid (Figure 10c). Therewithin, integrity of the body was held by connecting the edge of the shell by endplates. _Numerical results._ The physical nature of different models is qualitatively illustrated by deformed shapes, and a colour plot of the displacement magnitude of the cortical shell is shown in Figure 11. The displacement values, defined in millimetres, are illustrated in a unified colour scale. The first line of subfigures (Figure 11a-c) illustrates bonded shell-solid contact, while the next line of subfigures (Figure 11d-f) illustrates unbonded contact. The first column (Figure 11a,d) reflects results obtained with the large thickness equal to 0.5 mm, while the third column (Figure 11c,f) reflects results for most degradation of cortical shell. Characterising deformation shapes in a colour scale clearly illustrates the degradation degree.