# Analysis of Damage Models for Cortical Bone
## 3 Numerical Results and Discussion  
### Comparison in a One-Dimensional Problem  
To compare the performance of the models presented in the previous section, a one-dimensional version of both was implemented. To test the models, a typical tensile test was reproduced, since there is experimental data in the literature to evaluate the performance [18]. The one-dimensional model represents a bone fixed on one end and with an increasing displacement imposed on the other end. Stress was computed in postprocessing once the deformation was obtained.  
First, we solve the Fremond-Nedjar model with the following data:  
\[\begin{array}{ll}T=2,\quad\Omega=(0,1),\quad\Gamma_{D}=\{0,1\},\quad\Gamma_ {N}=\varnothing,\quad\boldsymbol{f}_{0}=\boldsymbol{0},\\ \boldsymbol{g}=\boldsymbol{0},\quad\lambda=0,\quad\mu=1.348\times 10^{10}, \quad\kappa^{*}=0.5,\quad\beta_{*}=0.01,\\ q^{*}=10^{5},\quad\lambda_{d}=17,\quad\lambda_{u}=4.2\times 10^{5},\quad \lambda_{w}=0,\end{array}\]  
and the initial condition:  
\[\beta_{0}(x)=1\quad\mbox{for all}\quad x\in(0,1).\]  
\begin{table}
\begin{tabular}{l c c c c} \hline \(h\downarrow k\to\) & **0.1** & **0.01** & **0.001** & **0.0001** \\ \hline \(2^{-2}\) & 13.5490 & 13.5613 & 13.561268 & 13.561268 \\ \(2^{-3}\) & 6.6596 & 6.6679 & 6.667863 & 6.667863 \\ \(2^{-4}\) & 3.3080 & 3.3054 & 3.305386 & 3.305386 \\ \(2^{-5}\) & 1.6641 & 1.6457 & 1.645661 & 1.645661 \\ \(2^{-6}\) & 0.8460 & 0.8210 & 0.821036 & 0.821036 \\ \(2^{-7}\) & 0.4370 & 0.4099 & 0.409942 & 0.409942 \\ \(2^{-8}\) & 0.2328 & 0.2046 & 0.204565 & 0.204565 \\ \(2^{-9}\) & 0.1315 & 0.1017 & 0.101652 & 0.101653 \\ \(2^{-10}\) & 0.0822 & 0.0496 & 0.049597 & 0.049596 \\ \hline \end{tabular}
\end{table}
Table 1: Numerical errors (\(\times 100\)) for some discretization parameters.  
That is, at initial time we assumed that the bone was completely healed. Moreover, since no mechanical forces were applied, the deformation is produced defining an imposed displacement at the right corner as,  
\[u(1,t)=5\times 10^{-3}\cdot t\quad\text{for all}\quad t\in[0,2].\]  
We note that the modifications needed to include this case into problem \(P\) were really straightforward, so the analysis performed in the previous section could be extended easily.  
Using the discretization parameters \(k=0.01\) and \(h=0.05\) in Figure 2 (up) the stress-strain curve obtained with the Fremond-Nedjar model is shown. As it can be seen in the evolution of the damage variable (down), the mechanical properties of the bone degrade as the strain increases, leading to the \"plastic\" region which corresponds with the damage regime of the stress-strain curve.  
Figure 2: Stress–strain curve obtained with the Fremond–Nedjar model in a tensile test (**up**) and evolution of the damage variable in the tensile test (**down**).  
Secondly, the [PERSON] et al. model was solved, so in Figure 3 the shape of the stress-strain curve seen before is reproduced again with the rheological model.  
Finally, both models were compared with experimental data obtained from [18]. Figure 4 shows the stress-strain curves obtained from both models against the experimental data (black dots). Both models show good agreement with the experimental results, with small differences in the linear part of the curve and the beginning of the damage range. The Fremond-Nedjar model shows better agreement in the linear part, but the curvature of the damage range fits worse than the rheological model. However, these small differences were not significant, since mechanical properties vary greatly from bone to bone, making very accurate fittings not useful.  
Figure 4: Comparative of both models with experimental data. Source of the experimental data: [18].  
Figure 3: Stress–strain curve obtained with the [PERSON] et al. model in a tensile test.  
In any case, both models were capable of reproducing the qualitative characteristics of bone mechanics. In particular, the modified Fremond-Nedjar model shows its capability to be used to model bone tissue in spite of it being proposed to model concrete structures.