# Analysis of Damage Models for Cortical Bone
## 2 Damage Models
### Fremond-Nedjar Model
\begin{array}{l}c(\beta_{n};\mathbf{u}_{n},\mathbf{u}_{n}-\mathbf{u}_{n}^{hk})-c(\beta_ {n}^{hk};\mathbf{u}_{n}^{hk},\mathbf{u}_{n}-\mathbf{u}_{n}^{hk})=c(\beta_{n}^{hk};\mathbf{u}_{ n}-\mathbf{u}_{n}^{hk},\mathbf{u}_{n}-\mathbf{u}_{n}^{hk})+c(\beta_{n}-\beta_{n}^{hk};\mathbf{u}_{ n},\mathbf{u}_{n}-\mathbf{u}_{n}^{hk}),\\ c(\beta_{n}^{hk};\mathbf{u}_{n}-\mathbf{u}_{n}^{hk},\mathbf{u}_{n}-\mathbf{u}_{n}^{hk})\geq C\| \mathbf{u}_{n}-\mathbf{u}_{n}^{hk}\|_{V}^{2},\end{array} \]  
using the fact that \(\beta_{n}^{hk}\in K^{h}\) (and so, \(\beta_{n}^{hk}\geq\beta_{*}\)) and the regularity \(\mathbf{u}\in\mathcal{C}([0,T];[H^{2}(\Omega)]^{d})\) we have  
\[\|\mathbf{u}_{n}-\mathbf{u}_{n}^{hk}\|_{V}^{2}\leq C\Big{(}\|\beta_{n}-\beta_{n}^{hk} \|_{Y}^{2}+\|\mathbf{u}_{n}-\mathbf{v}^{h}\|_{V}^{2}\Big{)}\quad\forall\mathbf{v}^{h}\in V ^{h}.\]  
Therefore, combining the previous estimates of both damage and displacement fields we conclude that  
\[ \begin{array}{l}\|\beta_{n}-\beta_{n}^{hk}\|_{Y}^{2}+\|\mathbf{u}_{n}-\mathbf{u}_{n}^ {hk}\|_{V}^{2}\leq C\Big{(}\|\beta_{0}-\beta_{0}^{h}\|_{Y}^{2}+\|\beta_{1}- \zeta_{1}^{h}\|_{Y}^{2}+\|\beta_{n}-\zeta_{n}^{h}\|_{Y}^{2}+k\sum_{j=1}^{n}\| \beta_{j-1}-\beta_{j-1}^{hk}\|_{Y}^{2}\\ \quad+k\sum_{j=1}^{n}\|\mathbf{u}_{j}-\mathbf{u}_{j}^{hk}\|_{V}^{2}+k\sum_{j=1}^{n}\| \beta_{j}-\zeta_{j}^{h}\|_{H^{1}(\Omega)}^{2}+k\sum_{j=1}^{n}\|\delta\beta_{j}- \dot{\beta}_{j}\|_{Y}^{2}+\|\mathbf{u}_{n}-\mathbf{v}^{h}\|_{V}^{2}\\ \quad+k\sum_{j=1}^{n}\|\phi(\mathbf{\varepsilon}(\mathbf{u}_{j}),\beta_{j})-\delta \beta_{j}+\kappa^{*}\Delta\beta_{j}\|_{Y}\ \|\beta_{j}-\zeta_{j}^{h}\|_{Y}+k^{-1}\sum_{j=1}^{n-1}\|\beta_{j+1}-\zeta_{j+1 }^{h}-(\beta_{j}-\zeta_{j}^{j})\|_{Y}^{2}\Big{)}.\end{array} \]Finally, keeping in mind that  
\[k^{-1}\sum_{j=1}^{n-1}\|\beta_{j+1}-\zeta_{j+1}^{h}-(\beta_{j}-\zeta_{j}^{j})\|_{ \gamma}^{2}\leq Ch^{2}\|\beta\|_{H^{1}(0,T;H^{1}(\Omega))^{\prime}}^{2}\]  
using the well-known approximation properties by finite elements (see [17]) and a discrete version of [PERSON]'s inequality, we obtain the desired linear convergence. The fully discrete scheme provided in problem \(\mathrm{VP}^{hk}\) has been solved using a penalty-duality algorithm, related to [PERSON]'s algorithm, for the numerical resolution of the discrete variational inequality. The discrete displacements have been obtained solving the discrete linear variational equation with the classical conjugate gradient method. We note that a similar scheme has also been employed for the numerical approximation of dynamic contact problems (see, for example, [16]). Moreover, the resulting algorithm has been implemented within the well-known code MATLAB in a 3.3 GHz PC (with 16 Gb of RAM memory), and a typical 1D run with parameters \(h=k=10^{2}\) took about 1.54 s of CPU time. In order to show the numerical convergence of the algorithm we solve Problem P with the following parameters:  
\[ \begin{array}{llll}T=2,&\Omega=(0,1),&\Gamma_{D}=\{0\},&\Gamma_{N}=\{1\}, &\boldsymbol{g}=\boldsymbol{0},\\ \lambda=0,&\mu=1.348\times 10^{10},&\kappa^{*}=0.5,&\beta_{*}=0.01,\\ q^{*}=10^{5},&\lambda_{d}=17,&\lambda_{u}=4.2\times 10^{5},&\lambda_{w}=0, \end{array} \]  
with the initial condition:  
\[\beta_{0}(x)=1\quad\text{for all}\quad x\in(0,1),\]  
and the volumetric force:  
\[\boldsymbol{f}_{0}(x,t)= \begin{cases}90\times 10^{6}\cdot t&\text{if }t<1,\\ 90\times 10^{6}&\text{if }t\geq 1.\end{cases} \]  
The numerical errors are given by  
\[E^{hk}=\max_{0\leq n\leq N}\left\{\|\boldsymbol{u}_{n}-\boldsymbol{u}_{n}^{hk }\|_{V}+\|\beta_{n}-\beta_{n}^{hk}\|_{Y}\right\},\]  
considering as \"exact solution\" \((\boldsymbol{u}_{n},\beta_{n})\) the one obtained for \(h=2^{-12}\) and \(k=10^{-5}\). The errors (multiplied by 100), obtained for different discretizations, are shown in Table 1 and depicted in Figure 1 against \(h+k\). As shown, the linear convergence of the algorithm stated in Theorem 1 seems to be achieved. Figure 1: Asymptotic constant error.