# Analysis of Damage Models for Cortical Bone
## 2 Damage Models
### Fremond-Nedjar Model
In this case, we use a uniform partition with step size \(k=T/N\) and nodes \(t_{n}=n\,k\) for \(n=0,1,\dots,N\). Therefore, using a combination of both implicit and explicit Euler schemes, the fully discrete approximations are considered as follows. Problem VP\({}^{hk}\). Find the discrete displacement field \(\mathbf{u}^{hk}=\{\mathbf{u}^{hk}_{n}\}_{n=0}^{N}\subset V^{h}\) and the discrete damage field \(\beta^{hk}=\{\beta^{hk}_{n}\}_{n=0}^{N}\subset K^{h}\) such that \(\beta^{hk}_{0}=\beta^{h}_{0}\) and, for all \(\mathbf{v}^{h}\in V^{h},\mathbf{\zeta}^{h}\in K^{h}\),  
\[\boldsymbol{\sigma}=2\,\beta\,\mu\varepsilon(\boldsymbol{u})+ \beta\,\lambda\text{Div}\,(\boldsymbol{u})\mathcal{I}\quad\text{in}\quad \Omega\times(0,T), \tag{2}\]7 \[\boldsymbol{\sigma}=2\,\beta\,\mu\varepsilon(\boldsymbol{u})+ \beta\,\lambda\text{Div}\,(\boldsymbol{u})\mathcal{I}\quad\text{in}\quad \Omega\times(0,T), \tag{2}\]8 \[\boldsymbol{\sigma}=2\,\beta\,\mu\varepsilon(\boldsymbol{u})+ \beta\,\lambda\text{Div}\,(\boldsymbol{u})\mathcal{I}\quad\text{in}\quad \Omega\times(0,T), \tag{2}\]9  
We remark that in problem VP\({}^{hk}\) the \"initial condition\" \(\mathbf{u}^{hk}_{0}\) for the displacement field must be calculated because it is not previously given. Thus, we take it as the solution to the corresponding discrete problem:  
\[-\text{Div}\,\boldsymbol{\sigma}=\boldsymbol{f}_{0}\quad\text{in} \quad\Omega\times(0,T),\]0We have the following theorem which states the linear convergence of these approximations under suitable additional regularity conditions. Assume that problem \(VP\) has a unique solution \((\mathbf{u},\beta)\) with the following regularity:  
\[ \begin{array}{l}\mathbf{u}\in\mathcal{C}([0,T];[H^{2}(\Omega)]^{d})\cap C^{1}([0,T] ;V),\\ \beta\in H^{2}(0,T;Y)\cap H^{1}(0,T;H^{1}(\Omega))\cap\mathcal{C}([0,T];H^{2}( \Omega)),\end{array} \]  
and let \((\mathbf{u}^{hk},\beta^{hk})\) be the solution to Problem \(VP^{hk}\). Then, it follows that the numerical approximation is linearly convergent; that is, there exists a positive constant \(C\), independent of the discretization parameters \(h\) and \(k\), such that  
\[\max_{0\leq n\leq N}\left\{\|\mathbf{u}_{n}-\mathbf{u}_{n}^{hk}\|_{V}+\|\beta_{n}-\beta _{n}^{hk}\|_{Y}\right\}\leq C(h+k).\]  
First, proceeding as in [16], we have the following estimates for the damage field, for all \(\xi^{h}=\{\xi_{j}^{h}\}_{j=0}^{n}\subset K^{h}\),  
\[ \begin{array}{l}\|\beta_{n}-\beta_{n}^{hk}\|_{Y}^{2}+k\sum_{j=1}^{n}\|\
abla (\beta_{n}-\beta_{n}^{hk})\|_{Y}^{2}\leq C\Big{(}\|\beta_{0}-\beta_{0}^{h}\|_ {Y}^{2}+\|\beta_{1}-\zeta_{1}^{h}\|_{Y}^{2}+\|\beta_{n}-\zeta_{n}^{h}\|_{Y}^{2} \\ \quad+k\sum_{j=1}^{n}\|\beta_{j-1}-\beta_{j-1}^{hk}\|_{Y}^{2}+k\sum_{j=1}^{n} \|\mathbf{u}_{j}-\mathbf{u}_{j}^{hk}\|_{Y}^{2}+k\sum_{j=1}^{n}\|\beta_{j}-\zeta_{j}^{ h}\|_{H^{1}(\Omega)}^{2}+k^{2}+k\sum_{j=1}^{n}\|\delta\beta_{j}-\dot{\beta}_{j}\|_{Y} ^{2}\\ \quad+k\sum_{j=1}^{n}\|\phi(\mathbf{\varepsilon}(\mathbf{u}_{j}),\beta_{j})-\delta \beta_{j}+\kappa^{*}\Delta\beta_{j}\|_{Y}\ \|\beta_{j}-\zeta_{j}^{h}\|_{Y}+k^{-1}\sum_{j=1}^{n-1}\|\beta_{j+1}-\zeta_{j+1 }^{h}-(\beta_{j}-\zeta_{j}^{j})\|_{Y}^{2}\Big{)},\end{array} \]  
where we used the notation \(\delta\beta_{j}=(\beta_{j}-\beta_{j-1})/k\) and the regularity of the continuous solution \(\beta\). Now, we obtain the estimates for the displacement fields. Subtracting the variational Equation (9) at time \(t_{n}\) for \(\mathbf{v}=\mathbf{v}^{h}\in V^{h}\subset V\) and the discrete variational Equation (11) we have  
\[c(\beta_{n};\mathbf{u}_{n},\mathbf{v}^{h})-c(\beta_{n}^{hk};\mathbf{u}_{n}^{hk},\mathbf{v}^{h} )\quad\forall\mathbf{v}^{h}\in V^{h},\]  
so it follows that, for all \(\mathbf{v}^{h}\in V^{h}\),  
\[c(\beta_{n};\mathbf{u}_{n},\mathbf{u}_{n}-\mathbf{u}_{n}^{hk})-c(\beta_{n}^{hk};\mathbf{u}_{n} ^{hk},\mathbf{u}_{n}-\mathbf{u}_{n}^{hk})=c(\beta_{n};\mathbf{u}_{n},\mathbf{u}_{n}-\mathbf{v}^{h} )-c(\beta_{n}^{hk};\mathbf{u}_{n}^{hk},\mathbf{u}_{n}-\mathbf{v}^{h}).\]  
Taking into account that  
\[