# Analysis of Damage Models for Cortical Bone
## 2 Damage Models
### Fremond-Nedjar Model
In any case, when damage becomes zero, the material is dense with microcracks and modelling it as elastic ceases to make sense (see [16] for details). Finally, we note that damage function \(\phi\) is given in (1), where constants \(\lambda_{d}\), \(\lambda_{u}\) and \(\lambda_{w}\) are constitutive parameters. Now, in order to obtain the variational formulation of Problem P, let \(Y=L^{2}(\Omega)\), \(H=[L^{2}(\Omega)]^{d}\) and \(Q=[L^{2}(\Omega)]^{d\times d}\) and denote by \((\cdot,\cdot)_{Y}\), \((\cdot,\cdot)_{H}\) and \((\cdot,\cdot)_{Q}\) the respective scalar products in these spaces. Moreover, let us define the variational space \(V\) as follows,  
\[V=\{\boldsymbol{v}\in[H^{1}(\Omega)]^{d}\,;\,\boldsymbol{v}=\boldsymbol{0} \quad\text{on}\quad\Gamma_{D}\},\]  
with the scalar product \((\cdot,\cdot)_{V}\), and norm \(\|\cdot\|_{V}\). Finally, let us define the convex set of admissible damage functions,  
\[K=\{\zeta\in H^{1}(\Omega)\,;\,\beta_{*}\leq\zeta\leq 1\,\text{ a.e. in }\Omega\}.\]  
By using [PERSON]'s formula and boundary conditions (5)-(7), we write the variational formulation of problem P.  
Problem VP. Find the displacement field \(\mathbf{u}:[0,T]\to V\) and the damage field \(\beta:[0,T]\to K\) such that \(\beta(0)=\beta_{0}\) and, for a.e. \(t\in(0,T)\), and for all \(\mathbf{v}\in V\) and \(\zeta\in K\),  
\[\boldsymbol{\sigma}=2\,\beta\,\mu\varepsilon(\boldsymbol{u})+ \beta\,\lambda\text{Div}\,(\boldsymbol{u})\mathcal{I}\quad\text{in}\quad \Omega\times(0,T), \tag{2}\]0 \[\boldsymbol{\sigma}=2\,\beta\,\mu\varepsilon(\boldsymbol{u})+ \beta\,\lambda\text{Div}\,(\boldsymbol{u})\mathcal{I}\quad\text{in}\quad \Omega\times(0,T), \tag{2}\]1  
where the bilinear functional \(c:K\times V\times V\to\mathbb{R}\) is defined as follows, for all \(\mathbf{v},\mathbf{w}\in V\) and \(\zeta\in K\),  
\[\boldsymbol{\sigma}=2\,\beta\,\mu\varepsilon(\boldsymbol{u})+ \beta\,\lambda\text{Div}\,(\boldsymbol{u})\mathcal{I}\quad\text{in}\quad \Omega\times(0,T), \tag{2}\]2  
The operator \(\mathrm{div}\) represents now the divergence of vector-valued functions, the element \(\mathbf{f}(t)\in V^{\prime}\) (as usual, \(V^{\prime}\) denotes the dual space of \(V\)) is obtained from [PERSON]' theorem as  
\[\boldsymbol{\sigma}=2\,\beta\,\mu\varepsilon(\boldsymbol{u})+ \beta\,\lambda\text{Div}\,(\boldsymbol{u})\mathcal{I}\quad\text{in}\quad \Omega\times(0,T), \tag{2}\]3  
and we recall that damage function \(\phi\) is defined in (1). We now consider a fully discrete approximation of problem \(VP\). To solve it numerically, a finite element scheme is used. This is done in two steps. First, we assume that the domain \(\overline{\Omega}\) is polyhedral and we denote by \(\mathcal{T}^{h}\) a regular triangulation in the sense of [17]. Thus, we construct the finite dimensional spaces \(V^{h}\subset V\) and \(E^{h}\subset H^{1}(\Omega)\) given by  
\[\boldsymbol{\sigma}=2\,\beta\,\mu\varepsilon(\boldsymbol{u})+ \beta\,\lambda\text{Div}\,(\boldsymbol{u})\mathcal{I}\quad\text{in}\quad \Omega\times(0,T), \tag{2}\]4 \[\boldsymbol{\sigma}=2\,\beta\,\mu\varepsilon(\boldsymbol{u})+ \beta\,\lambda\text{Div}\,(\boldsymbol{u})\mathcal{I}\quad\text{in}\quad \Omega\times(0,T), \tag{2}\]5  
where \(P_{1}(Tr)\) represents the space of polynomials of degree less or equal to one in the element \(Tr\), i.e., the finite element spaces \(V^{h}\) and \(E^{h}\) are composed of continuous and piecewise affine functions. Here, \(h>0\) denotes the spatial discretization parameter. Moreover, let \(K^{h}=K\cap E^{h}\) and assume that the discrete initial condition, denoted by \(\beta^{h}_{0}\), is given by  
\[\boldsymbol{\sigma}=2\,\beta\,\mu\varepsilon(\boldsymbol{u})+ \beta\,\lambda\text{Div}\,(\boldsymbol{u})\mathcal{I}\quad\text{in}\quad \Omega\times(0,T), \tag{2}\]6  
where \(\mathcal{P}^{h}\) is the classical finite element interpolation operator over \(E^{h}\) (see, e.g., [17]). Secondly, we consider a partition of the time interval \([0,T]\), denoted by \(0=t_{0}<t_{1}<\dots<t_{N}=T\).