# Analysis of Damage Models for Cortical Bone
## 2 Damage Models
### Fremond-Nedjar Model  
The damage model proposed by [PERSON] and [PERSON] in 1996 considers damage as an unknown of the problem (\(\beta\)) that varies between 1 (undamaged material) and 0 (completely damaged). This counter-intuitive definition accounts for the fact that the variable multiplies the mechanical properties of the material (the elastic modulus in the one-dimensional case) in such a way that, in a damaged material, they will be affected by this variable. As mentioned before, this model was presented for concrete structures, so some modifications to include bone behaviour were necessary. Since bone tissue is a living material that can heal over time, the assumption that this variable cannot recover, \(\beta<0\), is no longer true and it is removed from the initial formulation. The mathematical formulation of this model with the modifications to account for bone properties is defined in what follows. In [2] the damage source function \(\phi\) was defined by  
\[\phi(\varepsilon(\mathbf{u}),\beta)=\lambda_{d}\left(\frac{1-\beta}{\beta}\right) -\lambda_{u}|\varepsilon(\mathbf{u})|^{2}+\lambda_{w},\]  
where \(\lambda_{d},\,\lambda_{u}\) and \(\lambda_{w}\) are constitutive parameters. The second term becomes unmanageable mathematically when strains are very large, but then the whole model becomes inadequate, so we truncate the term as follows. Given \(q^{*}>0\), a sufficiently large strain energy truncation constant, let \(\Psi_{q^{*}}\) be given by  
\[\Psi_{q^{*}}(\boldsymbol{\tau})=\left\{ \begin{array}{ll}|\boldsymbol{\tau}|^{2 }&\text{if}\quad|\boldsymbol{\tau}|^{2}\leq q^{*},\\ q^{*}&\text{otherwise},\end{array} \right.\]  
where \(|\boldsymbol{\tau}|^{2}=\tau_{ij}\tau_{ij}\), and here and below, \(i,j=1,\ldots,d\) (\(d\) is the dimension of the problem), and a repeated index indicates summation. Therefore, we use the truncated damage source function:  
\[\phi(\varepsilon(\boldsymbol{u}),\beta)=\lambda_{d}\left(\frac{1-\beta}{\beta }\right)-\lambda_{u}\Psi_{q^{*}}(\varepsilon(\boldsymbol{u}))+\lambda_{w}. \tag{1}\]  
Finally, the mechanical problem is written as follows. Problem P. Find the displacement field \(\boldsymbol{u}=(u_{i})_{i=1}^{d}:\overline{\Omega}\times[0,T]\to\mathbb{R}^{d}\) and the damage field \(\beta:\overline{\Omega}\times[0,T]\to\mathbb{R}\) such that  
\[\boldsymbol{\sigma}=2\,\beta\,\mu\varepsilon(\boldsymbol{u})+ \beta\,\lambda\text{Div}\,(\boldsymbol{u})\mathcal{I}\quad\text{in}\quad \Omega\times(0,T), \tag{2}\] \[-\text{Div}\,\boldsymbol{\sigma}=\boldsymbol{f}_{0}\quad\text{in} \quad\Omega\times(0,T),\] (3) \[\dot{\beta}-\kappa^{*}\Delta\beta+\partial I_{[\beta_{*},1]}( \beta)\
i\phi(\varepsilon(\boldsymbol{u}),\beta)\quad\text{in}\quad\Omega \times(0,T),\] (4) \[\boldsymbol{u}=\boldsymbol{0}\quad\text{on}\quad\Gamma_{D}\times (0,T),\] (5) \[\frac{\partial\beta}{\partial\boldsymbol{\
u}}=0\quad\text{on} \quad\Gamma\times(0,T),\] (6) \[\boldsymbol{\sigma}\boldsymbol{\
u}=\boldsymbol{g}\quad\text{on} \quad\Gamma_{N}\times(0,T),\] (7) \[\beta(\boldsymbol{x},0)=\beta_{0}(\boldsymbol{x})\quad\text{in} \quad\Omega, \tag{8}\]  
where \(\Omega\) represents the bone, whose boundary \(\Gamma\) is assumed to be decomposed into two parts \(\Gamma_{D}\) and \(\Gamma_{N}\) such that \(\Gamma=\Gamma_{D}\cup\Gamma_{N}\), with \(meas(\Gamma_{D})>0\), \([0,T]\), for \(T>0\), denotes the time interval of interest, and let \(\text{Div}\,\) be the divergence of tensor-valued functions. \(\lambda\) and \(\mu\) are the classical [PERSON]'s coefficients and \(\kappa^{*}>0\) is a damage diffusion coefficient. Moreover, \(\beta_{0}\) is an initial condition for the damage field, \(\boldsymbol{f}_{0}\) and \(\boldsymbol{g}\) represent volume and traction forces, respectively, and we include the subdifferential of the indicator function \(I_{[\beta_{*},1]}\) into (4) to ensure that the damage function belongs to the interval \([\beta_{*},1]\). Here, \(\beta_{*}>0\) is a positive constant, assumed small, and it is introduced for mathematical reasons.