## 3 Damage-Healing Formulations
### Creep Damage-Healing Model for Salt Rock  
Rock salt is generally subjected to creep damage and cracks that result in the increase of the permeability of the material. Because damage results in inelastic flow in rock salt under hydrostatic compression, an extension of continuum damage approach to the healing of creep damage was developed in [96]. It was assumed that the macroscopic strain rate is influenced by the healing mechanism along with damage and creep. Anisotropic healing was also considered such that the conjugate stress measure for healing can be expressed as  
\[\sigma_{eq}^{h}=\frac{1}{3}\left(I_{1}-x_{10}\sigma_{1}\right), \tag{19}\]  
where \(I_{1}\), \(x_{10}\) and \(\sigma_{1}\) are the first invariant of the Cauchy stress, material constant and the maximum principal stress, respectively. It is considered that the healing is isotropic when \(x_{10}=0\) and anisotropic when \(x_{10}\
eq 0\). The kinetic equations of the healing were formulated based on an experimental observation that suggests two healing mechanisms can be activated in (Waste Isolation Pilot Plant) WIPP salt. The first mechanism assumes that the healing is present in a much smaller time period which results in unchanged damage variable, while the second mechanism assumes that the healing is present in a larger time period which reduces the damage variable. The healing variable proposed in [96] is considered to be the first-order kinetic equation expressed as  
\[h=\frac{\omega\sigma_{eq}^{h_{2}}H\left(\sigma_{eq}^{h_{2}}\right)}{\tau_{2}\mu} \tag{20}\]  
where \(\omega\), \(H\), \(\tau_{2}\), \(\mu\) are the damage variable, Heaviside function, time characteristic constant and the shear modulus, respectively. \(h_{2}\) describes the removal of damage. The healing variable defined in Equation (20) is characterized by the fact that only an individual healing mechanism can be simulated. This leads to the difficulty of finite element implementation of the healing model due to the challenging identification of the healing time involving non-relative loading histories. Thus, the starting time and the period of the healing become ambiguous. A simplified version of the healing model proposed in [96] was further developed in [97]. Instead of defining two separate healing mechanisms, a single healing mechanism based on damage, kinetic equation and equivalent stress was proposed. The simplified healing variable was defined with only one kinetic equation for one healing mechanism in contrast to the previous investigation in which one kinetic equation for each healing mechanism was defined. After modification of the kinetic equation, the following healing variable was obtained:  
\[h=\frac{\omega\left(\sigma_{eq}^{h}-\sigma_{b}\right)H\left(\sigma_{eq}^{h}- \sigma_{b}\right)}{\tau\mu} \tag{21}\]with  
\[\sigma_{b}=x_{7}|\frac{\sigma_{1}-\sigma_{3}}{x_{2}x_{7}}|^{1/x_{6}}, \tag{22}\]  
where \(\sigma_{1}\), \(\sigma_{3}\) and \(\sigma_{eq}\) are principal stresses and equivalent stress, respectively. \(x_{2}\), \(x_{6}\) and \(x_{7}\) are material constants. According to Equation (21), healing can be activated only when \(\sigma_{eq}>\sigma_{b}\). In addition, the healing reduces the volumetric strain to zero. The axial and lateral strains are also recovered under hydrostatic compression. In [98], a thermodynamic framework of CDHM was proposed in which the concept of healing surface and loading-unloading conditions were used. Rate-dependent and rate-independent formulations were also given and applied in the case of isotropic healing. The general thermodynamic framework was applied to study the healing crushed rock salt. The surface-based healing function takes the following expression:  
\[F=s:s-c_{B}\left[\left(B^{sp}\right)^{2}+\left(B^{d}\right)^{2}\right]+c_{S} \left(\omega\right)^{2}-F_{0}, \tag{23}\]  
where \(s\), \(B\), \(\omega\) and \(F_{0}\) are the Cauchy stress tensor, constrained modulus, surface energy per unit area, and the material positive constant, respectively.