## 2 Damage and Healing Configurations  
In this section, review of damage and healing variables is presented based on CDHM. According to CDM, the undamaged, damaged and effective material states defined by the undamaged cross-section \(S_{0}\), damaged cross-section \(S_{\varphi}\) and effective cross-section \(\bar{S}\) are respectively illustrated in Figure 1. The initial, damaged and effective configurations are also represented by their elastic modulus \(E_{0}\), \(E_{\varphi}\) and \(\bar{E}\), respectively. The damage variable can mostly be defined based on either cross-sectional or elastic stiffness reduction. The expressions of the damage variable based on cross-sectional reduction and elastic stiffness reduction are expressed as follows [86]:  
\[\varphi=\frac{S_{\varphi}}{S_{0}}\ \ \ with\ \ \ 0\leq\varphi\leq 1, \tag{1}\]  
\[\varphi=\frac{\bar{E}-E_{\varphi}}{E}, \tag{2}\]where \(\varphi\) is the damage variable and it takes the value \(\varphi=0\) when the material is undamaged and the value \(\varphi=1\) when the material is totally damaged. In [87], the authors defined the damage variable based on shear modulus G, Poisson's ratio \(\
u\) and the bulk modulus K reduction respectively as follows:  
\[\varphi=\frac{\bar{G}-\bar{G}}{G}, \tag{3}\]  
\[\varphi=\frac{\bar{\
u}-\
u}{\
u}, \tag{4}\]  
\[\varphi=\frac{\bar{K}-\bar{K}}{K}, \tag{5}\]  
where \(\bar{G}\), \(\bar{\
u}\) and \(\bar{K}\) are, respectively, the effective shear modulus, effective Poisson's ratio and the effective bulk modulus. As the damage is well-known to reduce the cross-section and material stiffness, the healing takes an opposite role of the recovering of the cross-section and material stiffness (see Figure 2). Then, the healing variable can be defined respectively as a function of the cross-section and material stiffness recovery follows [86]:  
\[h=\frac{S_{h}}{S_{\varphi}}\ \ with\ \ 0\leq h\leq 1, \tag{6}\]  
\[h=1-\frac{\bar{E}-E_{h}}{\varphi\bar{E}}, \tag{7}\]  
where \(h\), \(S_{h}\) and \(E_{h}\) represent the healing variable, healed cross-section and elastic stiffness, respectively. The values of \(h=0\), \(0<h<1\), and \(h=1\) represent the unhealed, partially healed, and fully healed material states, respectively. In [88], the authors defined two healing variables that reflect the healing effect in the case of coupled and uncoupled self-healing mechanics. The former mechanism assumes that the healing and damage occur simultaneously, while the latter one assumes that the healing is introduced when damage is constant (more details are given in Section 3.6). The expression of the healing variable in the case of coupled self-healing mechanism (\(h_{c}\)) is similar to Equation (7), while, for the uncoupled self-healing mechanism (\(h_{u}\)), it is written as follows [88]:  
\[h_{u}=\frac{S_{h}-S_{\varphi}}{S_{h}}\ \ \ with\ \ 0\leq h_{u}\leq 1. \tag{8}\]  
When damages are removed from (Figure 1c), the relationship between the nominal stress \(\sigma\) and effective stress \(\bar{\sigma}\) becomes as follows [43; 66; 76]:  
\[\bar{\sigma}=\frac{\sigma}{1-\varphi}. \tag{9}\]  
Figure 1: Undamaged, damaged and effective material states [89]. (Copyright, 2018, Journal of engineering mechanics).  
According to CDHM and following the configurations in Figure 2, Equation (9) becomes in the case of self-healing materials as follows [88; 90]:  
\[\bar{\sigma}=\frac{\sigma}{1-\varphi\left(1-h\right)}. \tag{10}\]  
Another relation of the nominal stress and effective stress was proposed in [91]. The authors assume that the material is totally healed when \(h=0\) and it is totally damaged when \(h=1\). This proposition takes the following expression:  
\[\bar{\sigma}=\frac{\sigma}{\left(\left(1-\varphi\right)+\varphi\left(1-h \right)\right)}. \tag{11}\]  
It is clear from Equation (10) that the effective stress is equal to the nominal stress when the material is fully healed (\(h=1\) & \(\varphi=0\)), while it approaches infinity when the material is totally damaged (\(h=0\) & \(\varphi=1\)).