## 3 Damage-Healing Formulations
### Damage-Healing Law for Concrete  
In the previous work [88], a continuum damage-healing model for autonomous and autogenous self-healing concrete was proposed. In this model, a time-dependent healing variable representing the opposite of the damage variable was proposed. Isotropic material was considered at macroscale  
Figure 8: Stress–strain response of the healed material with different rest periods [134]. (Copyright, 2013, Computational Mechanics).  
Figure 7: Microapsules-based self-healing mechanism: (**i**) crack formation; (**ii**) the crack breaks the microcapsules and the healing agent releases into the crack area; (**iii**) solidification of healing agent in contact with the catalyst [136]. (Copyright, 2012, Construction and Building Materials).  
subjected to tensile load. The concept of coupled and uncoupled self-healing mechanism was introduced. The uncoupled self-healing mechanism represents the autogenous self-healing, and the coupled mechanism represents the the autonomous self-healing. In addition, both mechanisms were applied in the case of the so-called nonlinear self-healing theory, and comparison with the linear self-healing theory was performed. The authors in [142] proposed the nonlinear self-healing theory. It concerns the generalized nonlinear and quadratic self-healing theories. It should be noted that the classical (linear) self-healing theory is represented in Equation (10). It is called linear because the equation is linear in \(h\). It was revealed that, in the case of small damage, the linear healing theory is a special case of the nonlinear healing theory. The configuration of the nonlinear healing theory is illustrated in Figure 9. Classical damage variable \(\varphi\) is used to describe the damaged material state in Figure 9a, while the healing variable h is used to describe the partially healed material state in Figure 9b. According to the nonlinear healing theory, a partial area of damage is subjected to healing as clearly shown in Figure 9b in which the healed area \(S_{h}\) is less than the damaged area \(S_{\varphi}\). The theory of decomposition of the damage variable was used to obtain the combined healing/damage variable \(\varphi_{hd}\). For more details on the decomposition theory, the reader can refer to [75; 143]. It should be noted that the combined healing/damage variable of the classical self-healing theory takes the following expression:  
\[\varphi_{hd}=\varphi\left(1-h\right). \tag{32}\]  
According to the nonlinear self-healing theory, Equation (32) becomes respectively in the case of generalized nonlinear and quadratic self-healing theories as follows:  
\[\varphi_{hd}=\ \frac{\varphi\left(1-h\right)}{1-h\varphi}, \tag{33}\]  
\[\varphi_{hd}=\varphi\left(1-h\right)\left(1+\varphi h\right). \tag{34}\]  
Equations (33) and (34) represent the expression of the nonlinear healing the generalization of Equation (32) in the case of generalized nonlinear and quadratic self-healing theories, respectively. Following the expressions in Equations (33) and (34), the equations of the effective stress of the generalized nonlinear and quadratic self-healing healing models are respectively expressed as follows:  
\[\sigma=\frac{\sigma\left(1-h\varphi\right)}{1-\varphi}, \tag{35}\]  
\[\sigma=\frac{\sigma}{1-\left(\varphi-h\varphi\right)\left(1+\varphi h\right)}. \tag{36}\]  
Figure 9: Configuration of the nonlinear self-healing theory [89].