## 3 Damage-Healing Formulations
### Unilateral-Effects-Based Models  
Damage material weakens the mechanical properties of the material. These properties can be recovered if the cracks close again. When the material is subjected to tensile loads and followed by compressive loads in the same direction, the cracks close in the compression domain and the material recovers its stiffness. This is known by the unilateral effects. The unilateral effect is simulated with the distinguishment between damage in tension and damage in compression using two damage variables. Using damage variable in tension and damage variable in compression, the loading mode for a diffuse network of identical microcracks is defined. The unilateral effect can be classified as a healing process due to its effect of crack closure. Many authors pointed out that taking into account the unilateral effect often leads to ambiguity in the computational analysis [151; 152]. In [153], the authors developed an isotropic 3D damage model for quasi-brittle materials that accounts for the microcracks closure. An anisotropic continuum damage framework accounting for the unilateral effect was proposed in [154]. In [155], the crack closure was simulated through the decomposition of the stress and strain tensors into positive and negative projection operators. [PERSON] and [PERSON] [156] proposed a thermodynamic framework to study the effect of the mechanical stress and temperature on the crack opening and closing in rocks, and crack closure was simulated through unilateral effect. The authors  
Figure 11: Stress–time response of the coupled damage healing model with different values of the material parameter [88]. (Copyright, 2018, Theoretical and Applied Fracture Mechanics).  
Figure 10: Stress–time response of the uncoupled damage healing model with various rest periods [88]. (Copyright, 2018, Theoretical and Applied Fracture Mechanics).  
in [157] proposed a micro-macro chemo-mechanical damage-healing model to simulate the evolution of the salt stiffness due to microcracks opening, closing and propagation. A unilateral effect was taken into account to simulate the crack closure and stiffness recovery under compression. The proposed model was found to be able to predict the stiffness recovery by the unilateral effect of crack closure. In [158], a nonlocal formulation of concrete damage model with unilateral effects. Unlike the use of spectral decomposition of stress or strain, the unilateral effect was simulated using the trace of the strain tensor. [PERSON] and [PERSON] [159] developed an inelastic-damage model that simulates the crack opening due to inelasticity and elastic modulus recovery due to crack closure. The crack closure was described using a scalar damage variable that is coupled with the Unitary Crack Opening (UCO). UCO is the internal variable that describes the inelastic strain. A function S called Cracks Opening Indicator was introduced in order to control the vanishing of the inelastic strain effect in the material when it is loaded under compression. It was assumed that the function S takes the value zero when the cracks are completely closed and takes the value of 1 when the cracks are completely opened. The expression of the function \(S\) was defined as  
\[S=1-\frac{F_{t}^{ac}}{F_{t}^{\sigma_{c}}}, \tag{40}\]  
where \(F_{t}^{ac}\) is the actual tension yield function value and \(F_{t}^{\sigma_{c}}\) is the tension yield function value corresponding to the crack closure stress \(\sigma_{c}\). It was also assumed that the material recovers its initial stiffness when \(F(\sigma_{ij})=F_{t}^{\sigma_{c}}\). It should be noted that the proposed model was not able to simulate complex problems that occur during complex unloading phase because the function \(S\) represents a scalar variable.