## 1 Introduction  
Brittle materials are subjected to microstructural degradations that lead to their failure. The material degradation is the result of the nucleation and growth of microvoids and microcracks. This phenomenon is expressed and termed by damage. In recent years, self healing materials have been used to repair damage in materials. Therefore, many investigations are conducted on self-healing materials. French Academy of Science discovered the self-healing theory in 1836. They found that calcium carbonate results from cement hydration on exposure to atmosphere that concerns the autogenous self-healing mechanism [1; 2]. This is due to hydration of cement or carbonation of calcium hydroxide [3]. The second category, called autonomous self-healing, was first proposed in [4]. The self-healing concept aims at automatically repairing the damages occurring in the material. Inspired from this idea, [PERSON] et al. [5] developed the continuum damage-healing mechanics (CDHM) for composited materials. CDHM represents the extension of the continuum damage mechanics (CDM) in which the healing effect is introduced into the constitutive equations. Furthermore, many investigations based on fracture mechanics and advanced finite element methods such as discrete particle/element method [6] and smooth particle hydrodynamics (SPH) [7] were also carried out on self-healing materials. For more details on these methods, the reader can refer to Refs. [8; 9; 10; 11; 12; 13] for the discrete particle/element method, Refs. [14; 15; 16; 17] for the SPH method, Refs. [18; 19; 20; 21; 22; 23; 24] for meshfree method, [25; 26; 27; 28; 29; 30] for the multiscale method, Refs. [31; 32; 33; 34; 35] for the phase field method, and Refs. [36; 37; 38; 39; 40; 41; 42] for advances in fracture mechanics. The presentation of CDM was first given by [PERSON] [43] in which the continuum damage mechanics framework was originally applied to handle the response of the creep failure of metal alloys. This framework was further developed by [PERSON] [44] in which the damage factor concept was introduced. CDM framework was further extended by many researchers who aimed at describing the process of damage [45; 46] where it was assumed that the material starts to rupture once the damage variable reaches a critical level. At the beginning of the application of CDM, much attention was given to the analysis of damage due to creep [47; 48; 49; 50]. Later on, further developments were carried out using the principles of continuum damage mechanics [51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66]. In general, damage mechanics interests in the study of the material in different scales, namely, microscopic, mesoscopic, macroscopic, and mixed scale (statistical method), in which the damage models are applied to describe the variation of the material properties and material failure due to crack initiation and propagation. The basic issue of CDM is to quantify the damage in the material. Many researchers defined the damage variable as the ratio of the number of damaged and total cross-section [67; 68], while other researchers used the concept of the effective stress to define the damage variable [69; 70]. Another method used to calculate the damage variable which is based on the elastic stiffness reduction was also proposed by [PERSON] [71], and investigated further by many researchers [72; 73]. The damage variable can be expressed as a scalar variable in the case of isotropic material and as a tensor in the case of anisotropic material [74; 75]. The description of the quasi-static behavior of ductile and brittle materials came subsequently on [76; 77; 78; 79; 80]. Application of conventional local damage models results in ill-posedness problems and strain localization due to the softening behavior of brittle materials which can be avoided using particular a simulation. This simulation can be performed using the developed nonlocal damage models of integral and gradient types [81; 82; 83].