## 1 Introduction  
In recent years, as the concept of sustainable composite materials has become increasingly popular, self-healing concrete (SHC) has emerged as an appealing topic as a potential solution to the sustainability challenge. Concrete is the most-used building material in the world, with about 2.5 tons per person per year, and is made especially attractive due to its cheap price. Concrete structures have been built since the discovery of Portland cement in the mid-nineteenth century. It is a quasi-brittle material that is strong in compression but weak in tension. That is why reinforcement is usually used with it for construction [1]. However, its weakest point is that it is subject to cracking and deterioration with time, and the repair of concrete buildings is costly, especially for infrastructures, which are difficult to access. Recently, biological systems have become the inspiration for material scientists, which has resulted in the creation of self-healing materials. There are many strategies and techniques for self-healing concrete and they are classified in different ways. However, the encapsulation-based self-healing concrete has lately received much interest because of its capacity to effectively heal fractures without human intervention, extending the operational life and lowering maintenance costs. The propagating cracks hit the capsule shells releasing the healing agents to heal the cracks in the vicinity of the damaged part, see Figure 1; as a result, it has promising prospects for infrastructures. Several experimental and numerical studies have been carried out in the literature to investigate the fracture of the capsules and the healing capability of the capsules. These studies have revealed that the bond between the capsule shell and the cement matrix is not always perfect and the efficiency of healing depends on the fracture of microcapsules and the release of the healing agent in order to heal the cracks [2,3]. In this regard, a recent overview of the debonding behavior of fiber-reinforced polymer (FRP) systems used for strengthening existing structures can be found in [4]. The concrete damage plasticity model (CDP) is integrated with finite element modeling to study the behavior and efficiency of CFRP as a strengthening technique for reinforced concrete beams subjected to blast loading, and the obtained results were in good correlation with experimental results [5,6]. Several numerical modeling methods have been developed recently to study the fracture mechanisms, such as the meshless methods, which have been used to study the fracture features of a marble semi-circular bend (SCB) specimen under a compressive loading condition. This showed that the geometrical definitions of the specimen can affect the fracture toughness [7]. An identification methodology of cohesive zone model (CZM) parameters has been proposed to study the debonding of adhesively bonded structures. It is based on the comparison of experimental results obtained with help from the digital image correlation (DIC) technique and FEM modeling [8]. Likewise, finite element analysis has been used to obtain the mixed-mode fracture parameters for sandstone specimens by introducing an inclined crack, whereas the fracture toughness was obtained experimentally [9]. A cohesive/volumetric modeling technique based on a diffuse interface fracture approach has been used to model debonding mechanisms in FRP-strengthened RC structural elements [10]. The extended finite element method (XFEM) and cohesive zone method (CZM) have been used to simulate the fracture and delamination in fiber metal laminates (FML). The obtained numerical results were in good agreement with the experimental observations [11]. An interesting review study has shown the potential of phase-field methods for modeling fracture recovery in self-healing concrete. Although this method still required further development before it can be used as a simulation tool [12].