## 5 Conclusions  
This study analyses four numerical models for simulating crack initiation and growth in heterogeneous materials with an interface phase, focusing on their implementation complexity and computational cost. The methods include the standard phase-field fracture, the cohesive phase-field fracture, the cohesive zone model, and a hybrid model where the cohesive zone handles interface debonding and the cohesive phase-field simulates matrix cracking. The conclusions are based on a 2D single inclusion model, considering that the interface thickness is comparable to other phases. The main findings are:  
* Considering that the hybrid model is the most accurate numerical framework for heterogeneous materials with interfaces where the thickness of the interface zone is much smaller than the other phases, the main finding of this study highlights the convergence of the CPFM model to the hybrid approach for a specific thickness of the interface zone within the CPFM numerical framework. As demonstrated, the CPFM approach is computationally less expensive than the hybrid model and can, therefore, serve as an efficient replacement. Moreover, due to the reduced computational cost of the CPFM approach, it can be effectively used to generate labeled data for training data-driven surrogate models to predict fractures in heterogeneous materials. However, it must be emphasised that if this approach is used as a replacement for the hybrid model, the thickness and material properties of the interface zone must first be calibrated and validated against experimental results within the CPFM framework. This makes the CPFM model less sensitive to interface thickness, making it a good option for multiphase materials with isotropic phases. See [36] for more information. * The SPFM model does not capture cohesive behaviour in semi-brittle materials in the softening zone, and its length scale parameter \(l_{c}\) is treated as an intrinsic material property, limiting its use in small-scale simulations. The CPFM model is more suitable for multiphase materials, especially when the interface thickness is comparable to other phases. In CPFM, the length scale parameter acts as a numerical property, making it ideal for small-scale simulations. It is also less computationally demanding and easier to implement than the hybrid approach. * Composite design with targeted properties can be achieved by adjusting inclusion arrangement, modifying interface properties, adding voids, or a combination of these factors. This study shows that by strategically arranging inclusions or interface properties, fracture toughness can be significantly improved while keeping all other parameters constant. Moreover, the advantages and limitations of each fracture models studied in this work can be described as follows: the SPFM approach provides a robust framework for macroscale simulation of brittle fracture in homogeneous materials. Its simplicity and ease of implementation have led to its widespread adoption for materials without significant heterogeneities. The length-scale parameter of the model is material dependent, which does not pose a major challenge for large scale simulations, but can lead to boundary effects in small-scale applications. Despite its advantages, the SPFM is limited in its ability to capture interfacial phenomena because it is restricted to macroscale problems and does not explicitly include cohesive behavior. The CPFM model extends the SPFM approach by incorporating cohesive behavior, enabling it to simulate fracture initiation and propagation in materials with interfaces or regions where cohesive interactions are prominent. This model is particularly well-suited to small-scale simulations as its length-scale parameter is independent of material properties. While the CPFM significantly improves the capabilities of the phase-field framework, it still relies on a diffuse crack representation, which can limit its accuracy in scenarios requiring sharp interface debonding simulations.