_applied_  
_sciences_  
_Article_  
Quasi-Static and Dynamic Crack Propagation by Phase  
Field Modeling: Comparison with Previous Results and  
Experimental Validation  
[PERSON] 1,*, [PERSON] 1, [PERSON] 2 and [PERSON] 1  
LA2 MP Laboratory, National Engineering School of S fax, University of S fax, S fax 3029, Tunisia;  
[EMAIL_ADDRESS] (Y.H. ); [EMAIL_ADDRESS] (B.Z.) Footnote 1: [[https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci)]([https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci))  
Footnote 2: [[https://www.mdpi.com/journals/pdf/10.3390/journals/pdf/10.3390/journals/pdf/10](https://www.mdpi.com/journals/pdf/10.3390/journals/pdf/10.3390/journals/pdf/10)]([https://www.mdpi.com/journals/pdf/10.3390/journals/pdf/10.3390/journals/pdf/10](https://www.mdpi.com/journals/pdf/10.3390/journals/pdf/10.3390/journals/pdf/10)). crack front in 'the strong discontinuity approach' [8]. The eXtended Finite Element Method (XFEM) [9] has been successfully used to study dynamic fracture propagation in elastic brittle [10] and ductile [11] materials. [PERSON] et al. [12] used the boundary element method (BEM) to predict crack propagation for fatigue fracture in mixed modes (I + II). Fractures are treated as discrete discontinuities in the methods mentioned above. Consequently, treating crack initiation or branching is complicated and frequently requires special criteria. Another approach is provided by non-local damage models [13; 14; 15; 16]. The thick level set approach calculates the fracture trajectory on the basis of the damage progression at the crack point, and the discontinuities are explicitly taken into account. However, discontinuities are not directly employed in other models, such as the phase field method. Rather, the crack effect is defined by a smooth transition. The completely damaged regions, as well as the undamaged material, are linked by a unique damage variable. The non-local damage models applied to [PERSON]'s theory are based on the variational formulation developed by [PERSON] and [PERSON] [17; 18] and implemented by [PERSON] et al. [19]. Complex fracture processes in two- and three-dimensional [20] solids, such as crack initiation, propagation, and branching, can be processed automatically without any additional ad hoc strategies. The monolithic scheme [21; 22; 23; 24; 25; 26] and the staggered scheme [19; 27; 28; 29; 30] are the two most important numerical methods adopted to simulate the mechanical problem and determine the fracture pattern. When the fracture propagates closely to an initial crack, monolithic algorithms eventually converge. However, these algorithms struggle to converge in the case of unstable crack initiation and propagation. The non-convexity of global energy and the jumps in the solution are the causes of this problem [31]. To avoid this, a variety of modifications [24; 31; 32] have been proposed. The staggered scheme is derived from the alternate minimization algorithm developed by [PERSON] et al. [19]. Two well-posed convex problems are solved separately. Even in unstable cases, this iterative method was shown to be very robust [29]. Therefore, staggered implementation was considered as the right approach to use in this paper. The only drawback of this technique is that it necessitates a relatively small time step [30]; thus, a high computational time is needed. To overcome this issue, several research works were adopted, such as detecting unstable propagation and reducing the external load variation [22; 23; 33; 34; 35; 30], deactivating damage variables until an equivalent stress reaches a critical value \(\sigma\)c [36], optimizing the assembly matrices [37], etc. Decomposing the stored elastic energy into two parts, one associated with a compressive strain and the other with a tensile one, is a common approach in solving the problem of crack propagation.