In this study, the influence of different damage criteria on crack propagation is studied. Based on the idea presented in Reddy and Srinivasa (2015), a reformulation of the conventional finite element method into an edge-based finite element network has been performed. This reformulation makes the introduction of damage criterion into the formulation easier and simpler than the traditional approaches. With the help of this reformulation, the global stiffness matrix obtained is evolutionary (with respect to that of the damage criterion). A Binary damage criterion is introduced, which takes minimum value (in this work it is taken as 0.001) or 1, value for each edge depending on the critical extensional strain value that the edge reaches. If an extensional edge strain exceeds the critical strain value, then a binary damage criterion value for that edge is set to minimum value. A Continuous damage criterion is developed and introduced into the formulation, the influence of the degree of the continuous damage criterion in the formulation is studied. The damage criterion was introduced both in the local and non-local mechanisms to study the influence of the damage criterion introduced and the non-locality of the system on crack initiation and crack propagation. This developed framework facilitates the study of the crack propagation for different classes of problems, depending on the chosen parameters, yet retaining the simplicity of the conventional FEA framework.