Modelling of crack propagation: Comparison of discrete lattice system and cohesive zone model

Lattice models are often used to analyze materials with discrete micro-structures mainly due to their ability to accurately reflect behaviour of individual fibres or struts and capture macroscopic phenomena such as crack initiation, propagation, or branching. Due to the excessive number of discrete interactions, however, such models are often computationally expensive or even intractable for realistic problem dimensions. Simplifications therefore need to be adopted, which allow for efficient yet accurate modelling of engineering applications. For crack propagation modelling, the underlying discrete microstructure is typically replaced with an effective continuum, whereas the crack is inserted as an infinitely thin cohesive zone with a specific traction-separation law. In this work, the accuracy and efficiency of such an effective cohesive zone model is evaluated against the full lattice representation for an example of crack propagation in a three-point bending test. The variational formulation of both models is provided, and obtained results are compared for brittle and ductile behaviour of the underlying lattice in terms of force-displacement curves, crack opening diagrams, and crack length evolutions. The influence of the thickness of the process zone, which is present in the full lattice model but neglected in the effective cohesive zone model, is studied in detail.