Research developments and engineering applications ofnumerical methods for simulating fracture process in quasi-brittlematerialsare reviewed and advantages and disadvantages of implicit and explicitrepresentations of strong discontinuities such as crack are discussed. Focus is placed on their construction features anddifferences, concerning concrete cohesive cracks with emphasis on elements with embeddeddiscontinuity model,on extended finite element method (XFEM) and on enriched finite elementtechnique. From the viewpoint of completeness, the element with embeddeddiscontinuity (EED) based on enhanced assumed strain can eliminate stresslock, enforce internal equilibrium conditions and reasonably represent acomplete separation at late stages of the fracture process, but the enrichmentfunctions applied in EED do not satisfy the compatibility conditions inthe cracked element. Thefact that the additional freedom introduced in the element level leadsto adependence of strain on both side of crack. Applying the additionalfreedom as aglobal freedom in XFEM, the enriched finite element based on the partition ofunity not only has the advatages that EED has, but also can avoid the strain dependenceon both side of crack. Certain limitations of remeshing and smeared crackmodel and the subtle difference between XFEM and the enriched finite element method arealso pointed out.