Cohesive zone models have been widely used to model interface crack initiation and propagation both in single-material media and bi-material systems. For single-material media with cohesive elements inserted into interface among segments, in order to ensure that the introduction of interface cohesive zone models does not affect the mechanical properties of single-material media before the softening stage of cohesive zone models, a selection criterion of stiffness of cohesive elements is proposed theoretically firstly based on the properties' equivalence. Taking the softening stage into account, the mechanical responses of the overall stress-strain relationship of single-material media, for the cases of stable increase of strain and snap-back instability of strain, are both obtained, and the related energy mechanism are investigated. For bi-material systems with cohesive elements at interface between two materials, the thickness-dependent failure characteristics of systems in uniaxial tension are found, which is attributed to the difference of the releasing rate of elastic strain energy in the materials with different thicknesses. Furthermore, as a more complex application of cohesive elements, based on the selection criterion proposed, failure behaviors of the ceramic coating/substrate systems under three-point bending are modeled by finite element method and inserting cohesive elements into the coating segments and the coating/substrate interface simultaneously. The simulation results indicate the transition of dominated failure mode from coating cracking to interface delamination with increasing coating thickness, and show faster damage of thick coating systems, agreeing with experimental results. The effects of interface strength and toughness of cohesive elements on failure are also revealed. These results can provide guidance for the application of cohesive elements, and help us better understand the overall failure behaviors of interface systems. (C) 2019 Elsevier Ltd. All rights reserved.