Coupled cohesive zone models for mixed-mode fracture: A comparative study R. Dimitri a,⇑ , M. Trullo a , L. De Lorenzis b , G. Zavarise a a Dipartimento di Ingegneria dell’Innovazione, Università del Salento, via per Monteroni, 73100 Lecce, Italy b Institut für Angewandte Mechanik, Technische Universität Braunschweig, Bienroder Weg 87, 38106 Braunschweig, Germany article info Article history: Received 9 July 2015 Accepted 8 September 2015 Available online 15 September 2015 Keywords: Cohesive zone modeling Contact Debonding Mixed-mode fracture Thermodynamics abstract This paper checks the consistency of some published exponential and bilinear mixed-mode cohesive zone models. The effect of coupling on traction-separation behavior and energy dissipation is investigated and the path-dependence of the debonding work of separation and failure domain is evaluated analytically and numerically. All selected models present several inconsistencies, except for the one by van den Bosch et al. (2006), which is, how- ever, not currently formulated within a thermodynamical framework but postulated in an ad-hoc manner. We thus propose a thermodynamically consistent reformulation of this model within damage mechanics, which holds monolithically for loading, unloading, deco- hesion and contact. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction Increasing attention has been devoted in the last decades to the use of cohesive zone models (CZMs) to study mixed- mode delamination, debonding, and, more generally, crack initiation and propagation within quasi-brittle materials or at material interfaces. This is due to the computational efficiency of these models and to their versatility for numerical implementation in many areas of computational mechanics. Cohesive models describe the traction-separation behavior of interfaces before and during fracture, and are characterized by two phases, i.e. an increase of the traction up to a peak value and a subsequent decrease to zero, which describe the crack initiation and the growth of cohesive surfaces until new traction-free surfaces appear. The basic concept of CZMs was proposed by Barenblatt [1,2] and Dugdale [3] as an alternative approach to singularity driven fracture mechanics, and has been extensively used in the literature to analyze the fracture process in a number of material systems such as concrete [4,5], polymers [6,7], ductile materials [8–10], ceramics [11], but also bimaterial systems such as polymer matrix composites [12–15] and metal matrix composites [16]. CZMs have been also used to simulate fracture under static [17–19], dynamic [11,20], and cyclic [21,22] loading conditions, delamination of layered composites at the micro- and macro-scale [23–26], delamination between a coating and a substrate [27], debonding of fiber reinforced polymer (FRP) sheets from concrete, masonry or steel substrates [28–34]. Despite the first models have been developed for single-mode fracture processes, cohesive fracture is expected to involve mixed-mode conditions, as observed in practice by experimental investigations performed on various types of lap joints [35], or interfaces between FRP sheets and flat [36,37] or curved substrates [38,39]. Mixed-mode CZMs can be classified as http://dx.doi.org/10.1016/j.engfracmech.2015.09.029 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Tel.: +39 0832 297177. E-mail address: rossana.dimitri@unisalento.it (R. Dimitri). Engineering Fracture Mechanics 148 (2015) 145–179 Contents lists available at ScienceDirect Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech