Cohesive zone model for high-cycle fatigue of composite bonded joints under mixed-mode I+II loading M.F.S.F. de Moura a,⇑ , J.P.M. Gonçalves b a Faculdade de Engenharia da Universidade do Porto, Departamento de Engenharia Mecânica, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal b IBM T.J. Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, NY 10598, USA article info Article history: Received 31 July 2014 Received in revised form 15 February 2015 Accepted 26 March 2015 Available online 2 April 2015 Keywords: Cohesive zone model High-cycle fatigue Adhesively bonded joints Mixed-mode I+II loading SLB test abstract A cohesive zone model to deal with high-cycle fatigue of composite bonded joints under mixed-mode I+II loading is developed. The model assumes a linear traction–relative dis- placements softening and uses a single parameter to simulate the cumulative damage resulting from static and fatigue loading. Finite element analyses considering the Single- Leg Bending test were conducted to verify the performance of the proposed method under different modified Paris laws and load ratios. It was verified that the model is able to capture all those effects with accuracy which proves the adequacy of the whole procedure concerning mixed-mode I+II fatigue/fracture characterization of composite bonded joints. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction The cohesive zone model (CZM) has been increasingly accepted as a valuable tool for the modelling of damage initiation and propagation. It has been commonly used in the context of finite element modelling where it usually appears as the con- stitutive law governing the behaviour of adjacent nodes in interface finite elements. Interface finite elements are usually zero thickness elements characterized by two surfaces of nodes where each node in one surface is attached to another node on the other surface. Before any deformation, the two surfaces of nodes can coincide (hence the zero thickness) and each pair of adjacent nodes have the same position. The CZM describes the relationship between stresses and relative displacements of each pair of adjacent nodes. It was initially proposed by Dugdale [1] and Barenblatt [2] and was first used in combination with the finite element method by Hillerborg et al. [3]. Until recently, the CZM together with the finite element method had been mainly used to model the initiation and propagation of damage in structures subjected to monotonic loads [4–11] or fast crack growth in brittle solids [12]. Examples of areas of application of this type of modelling on structures subjected to monotonic loads are delamination in laminated composite materials [4–7] and damage initiation and propagation in adhe- sively bonded joints [8–11]. Fatigue is an important type of loading in many structures including those that contain adhesively bonded joints. Consequently, it is necessary to develop methods that can accurately predict the fatigue strength of those structures and many studies have already been conducted in this area [13]. In recent years, the finite element method together with the CZM has been used to model the fatigue behaviour in adhesively bonded joints and in laminated composite materials. These models differ from those used in the study of structures under monotonic loads since, in addition to the CZM and http://dx.doi.org/10.1016/j.engfracmech.2015.03.044 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Tel.: +351 225081727. E-mail address: mfmoura@fe.up.pt (M.F.S.F. de Moura). Engineering Fracture Mechanics 140 (2015) 31–42 Contents lists available at ScienceDirect Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech