Mixed-mode I þ II continuum damage model applied to fracture characterization of bonded joints J.A.G. Chousal n , M.F.S.F de Moura Faculdade de Engenharia da Universidade do Porto, Departamento de Engenharia Mecˆ anica, Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal article info Article history: Accepted 17 October 2012 Available online 1 November 2012 Keywords: Bonded joints Fracture characterization Mixed-mode Continuum damage model abstract A continuum mixed-mode I þII damage model allowing simulation of damage initiation and propaga- tion in adhesive bonded joints is presented. The model is based on a stress criterion to identify damage onset and a fracture mechanics criterion to simulate gradual degradation of stiffness during loading. It is implemented in two-dimensional solid elements thus allowing identifying the volumetric shape of damaged regions in the vicinity of the crack tip. Additionally, the model is able to predict crack kinking which occurs as a function of the used criteria. The application of the proposed methodology in the context of pure modes (I and II) fracture characterization tests of bonded joints provide the evaluation of the influence of asymmetric propagation on the measured fracture energies. This phenomenon becomes more relevant with the increase of adhesive thickness. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction The use of adhesive joints in critical structures has been increasing due to several advantages intrinsic to this joining method. As a consequence, more demanding design methods are required in order to deal with several details influencing bonded joint mechanical behavior that are not accounted for in classical approaches. In the strength of materials based approaches the maximum stress or strain criteria are the most popular ones [1–4]. They are based on the assumption that failure occurs when one of the stress or strain tensor components attains the respective strength value. The referred criteria have a main difficulty when applied to the failure prediction of bonded joints. In fact, bonded joints with sharp corners are characterized by stress singularities at the end of overlapping regions. In order to overcome these drawbacks, the use of cohesive zone models (CZM) in fracture problems has become frequent in the most recent years. One of the most important advantages of cohesive models is related to its capacity to simulate onset and non-self- similar growth of damage. They are based on a softening relation- ship between stresses and relative displacements between crack faces, thus simulating a gradual degradation of material proper- ties. Generally, stress based and energetic fracture mechanics criteria are used to simulate damage onset and propagation, respectively. Cohesive damage models are usually based on interface finite elements [5–7] connecting plane or three- dimensional solid elements. Those elements are placed at the planes where damage is prone to occur but, in several cases, these critical regions can be difficult to identify in advance. Addition- ally, the application of CZM in the context of bonded joints frequently neglects adhesive thickness [7–9], thus not being able to capture the influence of asymmetrical propagation and crack different paths along adhesive thickness. Moreover, the referred approaches do not allow the simulation of adherends constraint on the fracture process zone development inside the adhesive. On the other hand, it is known that crack propagation in bonded joints often occurs under mixed-mode I þ II loading. In fact, typical applications of bonded joints frequently develop shear and peel stresses in critical regions. Furthermore, it is known that even in problems where one of the modes predominates, there is some mode-mixity that can influence the measured fracture energy. An example occurs when the double cantilever beam test is used to measure fracture energy under mode I loading. Since the crack can propagate near to an interface owing to stress concentrations induced by mismatch properties between adherends and adhe- sive [10,11] a geometrical asymmetry takes place. Depending on the adhesive thickness and material properties of the joint components (adherends and adhesive) a non-negligible mode II loading component can arise. This statement reinforces the need to develop accurate numerical methods accounting for mixed- mode I þ II crack growth in bonded joints. In order to overcome the referred drawbacks a continuum mixed-mode I þ II damage model is proposed. The model is implemented in two dimensional solid elements and combines stress and fracture based criteria to deal with damage onset and growth. The main advantage inherent to the proposed approach is Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/ijadhadh International Journal of Adhesion & Adhesives 0143-7496/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijadhadh.2012.10.014 n Corresponding author. Tel.: þ351 225081591; fax: þ351 225081584. E-mail address: jchousal@fe.up.pt (J.A.G. Chousal). International Journal of Adhesion & Adhesives 41 (2013) 92–97