Issue 39

M. A. Lepore et alii, Frattura ed Integrità Strutturale, 39 (2017) 191-201; DOI: 10.3221/IGF-ESIS.39.19 192 RVE dimension makes this approach versatile enough to deal with a plenty of practical problems, among which, as examples, we can cite the homogenization of an RVE as a step for multiscale modelling [4]; the damage and failure modelling of inhomogeneous micro-structured materials under quasi static load [5]; the damage evolution and failure modelling of homogeneous materials under creep conditions [6]. The cohesive zone modelling (CZM) approach can be considered as a special case of the CDM methodology in which the softening behaviour is confined to an interface domain and the RVE dimension is implicitly taken into account by the material law. Among the methodologies for studying decohesion processes, the CZM approach is relevant in many engineering applications, from the simple debonding of a cantilever beam [7], to the three-dimensional delamination modelling in a cracked FML full scale aeronautic panel [8]. This method is based on the definition of a cohesive zone law, i.e. the traction-separation relationship at the bonded surface of the joint. For basic modes of debonding, the CZM can be estimated with experimental data from suitable test campaigns. Once defined, both in a direct or indirect way, it can be implemented in a Finite Element code for simulating the debonding phase. Many cohesive traction-separation laws are reported in literature by using experimental, theoretical, and computational techniques. Among them, a large number concerns the response in normal (mode I) and shearing (mode II and III) direction with respect to the interface and under mixed mode I-II loading condition, whilst pure mode III or mixed mode II-III are less investigated. In order to evaluate experimentally the constitutive, peel, properties of adhesive and adherends, both metallic and composite, several standard test methods use the double cantilever beam (DCB) specimen under mode I fracture loading. Olsson et al. [9] developed an analytical solution for the determination of the constitutive properties of thin interphase layers, which provides with reliable results for non rigid adherends. Sorensen et al. [10] presented a traction-separation relation, based on the J-integral approach, for DCB. Valoroso et al. [11] proposed a deterministic identification of mode I cohesive parameters for bonded interfaces that overcomes the difficulties and limitations of ISO 25217 standard test. Stigh et al. [12] proposed an alternative experimental method to determine the complete cohesive relation of a thin adhesive layer loaded in peel with the same DCB specimen. Shen et al. [13], instead, implemented digital image correlation (DIC) technique to measure displacements at the crack tip in order to evaluate fracture parameters and determine cohesive relationships. Fernandes et al. [14] analysed mode II cohesive relationship of carbon–epoxy composite bonded joints using DIC and the direct method applied to the end notched flexure (ENF) test. Mode II and mixed mode I-II testing were also analysed in [15, 16]. A mode III traditional test methodology was presented in [17]. Cricrì et al. [18] presented a methodology, based on an innovative test device, for studying bonded elements subjected to pure mode III loading condition. The present work describes a straightforward strategy for modelling the structural bonding interface of a joint using a commercial FEM code, assuming that the loading modes are uncoupled. Such an approach is implemented in an algorithm and a validation study case is presented by using the software Abaqus. T HEORETICAL BACKGROUND n a 2D problem, the traction vector T , whose components  and  , respectively acting on the interface along normal n and tangential s direction, can be expressed in matrix form as: nn ns n sn ss s K K K K                           T K u (1) where K nn , and K ss are stiffness terms, while  n and  s are normal and tangential separations, respectively. In particular, separations are numerically equal to the jump displacements of the cohesive elements. The initial (undamaged state) stiffness components of K nn and K ss are respectively equal to the ratio between Young’s modulus and adhesive thickness and the ratio between shear modulus and adhesive thickness whereas K sn = K ns = 0. In general, during the phase of interface opening, the stiffness parameters vary, decreasing their values with damage increasing. Therefore, the damage law defines the stiffness terms. When  and  are considered uncoupled, the mixed elastic terms in Eq. (1), K ns and K sn , are set to null values. Many effective traction-separation laws valid for pure loading modes or uncoupled cohesive behaviours are available in literature. Among these, one of the simplest, sometimes also available in commercial FEM codes, is the bilinear (linear softening) relationship. Three points of the traction-separation curve define this model (origin, maximum and final points) whilst the area under the curve represents the critical fracture energy G c (Fig. 1). In the first section of the curve, two I

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