Issue 29

R. Dimitri et alii, Frattura ed Integrità Strutturale, 29 (2014) 266-283; DOI: 10.3221/IGF-ESIS.29.23 267 directly postulated in an ad hoc manner, which may lead to unphysical dissipation behavior. A comparative assessment of pure-mode CZMs can be found, for example, in [3-6] who investigated the influence of the shape of pure-mode relationships on the macroscopic mechanical response of material systems. A consistency check was performed by van den Bosch et al. [7] for the exponential model by Xu and Needleman [8], which was shown to describe unrealistically the mixed mode decohesion unless the energy release rates of both modes were assumed to be equal. This assumption, however, is not generally realistic, as already observed experimentally. Thereby, an adjusted non-potential-based exponential model was proposed through a modification of the coupling between normal and shear modes, to correct the unphysical behavioral features of the original model in the description of mixed-mode decohesion. However, several other mixed-mode models frequently used in numerical applications have never been analyzed in a similar fashion. A parametric analysis on the effect of the coupling parameters on stress distributions and energy dissipation can be useful to evaluate possible physical inconsistencies of a model, such as local abnormalities in the coupled elastic or softening mechanical response of the interface, incomplete dissipation of the fracture energy during decohesion, residual load-carrying capacity in normal or tangential traction after complete failure, etc. Advantages and/or shortcomings of each cohesive model, if explicitly assessed, may also justify the adoption of one model or another for numerical implementations. In this paper we analyze the improved Xu and Needleman (XN) mixed-mode exponential model proposed by [7], a model recently proposed by McGarry et al. [9], as well as two widely used linear mixed-mode CZMs (Högberg [10], Camanho et al. [11]). The basic question is whether these models realistically predict a mixed-mode debonding process, and whether they are consistent in both stress and energy terms. A parametric analysis on the effect of the coupling parameters on stress distributions and energy dissipation is performed in order to evaluate physical inconsistencies. Analytical predictions are also compared with results of numerical finite element models, where the interface is described with zero-thickness contact elements. A node-to-segment strategy incorporating decohesion and contact within a unified framework is here adopted [12]. A simple patch test is analyzed for the numerical prediction of mixed-mode interface debonding. The formulations are implemented and tested using the finite element code FEAP. In the second part of the paper, a new thermodynamically consistent mixed-mode CZM is proposed based on a modification of the XN improved model [8] proposed by van den Bosch [7]. Based on a predefined Helmoltz energy, the interface model is derived by applying the Coleman and Noll procedure, in accordance with the second law of thermodynamics, whereby the inelastic nature of the decohesion process is accounted for by means of damage variables. The model accounts monolithically for loading and unloading conditions, as well as for decohesion and contact. In order to verify the computational implementation, simple modes I and II are first introduced and mixed-mode tests are then employed to examine the fracture process under mixed-mode conditions. In the last case, a simple patch test and a matrix- fiber debonding test are provided as computational examples. S TATE OF THE ART REVIEW - COHESIVE ZONE MODELS his section describes the mixed-mode coupled cohesive laws chosen for the analytical and numerical comparative investigation. As already mentioned, tractions in the normal and tangential directions are linked to each other by means of a coupling parameter and/or an effective opening displacement. Traction and shear relate the normal and the tangential stresses, p N and p T respectively, with both the relative normal and tangential displacements, g N and g T , whereas the main parameters defining the traction-separation laws affect the initiation and steady-state propagation behavior. We start our study analyzing the improved XN exponential model [7] (henceforth denoted as CZM1), which has been already verified to be physically consistent, and herein considered as benchmark law for our comparative assessments between CZMs. This model is not derived from a potential and coincides with the original model in [8] only when the ratio between the tangential and the normal work of separation, denoted as q , is equal to one, with independent scaling factors applied to the normal and tangential equations to account for differences in the mode I and mode II interface strengths. The analytical expression of the interfacial stresses in the normal and tangential directions can be written in the following form 2 2 max max max max exp(1) exp exp N N N T N N N T p g g g p g g g                      (1) T