Issue 29

R. Serpieri et alii, Frattura ed Integrità Strutturale, 29 (2014) 284-292; DOI: 10.3221/IGF-ESIS.29.24 286 C OHESIVE - ZONE MODEL DERIVATION Multiscale formulation n this section the fundamental ideas behind the frictional cohesive-zone model proposed in Refs. [9, 10] are reported and discussed. For all the details of the derivation of the model and its implementation within an incremental nonlinear finite-element formulation, the reader is referred to the above references. An interface  is considered between two parts 1  and 2  f a body  . In this paper we will focus on two-dimensional problems whereby  is represented by a line. The model considers two scales: a macroscale where the interface is assumed to be smooth (Fig. 1(a)) and a microscale where the asperities of the fracture surface are considered (Fig. 1(b)). The terms microscale and macroscale are here used with aim to distinguish the smaller from the larger scale. In the context of the finite-element method, which is considered here, this implies the use of smooth interface elements to model the macroscale, whereby the details of the asperities do not have to be captured by the spatial discretization. Instead, at each integration point of each interface element, the cohesive law, which relates the relative displacement vector s to the interface stress σ , is determined by resolving a microscale problem for a Representative Interface Area (RIA). Figure 1 : Multi-scale approach linking a smooth interface at the (a) macro-scale level to the (b) RIA characterized by a periodic arrangement of microplanes. The RIA is assumed to be large enough to capture the details of the asperities. If the actual microscale is periodic, the repeating unit of the interface defines the RIA. In case it is not periodic or not perfectly periodic, the RIA can be taken large enough and with a geometry rich enough so that the response of the RIA can be considered statistically representative of the response of an ‘infinitesimally small’ area at the macroscale. In turn, this implies the assumption of ‘separation of scales’, which is typically required in multiscale homogenization [19]. While the approach could be theoretically implemented by representing the asperities through a curved profile, it is assumed that such profile is defined by a finite number  N of microplanes. A further assumption that is made in the model is that the two sides of the interface profile within the RIA are infinitely rigid. In other words, the deformation of the asperities, as well as their damage, wear and fracture, are not considered. This means that a single vector s characterizes the relative displacement between the two sides of the RIA. On the k th microplane a local reference system       , n t k k is defined,   n k and   t k representing the local normal and tangential directions, respectively. Hence, the relative displacement vector is decomposed on the k th microplane into its local mode-I component,     s n k kI s , and mode-II component,     s t k kII s . Thermodynamical formulation of the model On each microplane, the stress is obtained implementing the model developed by Alfano and Sacco [7]. The main idea behind such model is that an infinitesimal area dA of microplane can be decomposed into an undamaged part, whose I