PSI - Issue 33

Daniele Gaetano et al. / Procedia Structural Integrity 33 (2021) 1042–1054 Author name / Structural Integrity Procedia 00 (2019) 000–000

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1. Introduction In the last decades, the use of fiber-reinforced composite laminates has become widespread in engineering applications, due to their specific properties and their high versatility, such as the possibility of varying the composite properties by changing the volume fraction and/or the orientation of fibers in a single ply, as well as the ply stacking sequence. As is well-known, fiber reinforced composites may experience several failure mechanisms ranging from transverse cracking to fiber breakage. Other failure mechanisms can be induced also by instabilities phenomena, as previously analyzed by some of the authors (Greco et al., 2021b, 2018a, 2018b). In the technical literature, to simulate such damage events, both phenomenological and micromechanical approaches have been proposed (Herráez et al., 2015; Laws and Dvorak, 1988). Unfortunately, the former approaches are usually too approximate for complex applications of practical interest, where the latter ones are in general very expensive from a computational point of view, especially when highly nonlinear phenomena (such as fracture and/or contact) are considered. To avoid these issues, different approaches have been introduced, many of which are based on multiscale modeling, as those proposed by some of the authors for the prediction of the mechanical behavior of different composite materials and structures (Greco et al., 2020a, 2020b, 2014). The most used multiscale models are based on either hierarchical or semi-concurrent approaches (Belytschko and Song, 2010). The approaches of the first type, known as classical nonlinear homogenization approaches, in which a one-way coupling is established between the micro- and the macro-scale, are very efficient but in general not suitable for handling damage percolation and boundary layer effects. Besides, the approaches of the second type, relying on a two-way coupling between the different scales, are more versatile but suffer from convergence issues when applied to highly nonlinear problems characterized by a severe strain localization. To this end, enhanced computational homogenization techniques have been introduced. In particular, strain-gradient and continuous/discontinuous approaches have been widely used in the literature, by virtue of their capability to introduce a length scale into the overall constitutive response. However, one of their main drawbacks of semi-concurrent approaches consists in their high computational cost. Among all the existing multiscale fracture models, some are based on Linear Elastic Fracture Mechanics (LEFM) and are limited to a single propagating crack, so that they are unable to simulate the progressive micro-cracking coalescence, and therefore, the transition from a diffuse to a localized damage state (Greco et al., 2013; 2014; 2015). In the present work, a more effective two-scale cohesive finite element model is proposed, based on a hybrid continuous/discontinuous hierarchical homogenization, in which the micro- and macro-scales are only one-way coupled, thus assuring a high computational efficiency together with improved capabilities to capture strain localization phenomena. The proposed nonlinear homogenization is performed on a properly identified repeating unit cell (RUC), which is representative of the assumed periodic composite microstructure. The proposed model also assumes perfect separation between micro- and macro-scales, quasi-static loading conditions, small deformations, damageable fiber/matrix interfaces and breakable matrix modeled as a collection of cohesive elements embedded in a continuous phase. Thus, considered periodic boundary conditions imposed on the unit cell boundaries, a hierarchical homogenization scheme is applied with the aim of deriving the overall nonlinear response of the composite. To overcome the well-known ill-posedness of the homogenized macro-continuum problem in the softening regime, a further homogenization is performed at the interface level as a post-processing step. In other words, the discontinuous part of the numerically derived overall constitutive response is obtained after extracting the portion of the loading curve after the appearance of a macroscopic localization band. Finally, a homogenized traction-separation law is derived, after suitably defining a macroscopic cohesive traction vector and a macroscopic displacement jump. The main advantage of this approach is that a micromechanically-based traction-separation law is derived through off-line computations in a very efficient manner. In the present multiscale approach, multiple cracking at both microscopic and macroscopic scales is accounted for by adopting a cohesive/volumetric finite element approach, also used by some of the authors for the failure analysis of reinforced concrete elements (De Maio et al., 2019a). The layout of this paper is as follows. Section 2 contains the theoretical background of both cohesive/volumetric finite element approach and continuous/discontinuous nonlinear homogenization, which represent the main concepts needed to understand the numerical model developed in this work. Section 3 illustrates the fundamental steps of the proposed multiscale methodology, together with some computational details. In Section 4, the numerical results