PSI - Issue 41

Daniele Gaetano et al. / Procedia Structural Integrity 41 (2022) 439–451 Author name / Structural Integrity Procedia 00 (2019) 000–000

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homogenization (Massart et al. (2007), Belytschko et al. (2008), Nguyen et al. (2010), Verhoosel et al. (2010), Nguyen et al. (2011a), Nguyen et al. (2011b), Nguyen et al. (2012a), Nguyen et al. (2012b), Coenen et al. (2012), Bosco et al. (2015), Svenning et al. (2019)) schemes. All these approaches share as a feature the incorporation of a length scale into the macroscopic constitutive response of the equivalent homogenized material, thus inducing a nonlocal effect in the continuum description (see Luciano and Willis (2001), Tuna et al. (2020), Barretta et al. (2020) for additional details about the notion of nonlocality in the constitutive mechanical response). In particular, continuous/discontinuous homogenization approaches have been widely adopted within the so-called semi-concurrent multiscale methods, which have been proved to be very accurate for nonlinear composites exhibiting strain softening. All these methods share the key idea of establishing a “two-way” weak coupling between micro- and macro-variables, in the spirit of the well-know FE 2 method, as introduced in a pioneering work by Feyel (2003), but are often computationally too costly, if applied to real-life structural applications. In this work, a more efficient continuous/discontinuous hierarchical multiscale scheme is proposed for periodic microstructures, exploiting a general hybrid cohesive/volumetric approach in conjunction with a hybrid nonlinear homogenization approach. The proposed homogenization scheme is performed on a suitably identified Repeating Unit Cell (RUC), which is representative of the given microstructure. The main ingredient of the proposed multiscale model is a Diffuse Interface Model for representing the softening constitutive behavior at both the micro- and macro-scales, already proposed by some of the authors (please, see De Maio et al. (2019b), De Maio et al. (2020b) for additional details). In particular, the proposed multiscale approach is conceived to deal with the frequent case of Mode-I dominated quasi-brittle fracture, which is typical of globally isotropic heterogeneous materials (or, even, of transversely isotropic ones, as unidirectional fiber-reinforced composites). Under this particular assumption, a nonlinear homogenization step is performed to derive a complete homogenized constitutive response of the given microstructure, being subjected to a uniaxial tensile macro-strain path inducing pure Mode-I fracture. As post-processing outcomes, two independent homogenized constitutive responses are extracted: (i) a homogenized scalar damage evolution law, valid up to the occurrence of strain localization and derived under the assumption of isotropic damage behavior in the hardening regime; and (ii) a homogenized traction-separation law, which is able to take into account the softening behavior in an objective manner. The main advantage of this multiscale model relies in the possibility to microscopically derive the mechanical response of a given heterogeneous material in a very efficient manner, by performing off-line computations. 2. Description of the hybrid cohesive/volumetric multiscale finite element model In this section, the proposed hybrid continuous/discontinuous multiscale model is presented from both theoretical and numerical points of view. Such a model is based on a cohesive/volumetric finite element approach, already used by some of the authors in the framework of the failure analysis of many structures and materials (see, for instance, De Maio et al. (2019a), De Maio et al. (2020a), De Maio et al. (2021), Pascuzzo et al. (2021), De Maio et al. (2022), Gaetano et al. (2022), Greco et al. (2022)), here adopted at both microscopic and macroscopic scales, by assuming, as usual, small-strain kinematics and quasi-static loading conditions. Such a fracture approach, which relies on a generalization of classical cohesive zone models (Gálvez et al. (1998), Planas et al. (2003), Pascuzzo et al. (2020)), has the great advantage of avoiding complex remeshing operations and/or sophisticated moving mesh methodologies useful to accommodate the actual crack geometry representation in the finite element analyses (see, for instance, Ammendolea et al. (2021), Greco et al. (2021b)). 2.1. Theoretical formulation Let us consider a two-dimensional macroscopic solid Ω M with its boundary denoted by Γ M , the subscript M denoting the macro-scale (see Fig. 1a). The Neumann boundary N M M    is subjected to externally applied tractions M t , whereas the Dirichlet boundary \ D N M M M     is subjected to prescribed displacements M u . This solid is supposed to be susceptible to fracture phenomena, here represented by a Diffuse Interface Model (DIM), according to which damageable interfaces, collectively denoted as coh M  , are placed along all the internal mesh boundaries, in the spirit of well-known cohesive/volumetric finite element approaches (see, for instance, Blal et al. (2012)).