PSI - Issue 33

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

1045

4

1

1

1

  coh       u n , s V dV dS V V    



,   dV







(2)

V

V

c

d

where the macroscopic strain field  can be decomposed in a continuous and a discontinuous part, indicated with c  and d  , respectively, the latter being related to the displacement jump       u u u at the embedded cohesive micro cracks. It is well known that standard bulk homogenization is no longer valid for softening materials, meaning that, by considering distinct Representative Volume Elements (RVEs) with different sizes, i.e. containing different arrays of identic RUCs for periodic microstructures, nonobjective overall stress-strain responses are typically found (De Borst, 1991; Gitman et al., 2007; Kouznetsova et al., 2004; Pegon and Anthoine, 1997). In order to overcome such a drawback, in the spirit of several enhanced homogenization schemes introduced in recent years (Belytschko et al., 2008; Coenen et al., 2012; Comi et al., 2007; Nguyen et al., 2012, 2011; Phu Nguyen et al., 2010; Verhoosel et al., 2010), in this work a continuous/discontinuous homogenization scheme is considered, according to which, upon detection of strain localization, the homogenized macro-continuum is enriched by a discontinuity representing the macroscopic displacement jump equivalent to the displacement jump expected from the RUC (i.e., micro-scale) response (see the right portion of Fig. 1). In the present approach, since the periodicity directions are assumed constant during the RUC fracture simulation and only Mode-I crack propagation is considered, the macro-crack direction is assumed to be known in advance, and coinciding with the periodicity axis which is orthogonal to the applied tensile macro-stress. Hence, the normal n to this discontinuity must not be computed during the analysis. It follows that both the macroscopic cohesive traction coh t and the macroscopic displacement jump   u can be extracted from the microscopic fields by using the following relations: where coh  represents the equivalent macroscopic crack. The resulting homogenized traction-separation law still contains the response of micro-cracks experienced unloading during the macro-crack nucleation, so that only the softening branch of this law must be regarded as the microscopically informed constitutive behavior of the equivalent macroscopic crack. 2.2. Cohesive/volumetric finite element approach for cracking at the micro- and macro-scales In this work, multiple cracking at both the microscopic and macroscopic scales is accounted for by adopting a cohesive/volumetric finite element approach, an inter-element fracture approach firstly introduced by Xu and Needleman (Xu and Needleman, 1994). Such an approach is suitable for the simulation of multiple crack onset and propagation in different types of quasi-brittle materials, including some recent applications to brick masonry structures (Pepe et al., 2019), concrete and reinforced concrete structures (De Maio et al., 2020b, 2019b), as well as fiber reinforced concrete (FRC) structures enhanced with embedded nanomaterials (De Maio et al., 2020a). The resulting numerical model, here termed as Diffuse Interface Model (DIM), is a reliable and versatile framework for the accurate analysis of damage growth in several homogeneous and heterogeneous materials under general loadings, for which crack paths and/or patterns cannot be known a priori. To understand the main theoretical concepts underlying the adopted diffuse interface modeling approach, let us consider the computational domain h  depicted in the left part of Fig. 2, obtained by discretizing a given 2D continuum domain  with three-node finite elements of bulk type. The adopted DIM is obtained by introducing a finite set of zero-thickness cohesive elements, placed along the internal boundaries of the standard finite element mesh, whose nonlinear constitutive behavior is writable as:     coh   u n s dS V   coh coh 1 1 , d s V dV V V               t n u n    , (3)