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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where Γ eq is the equivalent localized crack, being oriented orthogonally to the applied macro-strain direction. The homogenized traction-separation law, as computed via the application of Eq. (2), still contains the hardening response, associated with the early crack propagation steps prior to strain localization. In order to exclude this contribution and, thus, to consider only the softening branch of this law, an energy-based split of the total homogenized traction-separation law is proposed (shown in Fig. 2a), leading to a proper definition of the localized macroscopic crack opening displacement loc M   u , as depicted in Fig. 2b.

Fig. 2. Derivation of the homogenized traction-separation law: (a) energy-based split of the total homogenized traction-separation law; (b) purely softening portion of the homogenized traction-separation law.

2.2. Numerical implementation The numerical implementation of the proposed hybrid cohesive/volumetric multiscale finite element model includes three operational steps:  Derivation of the undamaged moduli tensor via the solution of different linear perturbation problems around the undeformed configuration along pure macro-strain path directions.  Derivation of the homogenized damage evolution law, starting from the homogenized stress-strain relations obtained through a nonlinear bulk homogenization step along a given macro-strain path involving pure Mode-I crack propagation, according to the theoretical framework explained in Section 2.1.  Derivation of the homogenized traction-separation law for the embedded cohesive interfaces at the macroscopic scale, obtained by the projection of the nonlinear stress-strain relation derived in the previous step, after splitting it into a purely hardening and softening portions according to aforementioned energy-based technique, as shown in Fig. 2. It is worth noting that all these three steps are performed within an off-line computational stage, so that the overall damage evolution and traction-separation databases (i.e., the outcomes of the last two steps) can be used as material inputs for the subsequent on-line computational stages, to be applied for the “true” macroscopic nonlinear analyses, involving different geometries and boundary conditions. Finally, the numerical implementation of the entire procedure has been performed within the commercial finite element environment COMSOL Multiphysics, used in combination with ad-hoc codes written in MATLAB language to achieve advanced automation capabilities and to speed up all these sequential tasks.