PSI - Issue 66

Domenico Ammendolea et al. / Procedia Structural Integrity 66 (2024) 320–330 Author name / Structural Integrity Procedia 00 (2025) 000–000

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 Identifying a finite number of radial macrostrain path directions to discretize the macrostrain space.  Performing a nonlinear homogenization along each direction to derive a complete stress-strain database.  Interpolating the stress-strain points by means if linear shape functions.  Performing a localization analysis to extract the stress-strain response prior to strain localization. The second stage, called online stage, calculates the homogenized response of all the cohesive interfaces during the macroscale simulation, starting from the softening portion of the homogenized stress-strain law computed in the previous stage. The online stage is divided into the following steps, to be performed for each interface element and for each loading step of the macroscopic problem:  Computing the discontinuous part d  of the macrostrain tensor by using Eq. (6).  Computing the continuous part c  of the macrostrain tensor as the average macrostrain of the two bulk elements adjacent to the considered interface element.  Computing the macrostrain tensor c d      at the interface level.  Computing the macrostrain stress  by using the interpolated stress-strain database derived in the offline stage.  Computing the macroscopic traction coh   t n  . 4. Numerical applications The present numerical application concerns a pre-cracked 2D porous beam subjected to a Three-Point Bending (TPB) test involving pure Mode-I macroscopic fracture under plane-stress reduction, as depicted in Fig. 2(a). The underlying microstructure is made of a continuous matrix with embedded circular holes arranged in a rectangular array, whose Repeating Unit Cell (RUC) chosen for homogenization purposes is shown in Fig. 2(b).

Fig. 2. (a) Geometry and boundary conditions for the TPB test on a porous beam; (b) Repeating Unit Cell (RUC) adopted for periodic homogenization purposes. The governing geometric dimensions of this microstructure are the hole volume fraction 0.20   and the hole diameter 10 mm d  . The matrix has the following elastic bulk parameters, Young’s modulus 25 GPa E  and Poisson’s ratio 0.2   . Moreover, it has the isotropic damage model with implicit gradient regularization presented in Section 2.2 and characterized by the parameters: 5 0 3 10     , 0.999   , 3000   , and 2 3.5 mm c  . These values for the elastic and inelastic parameters are taken from Nguyen et al. (2011). The RUC depicted in Fig. 2(b) has been subjected to several independent radial macrostrain paths characterized by monotonically increasing values of the equivalent macrostrain 2 2 2 0.5 eq 11 22 12 ( )        . Then, after applying the nonlinear bulk homogenization procedure belonging to the offline computational stage presented in Section 3, complete macro-stress/strain curves have been derived for this RUC. Finally, the online computational stage of Section 3 has been validated by solving the macroscopic fracture problem for the heterogeneous beam modeled as a cohesive/volumetric finite element assembly, whose embedded interfaces are equipped with a Traction-Separation Law (TSL) to be found “on-the-fly” during the macroscale simulation. To investigate the influence of mesh topology on the overall fracture response of the considered porous structure obtained via the proposed multiscale approach, two different spatial discretizations have been considered inside a