PSI - Issue 66

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

324

5

These macroscopic quantities are well defined only prior to the occurrence of the macroscopic strain localization, after which the resulting macroscopic boundary value problem turns out to be ill-posed, as pointed out by Gitman et al. (2007). In this work, the macroscopic strain localization is detected through a rigorous localization analysis based on the singularity condition of the acoustic tensor:     det det 0 t     Q n C n (5) where t C is the homogenized tangent moduli tensor, whereas n denotes the potential macrocrack orientation. Once macrostrain localization is detected for a given macrocrack orientation n , it is possible to upscale the stress and strain fields at the RUC level to a general macroscopic Traction-Separation Law (TSL) of the anisotropic time, i.e.,   coh coh [[ ]],  t t u n . Specifically, the cohesive traction coh t can be simply found as the projection of the macrostress  , computed via the first of Eq. (4), on the macroscopic interface having normal n (which is a priori known from the given mesh at the macroscale, according to the adopted DIM approach). Instead, a unique definition for the macroscopic separation [[ ]] u is absent in literature, to the best of the authors’ knowledge. In the present work, [[ ]] u is computed from the discontinuous part d  of the macrostrain tensor, defined coherently with the regularized strong discontinuity kinematics proposed by Oliver and Huespe (2004):   1 [[ ]] d d s h    u n   (6) where d  is the band of the macroscopic discontinuity, h is the corresponding bandwidth, and   d   denotes the collocation function on d  . Moreover, following Nguyen et al. (2011), the bandwidth h can be computed as:



h

(7)





d

where  is the area of the considered 2D RUC, and d  is the length of the macrocrack embedded in the RUC. The latter can be estimated in a simplified manner in the case of rectangular RUC, by using the following formula proposed by Turteltaub et al. (2018):

 

 

l

l

min   

,

1

2 

(8)

d

2  e n e n 1 

where 1 l and 2 l are the RUC dimensions along the principal directions 1 e and 2 e , respectively. 3. Description of the proposed cohesive/bulk homogenization approach In this section, the proposed cohesive/bulk homogenization approach for 2D periodic structures is described. The adopted numerical implementation involves a two-stage process. The first stage, called offline stage, computes the overall constitutive behavior of the bulk, by means of a standard nonlinear homogenization performed on a given Repeating Unit Cell (RUC), being representative of the microstructure. The offline stage involves the following steps:  Defining the 3D macrostrain space, generated by the three in-plane strain components 11 22 12 , ,    , expressed with respect to the spherical coordinate system   , , t   , where the angular coordinates  and  represent the macrostrain direction and t is the monotonically increasing loading parameter along this direction.