PSI - Issue 66

Domentico Ammendolea et al. / Procedia Structural Integrity 66 (2024) 350–361 Author name / Structural Integrity Procedia 00 (2025) 000–000

355

6

2.3 Activation criterion The proposed methodology progressively activates finer regions within the computational domain during the numerical simulation as damage mechanisms develop in the coarse regions. This activation occurs when conditions defined by an appropriate activation criterion are satisfied. The definition of the activation criterion starts with the definition of the first failure surface for the Repeating Cell of the masonry adopted for describing the mechanical behavior of the masonry inside coarser regions (see Fig. 1-b). Such a failure surface identifies the set of macroscopic strain states associated with the onset of microscopic cracks inside the mortar joints comprised within the RC, thus representing an elastic limit domain for the RC. The first failure surface is constructed by first analyzing the RC subjected to a finite number of loading conditions (expressed in terms of macroscopic strain) and defined as follows:

1 sin cos sin sin cos               2 1 2 x y

(12)

1

xy

where,  is a loading multiplier that assesses the intensity of the imposed macrostrain, whereas the angles ( 1 2 ,   ) define an angular coordinate system describing the loading direction. In particular, such angles range 1 0 90     and 2 180 180      . The critical load multiplier  c, describing the crack onset condition for each loading direction defined by the angles ( 1 2 ,   ), is evaluated due to the linearity of the problem through a post-processing procedure using the following expression:

0

 

1 eq eq

(13)

c 



being, 0 1 eq  the value of the equivalent strain per unit  evaluated in the numerical analyses using Eq. (7). Each value of the critical load multiplier corresponds to a failure point of the failure domain in the strain space. The complete surface is obtained by interpolating the discrete failure points. The activation criterion used in the proposed model to refine the region of the computational domain affected by damage mechanisms is defined as follows: 0 eq t E    and

1 2 ( , )        , c i 1 2 ( , ) i

1,..,

macro

i

n



(14)

2

2

2

where n macro is the number of macro-elements inside the computational domain, y xy        represents the load multiplier of the i -th macro element, and  is a scale factor of the failure surface that enlarges or reduces the elastic domain of RC. 3 Numerical implementation The proposed adaptive multiscale method has been implemented within the commercially available Finite Element software COMSOL Multiphysics (COMSOL (2018)), which offers the opportunity to develop programming scripts through the built-in developer environment (Lonetti and Pascuzzo (2016), Greco et al. (2021), Pascuzzo et al. (2022a), Pascuzzo et al. (2022b), Ammendolea et al. (2023a), Ammendolea et al. (2023b), Ammendolea et al. (2024)). In particular, such a function has been used to create a homemade algorithm that manages the adaptive nature of the proposed multiscale model. The steps followed by the algorithm are illustrated in Fig. 2. i x