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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The present multiscale model, after being developed within the finite element simulation environment COMSOL Multiphysics, is validated by performing complete failure analyses on a pre-cracked porous beam experiencing matrix cracking, and by comparing the related numerical results with those arising from a direct simulation performed on a fully meshed specimen, here regarded as a reference analysis. 2. Theoretical background 2.1. Diffuse Interface Model (DIM) at the macroscopic scale The present multiscale strategy is intended to be used in synergy with a Diffuse Interface Model (DIM), already applied by some of the authors to simulate microscopic and macroscopic cracking in various structural applications (see, for instance, Greco et al. (2020), Gaetano et al. (2022a), Gaetano et al. (2022b), Greco et al. (2022), Pascuzzo et al. (2022b), Greco et al. (2024)), and here used to represent the equivalent cohesive/volumetric assembly as the result of the continuous/discontinuous homogenization process. This section aims to recall the key theoretical concepts on which the Cohesive/Volumetric Finite Element Method (CVFEM) is based. The computational domain obtained via a standard bulk finite element discretization of a given 2D macro-scale continuous domain  (the overbar symbol denoting a macroscopic quantity) is enriched by inserting cohesive interfaces coh  along all the internal mesh boundaries, as shown in Fig. 1.

Fig. 1. Schematic representation of the Cohesive/Volumetric Finite Element Method (CVFEM).

Under the assumptions of small displacements and quasi-static loads (volume forces f applied in t  ), the equilibrium problem of this composite mechanical system can be expressed in a variational manner as follows: find the kinematically admissible macroscopic displacement u such that: coh \   and surface forces t applied on the Neumann boundary









: d 

[[ ]] d  u

d

d

t

 f u 

 t u

 

 



 



(1)

coh

\  

\  





coh

coh

coh

t

for all virtual macroscopic displacements  u . In Eq. (1),  is the macroscopic stress tensor, s  u  is the macroscopic strain tensor ( s  denoting the symmetric part of the spatial gradient), coh t is the macroscopic interface traction vector, [[ ]] u is the macroscopic displacement jump, and  indicates the usual variational operator. It is worth noting that a nonstandard term appears in the left-hand side of Eq. (1), coinciding with the virtual work of the cohesive interface tractions. Furthermore, it is assumed that both bulk and cohesive finite elements of the macroscale mesh have nonlinear constitutive behaviors. In particular, bulk finite elements behave in pure hardening regime according to a generally anisotropic damage law, incorporating the loss in material stiffness due to microcrack nucleation and propagation. Such a constitutive law can be expressed in a classical Continuum Damage Mechanics (CDM) setting as :  C   , where C is the secant elastic moduli tensor, which depends on the internal damage state. In the absence of damage, C turns to be coincident with the (initial) undamaged moduli tensor 0 C . Since it has been pointed out by Cauvin and Testa (1999) that, in the general case, the internal damage state can be represented by a fourth-order tensor, in the present work, the secant elastic moduli tensor C itself has been chosen as the tensorial damage variable.