PSI - Issue 33

Daniele Gaetano et al. / Procedia Structural Integrity 33 (2021) 1042–1054 Author name / Structural Integrity Procedia 00 (2019) 000–000

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obtained via the proposed methodology are reported, together with suitable comparisons with direct numerical simulations, introduced for validation purposes. Finally, in Section 5, a discussion of the main numerical outcomes as well as some concluding remarks are presented. 2. Theoretical background In this work, a novel two-scale finite element model for the failure analysis of fiber-reinforced laminates has been proposed. This model relies on the combination of two main ingredients, i.e., a nonlinear continuous/discontinuous homogenization strategy and a cohesive/volumetric finite element approach. The theoretical concepts underlying these ingredients will be presented as follows. 2.1. Continuous/discontinuous nonlinear homogenization for periodic microstructures With reference to the left part of Fig. 1, let us consider a cracking heterogeneous body characterized by a periodic microstructure, being subjected to body forces f in the volume and to surface tractions t on the Neumann boundary and fixed on the Dirichlet one. Through standard periodic homogenization arguments, after a Repeating Unit Cell (RUC) is properly identified, the given body can be replaced by an equivalent homogenized body. The RUC can incorporate an arbitrary number of micro-cracks, collectively denoted by coh  , which can coalesce to form, after strain localization is occurred, one or more micro-cracks, denoted by coh  .

Fig. 1. Identification of the repeating unit cells (RUCs) of a periodic heterogeneous medium containing cohesive cracks before and after strain localization and representation of the related equivalent homogenized medium.

Under the assumption of perfect scale separation and according to the classical homogenization theory, the macroscopic stress and strain fields, indicated with  and  , can be obtained as averages of the microscopic counterparts and, specifically, can be defined in terms of boundary data of tractions t and displacement u , respectively, both referring to the RUC (i.e. microscopic) boundary value problem subjected to the so-called periodic boundary conditions:

1

1





(1)

dS u n . 

,

 t x

dS









s

V

V

V

V





In Eq. (1), x represents a material point of the given RUC, n is the outward normal at V  x , V denotes the measure of the volume occupied by the RUC, and  stands for the dyadic product (the subscript s denoting its symmetric part). After manipulating Eq. (1), the following expressions can be easily derived: