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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3. Description of the proposed two-scale failure model for fiber-reinforced laminates The proposed two-scale procedure, which is based on a continuous/discontinuous homogenization framework used in combination with a diffuse cohesive/volumetric finite element approach, consists in three operational steps, briefly summarized in the following. Please note that the present approach is conceived for application to the failure analysis of fiber-reinforced composite materials within undamaged isotropic behavior and involving prevalent Mode-I crack nucleation and propagation. In the first step, a proper Repeating Unit Cell (RUC) is chosen for the given periodic microstructure, and the undamaged overall moduli tensor is computed via a classical homogenization, which is based on the solution of different linear perturbation problems around the undeformed configuration along pure macro-strain path directions. For 2D problems, only nine components of the undamaged overall moduli tensor (0) ijkl C must be computed, so that only three macro-strain paths are considered, i.e., two uniaxial and a shear path. In the second step, a nonlinear homogenization along a given macro-strain path involving Mode-I fracture propagation is performed, with the aim of computing the damaged overall moduli tensor ( ) ijkl C D . In the present work, a simplified approach is pursued, consisting in assuming a perfect isotropic macro-behavior also in the damaged state. Therefore, the main outcome of this nonlinear homogenization step needed within the proposed numerical procedure is a scalar damage evolution function, which can be easily extracted from the macroscopic stress-strain relation along the given macro-strain path, as will be better clarified in Section 4. It follows that the damaged overall moduli tensor can be expressed as follows:   (0) ( ) 1 ijkl ijkl C D D C   , (7) and, then, stored as a first input material database for the characterization of the nonlinear bulk response. In the third step, the overall nonlinear stress-strain relation is projected to the macro-crack direction, supposed to be known in advance, as explained in Section 2.1. To this end, the two relations provided in Eq. (3) are employed. It follows that the homogenized stress-strain law (i.e.,    ), valid for the bulk, is converted into a homogenized traction-separation law (i.e.   coh  t u ). Since only the softening response is of interest for a damaging macroscopic body exhibiting strain localization, only the portion of the   coh  t u law after the strain localization onset point (coinciding with the detection of the maximum normal cohesive traction in the present simplified approach) is kept in this step. It follows that such a microscopically informed overall traction-separation law is stored as a second input material database for the characterization of the nonlinear interface response. All these three steps are performed in an off-line stage, i.e. prior to the “true” multiscale failure simulations, so that the two resulting databases can be used for several macroscopic nonlinear analyses (within the so-called on-line stage) performed on the same microstructured material, with different macroscopic geometric configurations subjected to different loading and boundary conditions. The entire numerical procedure described in the present section has been implemented within the commercial finite element software COMSOL Multiphysics (Comsol AB, 2019), used in combination with an ad-hoc MATLAB code, developed to automate all the aforementioned sequential tasks. 4. Simulation of transverse cracking and induced delamination in a Fiber Metal Laminate In this section, the proposed two-scale cohesive finite element approach is adopted to investigate the failure behavior of fiber-reinforced laminates subjected to transverse cracking and ply delamination under uniaxial tensile loading conditions. Moreover, all the subsequent predictions are used to assess the reliability of the proposed approach, as well as its numerical accuracy, in terms of global stress-strain response and related crack pattern. All the reported simulations have been conducted assuming a plane-strain 2D reduction, since the out-of-plane behavior of analyzed composites is not the subject of the present investigation. In detail, a Fiber Metal Laminate, made of stacked aluminum and glass/epoxy pre-preg layers, and known as GLARE TM , has been used for the present numerical experiments. This particular Fiber Metal Laminate was widely