PSI - Issue 41

Daniele Gaetano et al. / Procedia Structural Integrity 41 (2022) 439–451 Author name / Structural Integrity Procedia 00 (2019) 000–000

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Both bulk and interface macroscopic elements are equipped with microscopically derived constitutive relations, once a Representative Volume Elements (RVE) is suitably defined, accordingly with the supposed periodic nature of the underlying microstructure. In particular, the same Repeating Unit Cell (RUC), occupying the region Ω m and bounded by the surface Γ m (the subscript m denoting the micro-scale), is used for both bulk and interface behaviors, as shown in Figs. 1b and 1c. In the first case, the RUC problem is driven by an imposed macro-strain ε M , whereas in the latter case, the RUC problem is driven by an imposed macro-crack opening   M u . Without loss of generality, the RVE here considered is made of a continuous matrix (supposed to be damageable) and of a discontinuous phase (playing the role of reinforcement), bonded to each other by damageable interfaces. Reinforcement/matrix debonding is simulated by a Single Interface Model (SIM) while matrix cracking is described by the aforementioned Diffuse Interface Model (DIM). All the damageable interfaces at the microscopic level are indicated with coh m  .

Fig. 1. Cohesive/volumetric multiscale finite element model for periodic microstructures: (a) macroscopic boundary value problem; (b) bulk microscopic boundary value problem; (c) cohesive microscopic boundary value problem.

Under the hypotheses of scale separation and local periodicity, the macroscopic stress and strain fields, denoted as σ M and ε M respectively, can be defined in a standard manner as functions of boundary data of tractions t m and displacements u m , both referring the RUC problem under the action of the so-called periodic boundary conditions: 1 1 , dS dS       t x u n (1) In Eq. (1), x m denotes a material point inside the given RUC, n m is the outer unit normal at the RUC boundary, and the symbol  denotes the dyadic product (the subscript s indicating its symmetric part). The micro-to-macro transition based on the use of Eq. (1) loses its objectivity if applied beyond the appearance of strain localization phenomena. In this case, the localization band, whose width is supposed to be vanishing in a cohesive fracture representation, is described as a zero-thickness interface, equipped with a homogenized traction separation law. In the present work, since Mode-I dominated fracture is considered along predefined crack segments (according to the aforementioned Diffuse Interface Model adopted at the macroscopic scale), the macro-crack direction is assumed to be known in advance, being orthogonal to the applied tensile macro-strain at the RUC level. Therefore, the normal n M to the macroscopic discontinuity is not computed during the macroscopic analysis. The homogenized traction coh M t and separation   M u are extracted by the macroscopic stress and strain, as follows:   eq coh , M M M M s M M m        t n u n (2) m m M m m M m s m m m      