  This paper outlines a rigorous variational-based multilevel Global-Local
formulation for ductile fracture. Here, a phase-field formulation is used to
resolve failure mechanisms by regularizing the sharp crack topology on the
local state. The coupling of plasticity to the crack phase-field is realized by
a constitutive work density function, which is characterized through a degraded
stored elastic energy and the accumulated dissipated energy due to plasticity
and damage. Two different Global-Local approaches based on the idea of
multiplicative Schwarz' alternating method are proposed: (i) A global
constitutive model with an elastic-plastic behavior is first proposed, while it
is enhanced with a single local domain, which, in turn, describes an
elastic-plastic fracturing response. (ii) The main objective of the second
model is to introduce an adoption of the Global-Local approach toward the
multilevel local setting. To this end, an elastic-plastic global constitutive
model is augmented with two distinct local domains; in which, the first local
domain behaves as an elastic-plastic material and the next local domain is
modeled due to the fracture state. To further reduce the computational cost,
predictor-corrector adaptivity within Global-Local concept is introduced. An
adaptive scheme is devised through the evolution of the effective global
plastic flow (for only elastic-plastic adaptivity), and through the evolution
of the local crack phase-field state (for only fracture adaptivity). Thus, two
local domains are dynamically updated during the computation, resulting with
two-way adaptivity procedure. The overall response of the Global-Local approach
in terms of accuracy/robustness and efficiency is verified using single-scale
problems. The resulting framework is algorithmically described in detail and
substantiated with numerical examples.
