  While crack nucleation and propagation in the brittle or quasi-brittle regime
can be predicted via variational or material-force-based phase field fracture
models, these models often assume that the underlying elastic response of the
material is non-polar and yet a length scale parameter must be introduced to
enable the sharp cracks represented by a regularized implicit function.
However, many materials with internal microstructures that contain surface
tension, micro-cracks, micro-fracture, inclusion, cavity or those of
particulate nature often exhibit size-dependent behaviors in both the
path-independent and path-dependent regimes. This paper is intended to
introduce a unified treatment that captures the size effect of the materials in
both elastic and damaged states. By introducing a cohesive micropolar phase
field fracture theory, along with the computational model and validation
exercises, we explore the interacting size-dependent elastic deformation and
fracture mechanisms exhibits in materials of complex microstructures. To
achieve this goal, we introduce the distinctive degradation functions of the
force-stress-strain and couple-stress-micro-rotation energy-conjugated pairs
for a given regularization profile such that the macroscopic size-dependent
responses of the micropolar continua is insensitive to the length scale
parameter of the regularized interface. Then, we apply the variational
principle to derive governing equations from the micropolar stored energy and
dissipative functionals. Numerical examples are introduced to demonstrate the
proper way to identify material parameters and the capacity of the new
formulation to simulate complex crack patterns in the quasi-static regime.
