  In this paper, the Discontinuous Cell Method (DCM) is formulated with the
objective of simulating cohesive fracture propagation and fragmentation in
homogeneous solids without issues relevant to excessive mesh deformation
typical of available Finite Element formulations. DCM discretizes solids by
using the Delaunay triangulation and its associated Voronoi tessellation giving
rise to a system of discrete cells interacting through shared facets. For each
Voronoi cell, the displacement field is approximated on the basis of rigid body
kinematics, which is used to compute a strain vector at the centroid of the
Voronoi facets. Such strain vector is demonstrated to be the projection of the
strain tensor at that location. At the same point stress tractions are computed
through vectorial constitutive equations derived on the basis of classical
continuum tensorial theories. Results of analysis of a cantilever beam are used
to perform convergence studies and comparison with classical finite element
formulations in the elastic regime. Furthermore, cohesive fracture and
fragmentation of homogeneous solids are studied under quasi-static and dynamic
loading conditions. The mesh dependency problem, typically encountered upon
adopting softening constitutive equations, is tackled through the crack band
approach. This study demonstrates the capabilities of DCM by solving multiple
benchmark problems relevant to cohesive crack propagation. The simulations show
that DCM can handle effectively a wide range of problems from the simulation of
a single propagating fracture to crack branching and fragmentation.
