  A phase field model for ductile fracture considering Hencky strain and finite
J2 plasticity is presented using the nonlocal operator method. A variational
derivation of J2 plasticity at finite strain with a phase field model is
performed. The method includes a logarithmic strain tensor and an exponential
mapping in the plasticity evolution. A spectral decomposition based algorithm
for computing the first and second order derivatives of the composite matrix
function is implemented. A consistent tangential stiffness matrix is derived
and used in Newton-Raphson iterations. Several numerical examples are performed
to validate the method, including notched single-edged plates with brittle
fracture or ductile fracture and necking of a bar with/without phase field
model.
