  The prediction of crack initiation and propagation in ductile failure
processes are challenging tasks for the design and fabrication of metallic
materials and structures on a large scale. Numerical aspects of ductile failure
dictate a sub-optimal calibration of plasticity- and fracture-related
parameters for a large number of material properties. These parameters enter
the system of partial differential equations as a forward model. In this work,
we develop a step-wise Bayesian inversion framework for ductile fracture to
provide accurate knowledge regarding the effective mechanical parameters. To
this end, synthetic and experimental observations are used to estimate the
posterior density of the unknowns. To model the ductile failure behavior of
solid materials, we rely on the phase-field approach to fracture, for which we
present a unified formulation that allows recovering different models on a
variational basis. In the variational framework, incremental minimization
principles for a class of gradient-type dissipative materials are used to
derive the governing equations. The overall formulation is revisited and
extended to the case of anisotropic ductile fracture. Three different models
are subsequently recovered by certain choices of parameters and constitutive
functions, which are later assessed through Bayesian inversion techniques. To
estimate the posterior density function of ductile material parameters, three
common Markov chain Monte Carlo (MCMC) techniques are employed: (i) the
Metropolis-Hastings algorithm, (ii) delayed-rejection adaptive Metropolis, and
(iii) ensemble Kalman filter combined with MCMC. To examine the computational
efficiency of the MCMC methods, we employ the R-convergence tool.
