Abstract
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. Thus, an accurate estimation of the material parameters enables the precise determination of the material response in different stages, particularly for the post-yielding regime, where crack initiation and propagation take place. In this work, we develop a 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. A step-wise Bayesian inversion method is proposed to determine the posterior density of the material unknowns for a ductile phase-field fracture process. 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 hat{R}{-}convergence tool. The resulting framework is algorithmically described in detail and substantiated with numerical examples.
Highlights
Fracture in the form of evolving crack surfaces in ductile solid materials exhibits dominant plastic deformation
Another important observation is that the crack phase-field in Model 2 (M2) and Model 3 (M3) is more diffuse than Model 1 (M1)
Three common Markov chain Monte Carlo (MCMC) methods, namely the MH algorithm, the delayed rejection adaptive Metropolis (DRAM) algorithm, and ensemble-Kalman filter (EKF)-MCMC have been used to estimate the effective parameters in ductile fracture
Summary
Fracture in the form of evolving crack surfaces in ductile solid materials exhibits dominant plastic deformation. Typical examples are theories of gradient-enhanced damage [3,4,5,6], phase-field models [7,8,9], and strain gradient plasticity [10,11,12] Such models incorporate non-local effects based on length scales, which reflect properties of the material microstructure size with respect to the macro-structure size. Variational phase-field modeling is considered, which is a regularized approach to fracture with a strong capability to simulate complex failure processes This includes crack initiation ( in the absence of a crack tip singularity) [16,17,18], propagation, coalescence, and branching, without additional ad-hoc criteria [8,19,20]. A Bayesian estimation model (as an inverse model) is here used for the ductile fracture problem to provide accurate knowledge regarding the effective mechanical parameters
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