Bayesian inversion for unified ductile phase-field fracture

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 $$\hat{R}{-}convergence$$ R ^ - c o n v e r g e n c e tool. The resulting framework is algorithmically described in detail and substantiated with numerical examples.

Experimental and Numerical Evaluations of Localized Stress Relaxation for Vulcanized Rubber

“…These findings provide some guidelines for engineers to (i) calculate the localized stress response of rubber during rapid application of compressive strain and stress relaxation; and (ii) experimentally and numerically evaluate the influence of applied strain, geometry, and frictional contact on the stress relaxation behavior of rubber and viscoelastic polymeric components. Moreover, the present findings can be integrated into research works regarding the mechanical system, such as a Bayesian approach (i.e., a probabilistic method used to estimate the unknown parameters, and subsequently predicting the characteristic output), to estimate the material parameters for propagating fractures in elastic solids [ 22 , 23 ].…”

Influence of Moisture Content and Wet Environment on the Fatigue Behaviour of High-Strength Concrete

“…Hereby, is a non-negative degradation function due to the fatigue load, which renders the dissipated energy for a path-dependent quantity. The elastic energy density is formulated through the effective strain energy density with a monotonically decreasing quadratic degradation function (see [ 29 ]). Here, are denoted as Lamé constants.…”

“…The Macaulay bracket is used in this equation. For an alternative description of crack irreversibility, see [ 29 ]. For recent works on the phase-field modelling of fatigue failure behaviour, the interested reader is referred to [ 37 , 38 , 39 , 40 , 41 ].…”

Tie Rod-Equivalent Non-Linear Constitutive Law for Uniformly Loaded Cables

“…In the Table, density, elastic modulus, ultimate strength, and ratios between the ultimate strength and the elastic modulus are given for relevant fibers. The mechanical properties of the cable materials reported in Table 1 have been derived summarizing data coming from several sources: the interested reader can specifically refer not only to the relevant literature [30,37], but also to some web sources [38,39], which provide links to cable material producers, too. Looking at the Table, it clearly results that for most materials f t /E < 1%; for polybenzoxazole (PBO), and carbon fibers is f t /E = 1.5 − 2%; for aramid fibers, and liquid crystal aromatic polyester (LCP) fibers f t /E = 2.5 − 4.5%; while nylon and polyester fibers exhibit higher values.…”

“…It must be remarked that the proposed approach can be also a powerful tool to obtain reliable prior information, in view of the multifaceted implementation of advanced Bayesian updating techniques for the identification of mechanical parameters [30][31][32][33][34].…”