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This is a keynote lecture presented at the 2nd International Workshop on Plasticity, Damage and Fracture of Engineering Materials organized virtually from Middle East Technical University on 18-20 August 2021, Ankara, Turkey http://iwpdf.metu.edu.tr/ Please contact the corresponding author for your comments/questions. Corresponding (presenting) author: Laura De Lorenzis Department of Mechanical and Process Engineering, ETH Zürich, Switzerland [email protected] Authors: Laura De Lorenzis, Tymofiy Gerasimov, Corrado Maurini Abstract: The phase-field modeling approach to fracture has recently attracted a great deal of attention due to its remarkable capability to naturally handle fracture phenomena with arbitrarily complex crack topologies in three dimensions. On one side, the approach can be obtained through the regularization of the variational approach to fracture introduced by Francfort and Marigo in 1998, which is conceptually related to Griffith’s view of fracture; on the other side, it can be constructed as a gradient damage model with some specific properties. In this talk, the speaker highlights two very recent contributions to phase-field modeling of brittle fracture. The first part of the talk focuses on crack nucleation under multiaxial stress states. It is shown that the available energy decompositions, introduced to avoid crack interpenetration and to allow for asymmetric fracture behavior in tension and compression, lead to multiaxial strength surfaces of different but fixed shapes. Thus, once the intrinsic length scale of the phase-field model is tailored to recover the experimental tensile strength, it is not possible to match the experimental compressive or shear strength. The talk introduces a newly proposed energy decomposition that enables the straightforward calibration of a multi-axial failure surface of the Drucker-Prager type. The new decomposition, preserving the variational structure of the model, includes an additional free parameter that can be calibrated based on the experimental ratio of the compressive to the tensile strength (or, if possible, of the shear to the tensile strength), as successfully demonstrated on two data sets taken from the literature. The second part of the talk deals with brittle fracture in anisotropic materials featuring two-fold and four-fold symmetric fracture toughness. For these two classes, the talk introduces two newly proposed variational phase-field models based on the family of regularizations proposed by Focardi, for which Gamma-convergence results hold. Since both models are of second order, as opposed to the previously available fourth-order models for four-fold symmetric fracture toughness, they do not require basis functions of C1-continuity nor mixed variational principles for finite element discretization. For the four-fold symmetric formulation we show that the standard quadratic degradation function is unsuitable and devise a procedure to derive a suitable one. The performance of the new models is assessed via several numerical examples that simulate anisotropic fracture under anti-plane shear loading.

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