PSI - Issue 33

Umberto De Maio et al. / Procedia Structural Integrity 33 (2021) 954–965 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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Nomenclature d

scalar damage function DIM diffuse interface model ECM embedded crack model E

Young’s modulus of the bulk max  normal critical interface stress max  tangential critical interface stress I II , G G modal components of the energy release rate I II , c c G G mode-I and mode-II fracture energies d H  Heaviside function at the discontinuity d  0 0 , n s K K normal and tangential interfacial elastic stiffness parameters mesh L mesh size coh t cohesive traction vector coh coh , n s t t normal and tangential components of the cohesive traction vector u displacement field u displacement jump between the crack faces c u continuous part of the displacement field u d u discontinuous part of the displacement field u s  symmetric part of the gradient operator m  mixed-mode displacement jump 0 m  effective displacement jump at damage onset f m  effective displacement jump at total decohesion , n s   normal and tangential components of the displacement jump  arbitrary continuous function  Poisson’s ratio of the bulk

1. Introduction The cracking analysis in quasi-brittle materials is an ever-evolving research topic which involves several researchers to develop and/or improve fracture models for predicting the global structural response, in terms of load carrying capacity and crack patterns, of concrete structures. Depending on the crack representation, in the technical literature, such models are grouped in smeared and discrete fracture approaches. In particular, the smeared fracture models simulate the damage processes as a progressive loss of the material integrity through constitutive relations in which the mechanical effect of the crack growth is introduced with internal state variables which act on the degradation of the material elastic stiffness involving strain softening to describe the post-peak gradual decline of stress at increasing strain (Cervera and Chiumenti, 2006; Rots et al., 1985). Such models are judged very efficient in terms of numerical accuracy, but usually adopt different regularization techniques to prevent the ill-posedness of the associate BVPs due to the injection of the strain softening in the constitutive response (Červe nka et al., 2018; Comi and Perego, 2001; Fantuzzi et al., 2018). On the contrary, the discrete fracture models simulate the crack as an actual displacement discontinuity within or between the finite elements of a standard computational mesh, adopting a suitable cohesive traction-separation law to describe the crack propagation within the so-called fracture process zone (FPZ) (Lens et al., 2009; Hillerborg et al., 1976; Gálvez et al., 2002; Falk et al., 2001). This approach allows the crack patterns to be accurately predicted but suffers from mesh dependence issues, also requiring, in the case of complex fracture behavior, mesh update procedures. Within a finite element (FE) framework, two main modeling strategies can be found in the technical literature for simulating discrete cracks, i.e., intra-element and inter-element fracture models, according to which the crack can propagate either within the bulk elements or along the boundaries of the employed computational mesh, respectively. The inter-element models simulate the crack nucleation and propagation by means of specific interface elements, equipped with a traction-separation law, inserted between the bulk elements of a standard finite element mesh. In order