PSI - Issue 39

Andrea Pranno et al. / Procedia Structural Integrity 39 (2022) 688–699 Author name / Structural Integrity Procedia 00 (2019) 000–000

689

2

Nomenclature d

scalar damage function Young’s modulus of the bulk

E

, n s K K 0 0 normal and tangential interfacial elastic stiffness parameters mesh L mesh size  dimensionless normal stiffness parameter , n s t t normal and tangential components of the cohesive traction vector m  mixed-mode displacement jump , n s   normal and tangential components of the displacement jump  Poisson’s ratio of the bulk elements    displacement jump between the crack faces

1. Introduction Currently, due to its durability, affordability, and strong compressive strength, concrete is the most often used construction material in building and civil engineering (Biernacki et al. (2017)). Numerous efforts have been performed by scientists and researchers with the aim to design innovative concretes characterized by ductile behavior and high tensile strength. For instance, recently has been proposed numerous ultra-high performance concretes (UHPC) with high cement and binder content, small aggregate size, and low water/cement ratio achieving improved mechanical properties over the ordinary concrete in terms of both tensile and compressive strength (Azmee and Shafiq (2018); Fehling et al. (2014); Reda et al. (1999)). However, the increase in terms of compressive strength leads to a more brittle concrete that can be reinforced with discontinuous high-strength particles in order to increase its ductility and flexural strength (UHPFRC). The improvement of such mechanical properties was investigated in several experimental works. For instance, Wu et al. (2016) observed that the flexural strength in UHPC samples can be increased of about 21%, 47%, and 100% with the incorporation of 1 %, 2%, and 3% of straight steel fibers, respectively. Wille et al. (2012) highlighted that by reinforcing the UHPC samples with deformed steel fibers, instead of the straight ones, an increase of the tensile strength of about 60% can be obtained. In the last decade, numerous kinds of reinforcements at the nanoscale (such as graphene nanoplatelets, carbon nanotubes, nano-silica etc.) have been investigated due to their capacity to improve the compressive and the flexural strength, together with the fracture toughness, of the ultra-high performance concretes. The new investigated advanced concretes are thus characterized by different kinds of reinforcements acting at several length scales (nano-, micro- and macro-scales). For instance: Meng and Khayat (2016) showed an increase of the tensile strength of about 56% and an increase of the energy absorption capacity of about 187% with the addition of 0.3% of graphene nanoplatelets in UHPFRC samples; Wille and Loh (2010) highlighted that the addition of the 0.02% volume fraction of multiwalled carbon nanotubes can increase significantly the bond stresses between the cement matrix and the steel fibers; also, Kononova et al. (2016), by incorporating a small volume fraction of nanotubes, obtained an increase of the pull-out strength which is related to the improvement of the adhesion between the cement matrix and the microscopic reinforcement. Due to the strong separation of scales involved by fracture phenomena acting at both the nano and the macroscales, to the best author knowledge, such advanced concretes were scarcely investigated from the numerical point of view. In the past, the cracking analysis in fiber-reinforced concretes has been numerically investigated using different numerical models that can be grouped in discrete and smeared fracture approaches. The first method allows smearing the cracks over a computational domain in which, to take into account the presence of damaged material, the related mechanical properties are degraded. Such approaches inevitably neglect the discrete nature of the fracture phenomena, and they need to be supported by some regularization techniques (such as strain gradient (Peerlings et al. (1998)) and micropolar formulations (Leonetti et al. (2019))) to prevent the ill-posedness of the associated boundary value problems. On the other hand, the discrete fracture approaches describe the cracks as discontinuities between the