PSI - Issue 41

Umberto De Maio et al. / Procedia Structural Integrity 41 (2022) 598–609 Author name / Structural Integrity Procedia 00 (2019) 000–000

599

2

Nomenclature D

scalar damage variable Young’s modulus of the bulk

E

I II , G G modal components of the energy release rate I II , c c G G critical mode-I and mode-II fracture energies 0 0 , n s K K normal and tangential interfacial elastic stiffness parameters mesh L mesh size  dimensionless normal stiffness parameter  dimensionless tangential stiffness parameter coh t cohesive traction vector , c c   critical normal and tangential interface stresses m  mixed-mode displacement jump , n s   normal and tangential components of the displacement jump  Poisson’s ratio of the bulk    displacement jump between the crack faces

1. Introduction Cracking phenomena, including crack branching and coalescence, are expected in the concrete elements when a significant tensile stress state occurs. The concrete, belonging to the quasi-brittle materials, possesses a very small tensile strength thus resulting a relatively weak and brittle material under tension. To improve the tensile strength of concrete, mild reinforcements and prestressing steel are employed in the design of the structural members. Nevertheless, the concrete tensile strength is usually ignored in both design and numerical analysis of reinforced concrete (RC) structures since it provides a negligible contribution to the ultimate strength of the entire structure. However, the increasing need for the assessment of the strength and stiffness at both serviceability and ultimate limit state of existing RC structures, such as buildings and bridges, has encouraged researchers to develop advanced analytical and numerical models for analyzing the damage processes, including cracking and instabilities phenomena, of these structures ((Breysse, 2010; Lonetti et al., 2019; Lonetti and Pascuzzo, 2014)). As highlighted by the experiments, the cracking behavior in the RC structural elements is strongly influenced by several factors such as concrete tensile strength, concrete cover, embedded reinforcing materials (steel fibers and carbon nanotubes), and rebars together with their anchorage length and arrangement ((Lee and Kim, 2009; Alecci et al., 2016; Gribniak et al., 2016; Acierno et al., 2017; Barris et al., 2017)). In light of the experimental evidence, in order to obtain reliable numerical results, the bond behavior between rebars and surrounding concrete must be correctly simulated. To this end, analytical formulations, based on both fracture energy and bond stress-slip approach, have been proposed to describe the effects of the concrete/rebar bond behavior on the load-carrying capacity including the tension stiffening effect ((Gupta and Maestrini, 1990; Ouyang et al., 1997)). However, such analytical models provide reasonable results if simplified assumptions, such as uniformly distributed cracking and particular boundary conditions, are considered. These drawbacks have been overcome by adopting numerical modeling based on the finite element method (FEM) ((Roesler et al., 2007)). In particular, smeared and fracture approaches have been implemented in several numerical models to simulate the damage processes in RC structures. Smeared crack models capture the damage through suitably defined constitutive relations, thus smearing out all the discontinuities over the continuum ((Borst et al., 2004; Luciano and Sacco, 1998; Moshirabadi and Soltani, 2019)). Unfortunately, these approaches lead to ill-posed boundary value problems (BVPs), so they are susceptible to localization instabilities partially solved by inserting some localization limiters such as crack band models ((Červenka et al., 2018)), and nonlocal models ((Luciano, 2001; Barretta et al., 2015; Luciano and Willis, 2003)). Similar approaches are employed to capture microscopic instabilities and the buckling load of nacre-like composite materials and nano-beams, respectively ((Greco et al., 2021b; Pranno et al., 2022; Barretta et al., 2018, 2020; De Maio et al., 2020d)). On the other hand, the discrete fracture models assume that damage is lumped into main propagating cracks, and simulate the entire crack process by using inter- and intra-element cohesive modeling techniques ((De Maio et al., 2021)). These models, also employed to simulate the interfacial crack