PSI - Issue 41

2

Andrea Pranno et al. / Procedia Structural Integrity 41 (2022) 618–630 Author name / Structural Integrity Procedia 00 (2019) 000–000

619

Nomenclature d

scalar damage function

E

Young’s modulus of the bulk elements

0 0 , n s K K normal and tangential interfacial elastic stiffness parameters , 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 p n K normal interfacial plastic stiffness parameter c n K normal interfacial plastic stiffness parameter (in compression) 

scalar factor which takes into account the partial contact between the crack surfaces

1. Introduction The study on how the damage phenomena affect the static and dynamic mechanical response is fundamental to reduce the risk of catastrophic failures of structures (Greco et al., 2013; Skrame et al., 2022). It has been demonstrated, through the recent progress in the Structural Health Monitoring field (SHM), that damage phenomena can seriously affect the mechanical properties of the structures, specifically in complex engineering applications (such as bridges) where the structural integrity should be monitored in real-time to guarantee adequate level of safety (Bruno et al., 2016; Lonetti and Pascuzzo, 2014). For this reason, the development of mathematical models of structural systems is of critical importance for identifying effectively the onset and evolution of damage phenomena (Lonetti et al., 2019). It is widely known that by monitoring and analyzing variations in the static and dynamic mechanical behavior of structures that are induced by damage phenomena it is possible to identify structural damage. In general, natural vibration frequencies, natural vibration modes, modal damping factors, and other structural parameters are often considered as indicators of integrity of the structures based on their modal characteristics (Doebling et al., 1998; Salawu, 1997). For instance, the modal parameters are global quantities which can be measured with the experimental modal analysis “EMA” (Penny, 1988; Piersol et al., 2010; Silva and Maia, 1999) as a result of an elaboration process of the structural dynamic response when the structure is subjected to artificial excitations and the operational modal analysis “OMA” (Brincker and Ventura, 2015; Rainieri and Fabbrocino, 2014) when it is subjected to environmental excitations. As "model-based" damage identification techniques we refer to a variety of techniques for detecting damage by comparing the dynamic response of a structure, predicted by a mathematical model able to detect the damage occurrence, with its vibration response observed experimentally on the monitored structure. In this context it is clear the importance of adopting affective methodologies to simulate the occurrence of damage phenomena. Similar to the case of homogeneous solids, damage in plain concrete structural elements is mainly characterized by the presence of single cracks and notch defects (Lin, 2004; Shen and Pierre, 1994; Sinha et al., 2002). On the other hand, simulating damage phenomena in reinforced concrete structural elements is more complicated since their nonlinear behavior, arising from its heterogeneity (Luciano and Willis, 2003) as well as the different damage scenarios, makes them more challenging to investigate. To take into account concrete nonlinearities due to cracking effects and interactions between the constituent materials, such as adhesion between the concrete and reinforcement steel bars (De Maio et al., 2020a), advanced mathematical theories are commonly employed for instance those based on continuum and discrete fracture approaches (Luciano and Sacco, 1998; Greco et al., 2022; Pascuzzo et al., 2020; Greco et al., 2021a). Additional brittle failures can be caused in heterogeneous materials by onset of instabilities (De Maio et al., 2020; Greco et al., 2020b), commonly observed due to reinforcing fibers acting at the nano-, micro- or macro scales (Raffaele Barretta et al., 2018; Greco et al., 2020a, 2018a, 2018b) and due to the adoption of kind of reinforcement with advanced microstructures such as bioinspired, functional and metamaterials (Ammendolea et al., 2021; Greco et al., 2021b, 2020c; Pranno et al., 2022). Because of its current and potential uses in many types of reinforcements, theoretical examination of free vibration behaviors of micro-beams is also a developing research topic