PSI - Issue 66

Daniela Scorza et al. / Procedia Structural Integrity 66 (2024) 406–411 Author name / Structural Integrity Procedia 00 (2025) 000 – 000

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1. Introduction Nanostructures, such as nanoplates, nanobeams, nanotubes and so on, are nowadays widely used in several fields as, for instance, automotive, aerospace, biomedical and electronic sectors. This is mainly due to their unique electrical, thermal, magnetic, optical and mechanical properties. Moreover, it has been also observed that a superior fracture performance of nanostructures with respect to largescale structures (Qing and Tang, 2023), making the study of their fracture behavior of primary interest. Such an investigation can be performed by following two strategies: experimental characterisation and theoretical modelling. In the first case, despite the high level of reliability, performing experimental tests may be quite expensive and time consuming, whereas theoretical models, such as atomistic/molecular ones and Generalised Continuum Theories (GCTs), can be preferred as low-cost tools. By focusing our attention on nanobeams, accurate models, based on the GCTs, have been proposed in order to take into account the presence of cracks (Loya et al., 2009; Attia et al., 2024; Zhang and Qing, 2024). Among the GCTs, the Stress Driven nonlocal Model (SDM), originally proposed by Romano and Barretta (2017), has been recently applied to model the fracture behaviour of cracked nanobeams subjected to bending loads by some of the present authors (Scorza et al., 2021; Scorza et al., 2022; Scorza et al. 2023a; Scorza et al. 2023b). In particular, in Scorza et al. (2023a; 2023b), the mechanical behaviour of an edge-cracked nanobeam, subjected to both Mode I and Mixed Mode (I+II) loading, has been studied by modelling the cracked cross-section through a massless elastic rotational spring. According to the proposed procedure, the stiffness of such a spring has been determined by taking full advantage of both the Griffith’s energy criterion and the Linear Elastic Fracture Mechanics (LEFM). In the present paper, a novel nonlocal analytical model is proposed to take into account the effect of multiple cracks on the mechanical response of nanobeams. Section 2 is dedicated to the theoretical formulation, whereas details on the spring stiffness computation are reported in Section 3. A parametric analysis is presented in Section 4 and the main conclusions are outlined in Section 5.

Nomenclature i a

depth of the i th − crack , , B H L nanobeam width, height and length i c normalised distance between cracks d H

compliance of the i th − cracked cross-section

E

material elastic modulus geometric shape function

* f

stiffness of the i th − massless elastic rotational spring

i k c L i v

material internal characteristic length

transverse (vertical) displacement of the i th − beam segment

1 x L  = relative position of the first crack a H  = relative crack depth i  curvature of the i th − beam segment

2. Nonlocal analytical model for multiple cracked nanobeams 2.1. Differential governing equations for a nanobeam with n edge-cracks The novel analytical nonlocal model is based on the application of the SDM to the case of a straight nanobeam containing n edge-cracks subjected to a bending moment ( ) M x . Such a nanobeam has length L and constant cross-section B H  , with a bending stiffness EI (being E the elastic modulus and I the inertia moment). By assuming that the horizontal coordinate x of the local frame xyz is directed along the nanobeam axis, with its origin in the left-hand side of the nanobeam itself (that is, 0 x = and x L = identify the left- and the right-hand sides of the nanobeam, respectively), the i th − edge-crack, with depth i a , is located at the coordinate i x . According to the proposed analytical model, the nanobeam is split in correspondence of each i th − crack, thus obtaining 1 n + beam segments, which are assumed to be connected to each other through massless elastic rotational springs of stiffness