Nanomaterials 2021, 11, 2651. https://doi.org/10.3390/nano11102651 www.mdpi.com/journal/nanomaterials Article Nanobeams with Internal Discontinuities: A Local/Nonlocal Approach Daniela Scorza 1 , Sabrina Vantadori 2, * and Raimondo Luciano 1 1 Department of Engineering, University of Naples Parthenope, Centro Direzionale Isola C4, 80143 Naples, Italy; daniela.scorza@uniparthenope.it (D.S.); raimondo.luciano@uniparthenope.it (R.L.) 2 Department of Engineering and Architecture, University of Parma Parco Area delle Scienze 181/A, 43124 Parma, Italy * Correspondence: sabrina.vantadori@unipr.it Abstract: The aim of the present work is to extend the two-phase local/nonlocal stress-driven inte- gral model (SDM) to the case of nanobeams with internal discontinuities: as a matter of fact, the original formulation avoids the presence of any discontinuities. Consequently, here, for the first time, the problem of an internal discontinuity is addressed by using a convex combination of both local and nonlocal phases of the model by introducing a mixture parameter. The novel formulation here proposed was validated by considering six case studies involving different uncracked nano- beams by varying the constrains and the loading configurations, and the effect of nonlocality on the displacement field is discussed. Moreover, a centrally-cracked nanobeam, subjected to concentrated forces at the crack half-length, was studied. The size-dependent Mode I fracture behaviour of the cracked nanobeam was analysed in terms of crack opening displacement, energy release rate, and stress intensity factor, showing the strong dependency of the above fracture properties on both di- mensionless characteristic length and mixture parameter values. Keywords: energy release rate; internal discontinuity; mixture parameter; nanobeam; stress-driven integral model; stress intensity factor 1. Introduction In the last few decades, structures at the nanoscale level have gained an increasing interest in engineering applications. The reason for such concern from the scientific com- munity is mainly due to the outstanding mechanical, electrical, and thermal properties resulting from the nanoscale sizes [1]. The behaviour of materials at the nanoscale level is significantly different from that exhibited by the same materials at the macroscale level [2–6]. In order to both analyse the so-called size effect and properly evaluate the size-dependent properties, two approaches may be followed: experimental characterisation [7–9] and theoretical modelling [10–26]. For instance, the tensile yield strength of a gradient nano-grained (GNG) surface layer in a bulk coarse-grained (CG) rod of a face-centred cubic Cu was investigated by Fang et al. [7], who observed an increment of strength of about 100% with respect to that of a CG Cu. A study on ductility and strain hardening on a sandwich sheet structure, composed by a CG core between two GNG layers, was conducted by Wu et al. [8].Also, the fracture prop- erties of single-crystalline copper nanowires have been investigated by performing uni- axial tensile tests through a micromechanical device inside a scanning electron micro- scope chamber [9]. It was observed that the fracture strength was much higher than that of bulk copper and that both ductile and brittle-like fracture modes were present in the same batch of fabricated nanowires, depending on their diameters [9]. However, despite their high level of reliability, performing experimental tests at the nanoscale may be quite expensive and time consuming, leading to often prefer theoretical Citation: Scorza, D.; Vantadori, S.; Luciano, R. Nanobeams with Internal Discontinuities: A Local/Nonlocal Approach. Nanomaterials 2021, 11, 2651. https://doi.org/10.3390/nano1110265 1 Academic Editor: Ali Farajpour Received: 25 August 2021 Accepted: 8 October 2021 Published: 9 October 2021 Publisher’s Note: MDPI stays neu- tral with regard to jurisdictional claims in published maps and institu- tional affiliations. Copyright: © 2021 by the authors. Li- censee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and con- ditions of the Creative Commons At- tribution (CC BY) license (http://crea- tivecommons.org/licenses/by/4.0/).