Nanobeams with Internal Discontinuities: A Local/Nonlocal Approach

Abstract

The aim of the present work is to extend the two-phase local/nonlocal stress-driven integral 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 nanobeams 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 dimensionless characteristic length and mixture parameter values.

Nonlocal analysis of edge‐cracked nanobeams under Mode I and Mixed‐Mode (I + II) static loading

“…The beam is split in two parts: the first part (named left part), with abscissa 0 ≤ x ≤ x d , subjected to the bending moment M 1 x ð Þ, and the second part (named right part), with abscissa x d < x ≤ L, subjected to the bending moment M 2 x ð Þ. The SDM governing equations, according to the differential formulation, 37,38,40 are written for a Euler-Bernoulli nanobeam as…”

“…41 Two constitutive boundary conditions and two constitutive continuity conditions are added to the above governing equations. 37,38,40 However, it is in general more convenient to rewrite Equation (1) in terms of the transversal displacement, v x ð Þ, by exploiting the well-known relationship…”

“…and by deriving Equation (1) two times with respect to the axial coordinate x. Therefore, the differential governing equations become v • Constitutive Continuity Conditions (CCCs) at the internal discontinuity (x ¼ x d ) [37][38][39] :…”

“…In the context of cracked nanostructures under static loading, that is, the loading condition examined in the present research work, both atomistic modeling [19][20][21][22][23][24][25][26][27][28] and GCTs [29][30][31][32][33][34][35][36][37][38][39] are put forward into several studies to investigate their fracture behavior, being the experimental investigation still a challenge, due to technical difficulties connected with the nanoscale sizes involved.…”

“…Due to the fact that, to the best knowledge of the present authors, the SDM has not been applied yet to edge-cracked nanobeams under Mixed-Mode (I + II) static loading (although longitudinal cracked nanobeams have been already investigated [37][38][39] ), in the present paper both Mode I and Mixed-Mode (I + II) static bending behavior of an edge-cracked nanobeam is analyzed by using the above model. More precisely, the cracked nanobeam is modeled by using a modification of the classical cracked-beam theory, consisting in dividing the beam into two beam segments connected through a massless elastic rotational spring, located at the cracked section.…”