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

Abstract

In the present paper, the behavior of an edge‐cracked nanobeam under both Mode I and Mixed‐Mode (I + II) static bending loading is analyzed by using the stress‐driven nonlocal model (SDM), being the SDM applied for the first time in the case of Mixed‐Mode loading. 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 cross section. The bending stiffness of the cracked cross section is computed by exploiting: the Griffith energy criterion and the conventional linear elastic fracture mechanics. Cracks under both Mode I and Mixed‐Mode (I + II) loading are considered. Finally, an edge‐cracked cantilever nanobeam subjected to a transversal point load at the free end is analyzed. The static deflection is calculated and discussed for different values of relative crack depth, crack location, and crack orientation.

A nonlocal elasticity theory to model the static behaviour of edge-cracked nanobeams

“…1(b)). According to the SDM and under the Euler-Bernoulli hypothesis, the transverse displacement due to loads acting perpendicular to the beam axis can be determined by the following two equations [46] together with two constitutive boundary conditions and six continuity conditions:…”

“…An example of the calculation of such a size is reported in Ref. [46], related to a single crystal of a silicon diamond-cubic structure, by obtaining a value of 15nm. For crack lengths greater than such a value, the LEFM continuum assumption, which postulates the presence of a large number of atoms in the vicinity of the crack tip, is not violated.…”

“…Although the critical crack size should be calculated for the examined material, it can be assumed a value equal to 15nm (that is, that obtained for silicon in Ref. [46]), having the aluminium a face-centered-cubic crystal structure (10 atoms) with a lattice parameter equal to 0.40478nm versus a diamond-cubic structure (8 atoms) and a lattice parameter of 0.543nm of the silicon.…”

“…In particular, two aspects have to be pointed out: -as far as the uncracked microbeams are concerned, the formulation presented in Ref. [46] is applied, since it represents a particular case of that here proposed; -as far as the cracked microbeams are concerned, the formulation presented in Section 2 is applied, where f II =0 (being the crack loaded under pure Mode I) and f I is assumed to be equal to that reported in Tab. 1 for =0° [56].…”

“…Therefore, in the present paper, the mechanical behaviour of edge-cracked nanobeams under Mixed-Mode loading is analytically investigated by exploiting the aforementioned SDM within the Euler-Bernoulli beam theory. In particular, according to the proposed formulation [46], the nanobeam is divided into two beam segments, linked through a massless spring at the cracked cross-section. The spring stiffness is computed by employing firstly the Griffith energy criterion, and then the conventional Linear Elastic Fracture Mechanics (LEFM), as long as the crack length is assumed to be greater than a critical crack size, so ensuring that LEFM does not breakdown.…”