On well-posedness of two-phase nonlocal integral models for higher-order refined shear deformation beams

Abstract

Due to the conflict between equilibrium and constitutive requirements, Eringen’s strain-driven nonlocal integral model is not applicable to nanostructures of engineering interest. As an alternative, the stress-driven model has been recently developed. In this paper, for higher-order shear deformation beams, the ill-posed issue (i.e., excessive mandatory boundary conditions (BCs) cannot be met simultaneously) exists not only in strain-driven nonlocal models but also in stress-driven ones. The well-posedness of both the strain- and stress-driven two-phase nonlocal (TPN-StrainD and TPN-StressD) models is pertinently evidenced by formulating the static bending of curved beams made of functionally graded (FG) materials. The two-phase nonlocal integral constitutive relation is equivalent to a differential law equipped with two restriction conditions. By using the generalized differential quadrature method (GDQM), the coupling governing equations are solved numerically. The results show that the two-phase models can predict consistent scale-effects under different supported and loading conditions.

Nonlocal Elasticity for Nanostructures: A Review of Recent Achievements

“…Solutions to the relative elastostatic problem can be found in [87]. It is worth noting that the outcomes contributed in [43] amend the erroneous statements in [88] regarding the ill-posed nature of the structural problem of third-order beams based on the stress-driven nonlocal model.…”

Free damping vibration of functionally graded porous viscoelastic nonlocal microbeam with thermal effect

“…Through Eringen’s nonlocal differential model (ENDM) is extendedly applied to address the size effect of microstructures, inconsistent size-dependent responses are obtained for tensile bar (Benvenuti and Simone, 2013; Pisano and Fuschi, 2003) and flexural beams (Li et al, 2015a; Reddy and Pang, 2008; Zhang et al, 2019a; Zhang and Qing, 2020; Tang and Qing, 2021). In addition, it is also reported that ENDM would lead to ill-posed mathematical formulation for high-order shear deformation beams(Zhang and Qing, 2021b, 2021c; Zhang et al, 2021) and plates(Peddieson et al, 2003; Qing, 2022), since the order of differential governing equations is higher than the number of boundary conditions.…”

Well-posed two-phase nonlocal integral models for free vibration of nanobeams in context with higher-order refined shear deformation theory

“…As another effective strategy, Romano et al (Romano and Barretta, 2017;Romano et al, 2017a) proposed a novel nonlocal constitutive formulation of stressdriven type, which assumes the elastic strain as the integral convolution between the stress field and the nonlocal kernel. Such a strategy also leads to well-posed result for bounded structures, and a consistently stiffening effect can be obtained when increasing nonlocal length-scale parameters (Apuzzo et al, 2017;Barretta et al, 2018b;Barretta et al, 2019b;Pinnola et al, 2020;Russillo et al, 2021;Zhang and Qing, 2021b;Zhang et al, 2020). Subsequently, the stressdriven type two-phase local/nonlocal mixed integral model (Apuzzo et al, 2020;Barretta et al, 2018a) is also formulated by combining the purely stress-driven nonlocal model with the local elasticity theory.…”