A research into bi-Helmholtz type of nonlocal elasticity and a direct approach to Eringen’s nonlocal integral model in a finite body

Abstract

There are exhaustive reports revolving around the nonlocal differential beam models of micro- and nano-electromechanical systems (MEMS–NEMS), carbon nanotubes (CNTs), nanomaterials, etc., since the nonlocal continuum theory is considered a viable option for shedding light on size effect phenomena. A constitutive equation with a second-order differential operator, Helmholtz type, is employed in these studies. However, these models do not produce quadratic, self-adjoint energy functionals and give rise to paradoxes in static and dynamical problems. The transformation of Eringen’s integral constitutive equation into a differential one is not an injective process in a finite domain and that is probably responsible for the inconsistencies and paradoxes raised. This work researches into the adequacy of a higher-order differential operator, bi-Helmholtz type, applied to engineering problems. The bi-Helmholtz-type operator is more effective for describing wave dispersion of atomic models than the Helmholtz type. Eringen’s nonlocal integral stress model is also looked into beams for both types of kernels. The nonlocal integral model’s key benefit is the energy consistent formulas produced, whereas its main drawback lies in the handling of the 1st kind Fredholm governing equation. Exploiting the modified nonlocal attenuation function (kernel) normalized in a finite domain, the 1st kind Fredholm integral equation is directly transformed into a 2nd kind one. Furthermore, the modified kernel successfully handles the physical inconsistencies in a finite domain, such as the reversion of the nonlocal integral stress model to the classic-local model when the nonlocal parameter tends to zero. Our research objective is to explore bi-Helmholtz operator’s application and the corresponding kernel in the nonlocal integral model. The results deduced and the conclusions reached can be considered adequate for the nonlocal integral form’s suitability for dealing with problems in nanoscale.