Nonlocal gradient integral models with a bi-Helmholtz averaging kernel for functionally graded beams

Abstract

In this paper, we develop the bi-Helmholtz averaging kernel-based variationally nonlocal strain and stress gradient (NStrainG and NStressG) integral models to study the size-dependent bending of functionally graded (FG) beams. The integral constitutive relation is transformed into a higher-order differential form while four non-standard constitutive boundary conditions are derived in detail. The Laplace transform technique is utilized to derive closed-form solutions. Interestingly, we find that, unlike the Helmholtz kernel case, the local stress contribution must be included in the bi-Helmholtz kernel-based NSrainG model to avoid the conflict between equilibrium and constitutive requirements. In the numerical simulations, a series of examples are presented to study the size-effects on the bending deformation of the beams with various boundary types. Numerical results show that the bi-Helmholtz kernel-based nonlocal gradient theories are capable of describing both softening and stiffening size-effects in small structures by suitably tuning the nonlocal and gradient relevant parameters involved in nonlocal gradient models. Moreover, the effect of FG parameter is also presented.