On well-posed integral nonlocal gradient piezoelectric models for static bending of functionally graded piezoelectric nanobeam

Abstract

In this paper, we propose general strain- and stress-driven integral nonlocal gradient piezoelectric models (INGPM), which are capable of distinguishing the softening and toughening size-effects due to elasticity and piezoelectricity. Based on differential calculation, INGPM can be transformed equivalently to differential forms with four constitutive constraints. Due to absence the nonlocal and gradient data, the simplified INGPMs with identical elastic and piezoelectric size-dependent parameters are applied to formulate the static bending of functionally graded piezoelectric Euler-Bernoulli nanobeam. The governing differential equations and boundary conditions as well as constitutive constraints are deduced and expressed in nominal forms. It is found that both strain- and stress-driven INGPMs would lead to well-posed mathematical formulation because the total differential order of the differential equations equal to the number of boundary conditions and constitutive constraints. Neglecting the constitutive constraints, the strain-driven INGPM turns to be differential nonlocal gradient piezoelectric model (DNGPM), which is widely applied to study the size-effect of nanostructures. However, it shows that DNGPM would result in ill-posed mathematical formulation, because the total number (12) of differential orders for governing equations is larger than the number (6) of boundary conditions. The general differential quadrature method (GDQM) is utilized to discretize the differential governing equations, boundary conditions and constitutive constraints. Numerical results show that bending deflections increase consistently with the increase of nonlocal parameter and the decrease of gradient parameter for strain-driven INGPM and the decrease of nonlocal parameter and the increase of gradient parameter for stress-driven INGPM, respectively.