Theoretical analysis for static bending of Euler–Bernoulli using different nonlocal gradient models

Abstract

In this work, the bending response of Euler–Bernoulli beam is analyzed with three nonlocal strain gradient model with high-order boundary conditions and modified nonlocal strain and stress gradient models, in which the integral constitutive equation is transformed to an equivalent differential equation with two extra constraint equations. The corresponding differential equations are solved uniquely by the Laplace transformation and its inverse version. Under different boundary and loading conditions, the inconsistency of a nonlocal strain gradient model is examined extendedly, and consistent softening and toughening responses can be observed while using modified nonlocal strain and stress gradient model.