Green’s functions of size-dependent Timoshenko beams: Gradient elasticity versus stress-driven nonlocal theories

Abstract

In this article, Timoshenko elastic nano-beams modeled by three different models (stress-driven nonlocal integral model, strain gradient elastic model without transversal strain gradient, and strain gradient elastic model with transversal strain gradient effect) are analyzed by the help of Green’s function method. Conventionally, the governing boundary value problem (BVP) for Timoshenko nano-beams consists of two differential equations (DEs) and six boundary conditions (BCs) with two unknown dependent variables, i.e., (1) the cross-sectional rotation function, and (2) the transverse displacement field. In the stress-driven model, this multi-field problem is converted into a new single-field BVP including a single higher order DE with only one dependent variable. Obviously, this recent problem needs some new BCs which are rigorously derived in the second step of formulation. The third, and in fact, the last step of the proposed formulation is to solve the new organized BVP by the means of Green’ function. However, in the strain gradient model, the BVP consists of two six-order DEs with two unknown functions. Also, this problem includes twelve boundary conditions which eight of them are “decoupled” and four of them are “coupled” which should be solved simultaneously. To the best of knowledge to the researchers, the current study is a first successful attempt to solve a BVP with some “coupled” boundary conditions by Green’s function. As an instance, the Green’s function of simply supported Timoshenko nano-beams is obtained for both (1) stress-driven nonlocal integral model, and (2) strain gradient elastic model which, respectively, contain symmetric and non-symmetric kernels. Alongside the graphical presentation of the Green’s functions, the results of all models are presented for several loading including the distributed loads expressed by the help of polynomials with higher degrees (equal or more than two) and sinusoidal distributed loading.