Application of the Green's function method for static analysis of nonlocal stress-driven and strain gradient elastic nanobeams.

• The static response of a generally restrained stress-driven nonlocal Euler-Bernoulli nanobeam (GRNB), subjected to arbitrary loads is theoretically and numerically investigated. • It is shown that the gradient elasticity theory is a stress-driven nonlocal model with unsymmetrical exponential kernel normalized along the finite beam. • The stress-driven nonlocal and the strain gradient elasticity theories are ruled by a similar 6th order differential equation, with distinct higher-order boundary conditions. • The Green's function method is handled to formulate the problem in a systematic integral approach. • Both theories give very close results and are associated to a stiffening effect of the small-scale effect. This paper focuses on the static analysis of a generally restrained Euler-Bernoulli nanobeam (GRNB), which is subjected to arbitrary distributed or concentrated loads. The small length scale effect is introduced through a strongly nonlocal integral elastic law called the stress-driven nonlocal Euler-Bernoulli beam model, and a weakly nonlocal elastic law called the strain-gradient Euler-Bernoulli beam model. The stress-driven nonlocal theory and the strain gradient elasticity theory are both employed, to formulate the differential equation governing the Euler-Bernoulli nanobeams. Both theories are ruled by a similar 6th order differential equation, with distinct higher-order boundary conditions. The paper demonstrates that the strain gradient theory can be also reformulated as a stress-driven nonlocal theory with unsymmetrical kernel, in contrast to the initial stress-driven nonlocal theory which has symmetrical exponential kernel. Additionally, some theoretical backgrounds about both nonlocal elasticity theorems are provided. Exact solutions for both nanobeam models are presented, using Green's function method. The parameters of the Green's functions are derived for each beam theory, and then utilized to establish the static displacement function of both nanobeam theories. Exact solutions are provided for the bending of both nanobeams with various end boundary conditions, that account for the small length scale effects. The paper also compares the responses of the stress-driven Euler-Bernoulli beam and the strain gradient Euler-Bernoulli beam, showing close deflections and similar effects of increasing the nonlocal parameter on beam stiffness. A major difference highlighted between the two theories is that the strain gradient theory predicts a homogeneous strain state for homogeneous stress state, as opposed to the stress-driven theory. [ABSTRACT FROM AUTHOR]