Bending analysis of nanoscopic beams based upon the strain-driven and stress-driven integral nonlocal strain gradient theories

Abstract

Strain gradient and nonlocal influences have important roles in the mechanical characteristics of structures as they are scaled down to nanoscopic dimensions. In this research, an attempt is made to capture both effects on the static bending response of nanoscopic Bernoulli–Euler beams by means of a comprehensive model. For this purpose, the strain gradient theory of Mindlin is combined with the integral (original) form of Eringen’s nonlocal theory. Also, the integral nonlocal formulation is written on the basis of both strain-driven and stress-driven versions of nonlocal theory. This mixed nonlocal strain gradient formulation is capable of reducing to the size-independent and combination of integral nonlocal strain gradient family theories by employing simple substitutions. Through constructing finite difference-based differential and integral matrix operators, numerical solution approaches are developed to obtain the deflection of nanobeams under various end conditions. In addition, for the solution of governing equation of strain-driven nonlocal strain gradient theory, an efficient technique is presented that is applied to the variational statement of the problem in a direct approach. Selected numerical results are provided to explore the simultaneous influences of strain gradient terms and nonlocality based on different theories on the static bending response of beams. It is revealed that the developed size-dependent model has the ability of considering strain gradient and nonlocal influences in the most general way.