Perturbation approach to Eringen's local/non-local constitutive equation with applications to 1-D structures

Eringen's two-phase local/non-local constitutive equation is preferred over its full non-local counterpart due to mathematical simplifications it provides. Then again, an integro-differential equation must be solved, which requires rigorous examination of the existence of an exact solution in certain forms. For this purpose, some additional constraints are attained to strain field for the sake of an exact solution which may be in contrast with the balance equations. It is the aim of this study to look for possible approximated solutions in series by a perturbation approach. Indeed, we find that response of structures with non-local constitutive relation may be approximated by a set of local elasticity problems, the existence and uniqueness of which are ensured. The present approach does not require any more conditions than physical boundary conditions, such as constitutive boundary conditions. It is applied to simple one-dimensional structural elements, and numerical evidence on possible convergence of the series expansion is provided. Some structural problems of bars and beams, which may be the simplified models of nanostructures in modern engineering applications, are discussed and solutions to them are given in closed-form.