One-dimensional nonlocal and gradient elasticity: Assessment of high order approximation schemes

•We examine high-order approximation schemes for nonlocal elastic problems.•We consider the integro-differential and the strain gradient formulations.•The accuracy of the approximation schemes is assessed through 1D examples.•High continuity of the approximation decreases the error in energy norm.

We investigate the application and performance of high-order approximation techniques to one-dimensional nonlocal elastic rods. Governing equations and corresponding discrete forms are derived for the integro-differential formulation proposed by Eringen and the laplacian-based strain gradient formulation developed by Aifantis and coworkers. Accuracy and convergence rate of the numerical solutions obtained with Lagrange, Hermite, B-spline finite elements and C∞ generalized finite elements are assessed against the corresponding analytical solutions.