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.