Benchmarks in nonlocal elasticity defined by Eringen's integral model

In this paper we present low-residual approximate solutions for nonlocal 1D and 2D elasticity problems defined according to Eringen's integral model. The benchmarks in the 1D cases are defined by prescribing the stress field while the unknown fields are the strains or the displacements. For the 2D cases we define problems with equilibrated tractions and evaluate the approximate displacement field. Meanwhile a Fourier series as well as a set of Chebyshev polynomials are used as the basis functions for the main unknown fields. We increase the number of the approximation functions to decrease the norm of the residuals and repeat the procedure until reasonable accuracy is obtained for the final solution. Since the procedure is very time consuming, in some benchmark problems we present the calculated coefficients and in some other we give some point-wise values for further use. The results presented in this paper are particularly useful for the validation and convergence studies when numerical methods are to be used for the solution of the nonlocal elasticity problems.