On integral and differential formulations in nonlocal elasticity

TL;DR

This paper compares integral and differential formulations in nonlocal elasticity, revealing limitations of Eringen's differential model, and introduces an effective boundary condition accounting for nonlocal boundary layers, with implications for surface wave solutions.

Contribution

It demonstrates that Eringen's differential model may be ill-posed and develops an asymptotic approach to derive effective boundary conditions including nonlocal boundary layer effects.

Findings

Differential model solutions do not satisfy the original nonlocal equations.

Effective boundary conditions incorporate nonlocal boundary layer effects.

Nonlocal correction to classical solutions can be significantly larger than in equations of motion.

Abstract

The paper is concerned with comparative analysis of differential and integral formulations for boundary value problems in nonlocal elasticity. For the sake of simplicity, the focus is on an antiplane problem for a half-space for an exponential kernel. First, a surface loading in the form of a travelling harmonic wave is studied. This provides a counter-example, revealing that within the framework of Eringen's theory the solution to the differential model does not satisfy the equation of motion in nonlocal stresses underlying the related integral formulation. A more general differential setup, starting from singularly perturbed equations expressing the local stresses through the nonlocal ones, is then investigated. It is emphasized that the transformation of the original integral formulation to the differential one in question is only possible provided that two additional conditions on nonlocal stresses hold on the surface. As a result, the formulated problem subject to three boundary conditions appears to be ill-posed. Next, the asymptotic solution of the singularly perturbed problem, subject to a prescribed stress on the boundary, together with only one of the aforementioned extra conditions, is obtained at a small internal size. Three-term expansion is obtained, leading to a boundary value problem in local stresses over interior domain. The associated differential equations are identical to those proposed by Eringen, however, the derived effective boundary condition incorporates the effect of a nonlocal boundary layer which has previously been ignored. Moreover, the calculated nonlocal correction to the classical antiplane problem for an elastic half-space, coming from the boundary conditions is by order of magnitude greater than that appearing in the equations of motion. Finally, it is shown that the proposed effective condition supports an antiplane surface wave.