On integral and differential formulations in nonlocal elasticity

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 with prescribed shear stress along the surface. In addition, 1D exponential kernel depending on the vertical coordinate is considered.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 also investigated. It is emphasised 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, in line with earlier observations for equilibrium of a nonlocal cantilever beam.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. Such simplification may be justified when only one of the stresses demonstrates nonlocal behaviour; a similar assumption has been recently made within the so-called dilatational gradient elasticity. 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.