On integral and differential formulations in nonlocal elasticity

The paper is concerned with comparative analysis of differential and integral formulations for boundary value problems in nonlocal elasticity. For the sake simplicity, focus on an antiplane problem a half-space prescribed shear stress along surface. In addition, 1D exponential kernel depending vertical coordinate considered. First, surface loading form travelling harmonic wave studied. This provides counter-example, revealing that within framework Eringen’s theory solution to model does not satisfy equation motion stresses underlying related formulation. A more general setup, starting from singularly perturbed equations expressing local through ones, also investigated. It emphasised transformation original formulation one question only possible provided two additional conditions hold As result, formulated subject three appears be ill-posed, line earlier observations equilibrium cantilever beam. Next, asymptotic problem, boundary, together aforementioned extra conditions, obtained at small internal size. Such simplification may justified when demonstrates behaviour; similar assumption has been recently made so-called dilatational gradient Three-term expansion obtained, leading over interior domain. associated are identical those proposed by Eringen, however, derived effective condition incorporates effect layer which previously ignored. Moreover, calculated correction classical elastic half-space, coming order magnitude greater than appearing motion. Finally, it shown supports wave.