In this paper, the self-adjointness of Eringen’s nonlocal elasticity is investigated based on simple one-dimensional beam models. It is shown that Eringen’s model may be nonself-adjoint and that it can result in an unexpected stiffening effect for a cantilever’s fundamental vibration frequency with respect to increasing Eringen’s small length scale coefficient. This is clearly inconsistent with the softening results of all other boundary conditions as well as the higher vibration modes of a cantilever beam. By using a (discrete) microstructured beam model, we demonstrate that the vibration frequencies obtained decrease with respect to an increase in the small length scale parameter. Furthermore, the microstructured beam model is consistently approximated by Eringen’s nonlocal model for an equivalent set of beam equations in conjunction with variationally based boundary conditions (conservative elastic model). An equivalence principle is shown between the Hamiltonian of the microstructured system and the one of the nonlocal continuous beam system. We then offer a remedy for the special case of the cantilever beam by tweaking the boundary condition for the bending moment of a free end based on the microstructured model.

In this paper, we calibrate Eringen's small length scale coefficient e0 for an initially stressed vibrating nonlocal Euler beam via a microstructured beam modelled by some repetitive cells comprising finite rigid segments and elastic rotational springs. By adopting the pseudo-differential operator and Padé's approximation, an analytical solution for the vibration frequency in terms of initial stress may be developed for the microstructured beam model. When comparing this analytical solution with the established exact vibration solution from the nonlocal beam theory, one finds that the calibrated Eringen's small length scale coefficient e0 is given by where σ0 is the initial stress and is the mth mode buckling stress of the corresponding local Euler beam. It is shown that e0 varies with respect to the initial axial stress, from at the buckling compressive stress to when the axial stress is zero and it monotonically increases with increasing initial tensile stress. The small length scale coefficient e0, however, does not depend on the vibration/buckling mode considered.

This paper is focused on the buckling and the vibration analyses of microstructured structural elements, i.e., elements composed of repetitive structural cells. The relationship between the discrete and the equivalent nonlocal continuum is specifically addressed from a numerical and a theoretical point of view. The microstructured beam considered herein is modeled by some repetitive cells composed of finite rigid segments and elastic rotational springs. The microstructure may come from the discreteness of the matter for small-scale structures (such as for nanotechnology applications), but it can also be related to some larger scales as for civil engineering applications. The buckling and vibration results of the discrete system are numerically obtained from a discrete-element code, whereas the nonlocal-based results for the equivalent continuum can be analytically performed. It is shown that Eringen’s nonlocal elasticity coupled to the Euler-Bernoulli beam theory is relevant to capture the main-scale phenomena of such a microstructured continuum. The small-scale coefficient of the equivalent nonlocal continuum is identified from the specific microstructure features, namely, the length of each cell. However, the length scale calibration depends on the type of analysis, namely, static versus dynamic analysis. A perfect agreement is found for the microstructured beam with simply supported boundary conditions. The specific identification of the equivalent stiffness for modeling the equivalent clamped continuum is also discussed. The equivalent stiffness of the discrete system appears to be dependent on the static-dynamic analyses, but also on the boundary conditions applied to the overall system. Satisfactory results are also obtained for the comparison between the discrete and the equivalent continuum for other type of boundary conditions.

This paper is concerned with the bending response of nonlocal elastic beams under transverse loads, where the nonlocal elastic model of Eringen, also called the stress gradient model, is used. This model is known to exhibit some paradoxical responses when applied to beams with certain types of boundary conditions. In particular, for clamped-free boundary condition, this nonlocal model is not able to predict scale effects in the presence of concentrated loads, or it leads to an apparent stiffening effect for distributed loads in contrast to other boundary conditions for which softening effect is observed. In the literature, these paradoxes have been resolved by changing the kernel of the nonlocal model or by modifying the standard boundary conditions. In this paper, the paradox is solved from the nonlocal differential model itself via some related discontinuous nonlocal kinematics. It is shown that the kinematics related to the nonlocal constitutive law lead to the use of moment or shear discontinuities. With such a nonlocal differential model coupled with the nonlocal discontinuity requirements, the beam effectively shows a softening response irrespective of the boundary conditions studied, including the clamped-free boundary conditions, and thereby resolves the paradox. The model is also compared to lattice-based solutions where an excellent agreement between the present nonlocal model and the lattice one is obtained. Finally, the stress gradient model is shown to be cast in a stress-based variational framework, which coincides with a Timoshenko-type model where the shear effect is shown to play the nonlocal role.

This paper presents the determination of Eringen's small length scale coefficient e₀ for buckling of nonlocal Timoshenko beam from a microstructured beam model. The microstructured beam model is composed of discrete rigid elements (of equal length), which are connected by rotational and shear springs that model the bending and shearing behaviors in a beam. The exact solution of e₀ is given for nonlocal Timoshenko beam with small length scale term appearing in the normal stress-strain relation only. It is shown that e₀ approaches $1/\sqrt{12}\approx 0.289$ which coincides with the one calibrated for nonlocal Euler beams.