A non-local asymptotic theory for thin elastic plates

Abstract

The 3D dynamic nonlocal elasticity equations for a thin plate are subject to asymptotic analysis assuming the plate thickness to be much greater than a typical microscale size. The integral constitutive relations, incorporating the variation of an exponential nonlocal kernel across the thickness, are adopted. Long-wave low-frequency approximations are derived for both bending and extensional motions. Boundary layers specific for nonlocal behaviour are revealed near the plate faces. It is established that the effect of the boundary layers leads to first-order corrections to the bending and extensional stiffness in the classical 2D plate equations.

Free vibrations of nonlocally elastic rods

“…and c 1 , c 2 are arbitrary constants, the substitution of (22), (23) into (20) does not allow to find neither stress distribution σ(x) for the CC conditions, nor the natural modes and frequencies, if some of the ends are free.…”

“…with respect to the eigen-frequency ω. As ξ 1 → 0, we have α → λ and β → ∞, where λ is defined by (23), and solution (33) transforms to a solution corresponding to PNIM. However, this limiting passage may lead to considerable computational difficulties when solving the transcendental frequency equation (35) containing small parameters.…”

Asymptotic derivation of nonlocal beam models from two-dimensional nonlocal elasticity

“…Different approaches exist to derive the governing differential equation for thin and thick beams. Among them, the asymptotic models have attracted much attention over the last few decades (one may refer to different papers in the literature [15][16][17][18][19]). In this case, using a reduction method, arbitrarily chosen variables (displacements, stresses, strains) are expanded in power series in term of small parameters, in most cases the thickness ratio [15] or the nonlocal parameter [16].…”

“…Among them, the asymptotic models have attracted much attention over the last few decades (one may refer to different papers in the literature [15][16][17][18][19]). In this case, using a reduction method, arbitrarily chosen variables (displacements, stresses, strains) are expanded in power series in term of small parameters, in most cases the thickness ratio [15] or the nonlocal parameter [16]. Equations are derived at different order of the small parameter [17][18][19][20], the accuracy of the model increasing with the order of the expansion.…”

“…In the literature, some authors have developed an asymptotic expansion based on a one-field kinematic field, whereas others used a mixed field with stress and displacement expression for the asymptotic method. Furthermore, recently, Chebakov et al [16] proposed an asymptotic derivation of the thin plate equations by using the integral formulation of the nonlocal stress-strain constitutive law given initially by Eringen and Edelen [9]. To the best of the authors' knowledge, the asymptotic derivation of the nonlocal thin and thick beam equations has not been performed in the literature.…”