Asymptotically correct boundary conditions for the higher-order theory of plate bending

Abstract

The construction of refined boundary conditions for the case of edge loading, corresponding to the theory with modified inertia in respect to the order of asymptotic approximation, is considered. The methods of investigation are based on the works of A. L. Goldenveizer and A. V. Kolos. The solution of three-dimensional (3D) problem is constructed as a superposition of the long-wave approximation and boundary layers. The traction on the edge is presented as generalized Fourier expansions, using the Legendre polynomials. This approach allows to obtain explicit expressions for the coefficients of boundary layers via an iteration procedure, in course of which the boundary conditions are satisfied with a pure asymptotic error. As a result, the refined boundary conditions (RBCs) are constructed. Effective stress resultants and couples are introduced, the coefficients of which (the Goldenveizer–Kolos constants) are the simple polynomial functions of Poisson’s ratio. In addition, the contribution of boundary layers to edge displacements is determined. The comparison of the dispersion curve for the edge wave, calculated on the basis of the theory with modified inertia and RBC, with the 3D solution show good agreement in the 10 times wider frequency range than that of the Kirchhoff’s theory. The same result is obtained for the amplitude of a transient edge wave, excited by edge loading. It is also shown that with making use of RBC one can consider the action of a self-equilibrated edge load in the framework of two-dimensional (2D) theory.

Asymptotic long-wave model for a high-contrast two-layered elastic plate

“…We mention contributions [38, 39], where asymptotically correct decay boundary conditions for anti-plane waves were derived using the Saint-Venant’s principle accounting for the contrast properties of a laminate. It is also worth mentioning the paper by Wilde et al [44], in which, using asymptotic approach, the refined boundary conditions for the higher-order theory of plate bending in the case of edge loading were obtained.…”

“…∂t 2 to the asymptotic relation (57), with equations ( 24), (44), and (58) taking into account, and omitting subscripts, we obtain the following differential equation for w:…”

Decay Conditions for Antiplane Shear of a High-Contrast Multi-Layered Semi-Infinite Elastic Strip

“…The particular value of this approach is related to the fact that it allows for important generalizations using perturbation along a small parameter. For example, plane and anti-plane boundary layers are constructed on the basis of this method, resulting in the derivation of asymptotic boundary conditions in elastic plates and shells, e.g., see [3][4][5] and also [6,7] and references therein. Moreover, it is possible to derive low-frequency corrections for static decay conditions, corresponding to dynamic generalizations of the Saint-Venant's principle for a semi-infinite strip; see [8,9].…”