On the correctness condition in boundary value problems of gradient elasticity theory


DOI: 10.34759/trd-2021-120-02

Аuthors

Lurie S. A.1*, Shramko K. K.2**

1. Institute of Applied Mechanics of Russian Academy of Science, IAM RAS, 32a, Leninskii av., Moscow, В-334, GSP-1, 119991, Russia
2. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: salurie@mail.ru
**e-mail: konstantin_home@mail.ru

Abstract

The main subject of study in this article concerns the variational formulation of gradient models of elasticity and specific symmetry conditions in gradient theories of elasticity of a general form. Special attention is being paid to the symmetry conditions, which are a consequence of the fact that it is possible to change the order of differentiation in the components of the displacement gradient tensor in the potential energy density expression. First, the conditions of symmetry in the theory of elasticity and gradient elasticity are discussed, the features of the properties of symmetry in gradient models are noted in comparison with the classical theory of elasticity. Then the variational statements of gradient elasticity and the structure of the boundary conditions following from this statement are briefly discussed. Finally, the conditions for the formulation correctness of the of boundary value problems and their connection with the symmetry property of gradient elastic modules are established. This article presents the symmetry conditions analysis for components of the generalized elastic properties tensor of gradient models of elasticity. It is demonstrated that one of the conditions is related to the correctness of boundary value problems of gradient elasticity theories. It is established for the first time that physically and energetically insignificant components of the elastic modulus tensor may appear on the surface under boundary static conditions for stresses (with variations in displacements) and under boundary conditions on the contours of edges, which can lead to significant errors in solving applied problems. A procedure allowing always obtaining correct boundary conditions for arbitrary optioins of gradient elasticity theories is presented.

Keywords:

combination of movements, circular movements, elliptical trajectory, circular trajectory, multiples of speed

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