On an Extended Family of Quasi-equivalent Models of the Gradient Elasticity Theory

Abstract

There are investigated the formulations of boundary value problems in the Mindlin-Tupin gradient theory characterized by a higher differential order of equilibrium equations and a varied spectrum of boundary value problems, formulated both on a piecewise smooth surface and on the edges of this surface. We consider the possibility of simplifying boundary value problems by eliminating boundary conditions at the edges by introducing an extended spectrum of gradient applied models in the class of equivalent models having the same potential energy density. For this purpose, we investigate the variational statements of boundary value problems, which establish admissible kinematic connections on the surface in the form of linear combinations of the displacement vector and the first derivatives of displacements (both normal and tangential). Classes of gradient models obtained by introducing kinematic constraints on the surface, in which there are no boundary conditions at the edges, are indicated. These include models built by introducing kinematic constraints on the displacement vector and some special classes of models in which the kinematic constraints on the surface are set to the derivatives of displacements.