The Strain Gradient Elasticity Theory in Orthogonal Curvilinear Coordinates and its Applications

Abstract

AbstractThe strain gradient elasticity theory including only three independent material length scale parameters has been proposed by Zhou et al. to explain the size effect phenomena in micro scales. In this paper, the general formulations of strain gradient elasticity theory in orthogonal curvilinear coordinates are derived, and then are specified for the cylindrical and spherical coordinates for the convenience of applications in cases where orthogonal curvilinear coordinates are suitable. Two basic problems, one is the twist of a cylindrical bar and the other is the radial deformation of a solid sphere, are analyzed under the cylindrical and spherical coordinates, respectively. The results reveal that only the material length scale parameter l2 enters the torsion problem, while completely disappears in the problem of radial deformation of a sphere. The size effect of radial deformation of a solid sphere is controlled by the material length scale parameters l1 and l2. In addition, for the incompressible solid sphere especially, only the material length scale parameter l1 enters this radial deformation problem by neglecting the strain gradient terms associated with hydrostatic strains. Predictably, the present paper offers an alternative avenue for measuring the three independent material length scale parameters from bar twisting and sphere expansion tests.