Dilatation gradient elasticity theory

Abstract

A simplified version of the strain gradient elasticity theory, in which all gradient effects are related to the first scalar invariant of the infinitesimal strain tensor, i.e., to the dilatation, is developed. Two variants of the theory with different forms of boundary conditions are derived using the variational approach. The first variant is derived taking into account independence of the dilatation variation on the body surface and it has simplified traction boundary conditions formulated only with respect to the total stress tensor. The second variant is derived following a general procedure exploiting the surface divergence theorem which results in a more complex form of boundary conditions on the body surfaces and edges. Correctness of the presented formulations of the theory is discussed. Examples of analytical solutions for the problems of pure bending, pressurized sphere and radial vibrations of sphere are obtained and compared for both variants of the theory.