On variational principles in coupled strain-gradient elasticity

Abstract

Strain-gradient elasticity is a special case of high-gradient theories in which the potential energy density depends on the first and second gradient of the displacement field. The presence of a coupling term in the material law leads to a non-diagonal quadratic form of the stored energy, which makes it difficult for the derivation of fundamental theorems. In this article, two variational principles of the minimum of potential and complementary energies are argued in the context of the coupled strain-gradient elasticity theory. The basis of the proofs of both variational principles is the equivalent transformation of the stain and strain-gradient energy density that allows to avoid the complication related to the presence of the fifth-rank coupling tensor in the equation for the potential energy density and leads to diagonalization of the quadratic form of the stored energy. This transformation enables to inverse Hook’s law, to determine compliance tensors, and to obtain closed-form relation for the complementary energy. After that the proofs of both principles of a minimum of potential and complementary energies are provided in the usual manner adopted in the classical theory of elasticity.