A unifying variational framework for stress gradient and strain gradient elasticity theories

• Variational principles for stress and strain gradient elasticity.
• Higher order boundary conditions for stress gradient elasticity.
• The h.o. boundary conditions relax for stress gradient beams and plates.

Stress gradient elasticity and strain gradient elasticity do constitute distinct continuum theories exhibiting mutual complementary features. This is probed by a few variational principles herein presented and discussed, which include: i) For stress gradient elasticity, a (novel) principle of minimum complementary energy and an (improved-form) principle of stationarity of the Hellinger–Reissner type; ii) For strain gradient elasticity, a (known) principle of minimum total potential energy and a (novel) principle of stationarity of the Hu–Washizu type. Additionally, the higher order boundary conditions for stress gradient elasticity, previously derived by the author (Polizzotto, Int. J. Solids Struct. 51, 1809–1818, (2014)) in the form of higher order boundary compatibility equations, are here revisited and reinterpreted with the aid of a discrete model of the body's boundary layer. The reasons why the latter conditions need to be relaxed for beam and plate structural models are explained.