Variational formulations and extra boundary conditions within stress gradient elasticity theory with extensions to beam and plate models

Abstract

• Extension to stress gradient elasticity of two variational principles of class elasticity. • Extra boundary conditions as congruence conditions at the boundary. • Extension to stress gradient elasticity of Timoshenko beam and Kirchoff plate. • Gradient-induced boundary conditions for beams and plates in stress gradient elasticity. • Comparison of stress gradient elasticity theories by Forest and Sab and by Polizzotto. The principle of minimum total potential energy and the primary principle of virtual power for stress gradient elasticity are presented as kinematic type constructs dual of analogous static type principles from the literature (Polizzotto, 2014; Polizzotto, 2015a). The extra gradient-induced boundary conditions are formulated as “boundary congruence conditions” on the microstructure’s deformation relative to the continuum, which ultimately require that the normal derivative of the stresses must vanish at the boundary surface. Two forms of the governing PDEs for the relevant boundary-value problem are presented and their computational aspects are discussed. The Timoshenko beam and the Kirchhoff–Love plate theories are extended to stress gradient elasticity under the assumption that stress gradient effects do not propagate transversally. It is shown that for beam models no extra gradient-induced boundary conditions are required, whereas for plate models such conditions must be enforced at the contour surface of the plate, where the normal derivative of the stress resultants are required to vanish. Appendix A is devoted to some basic aspects of the mechanics of the microstructure; Appendix B to a comparison between the present theory and an analogous theory from the literature (Forest and Sab, 2012).