A hierarchy of simplified constitutive models within isotropic strain gradient elasticity

Simplified isotropic models of strain gradient elasticity are presented, based on the mutual relationship between the inherent (dual) gradient directions (i.e. the gradient direction of any strain gradient source and the lever arm direction of the promoted double stress). A class of gradient-symmetric materials featured by gradient directions obeying a reciprocity relation and by 4 independent h.o. (higher order) coefficients is envisioned, along with the sub-classes of hemi-collinear materials (3 h.o. coefficients, gradient directions in part coincident), collinear materials (2 h.o. coefficients, equal gradient directions) and micro-affine materials (1 h.o. coefficient, behavioral affinity at micro- and macro-scale, coincident with the Aifantis model). All models comply with the energy positive definiteness conditions. The boundary-value problem for the wide class of gradient-symmetric materials is governed by a set of Poisson-Helmholtz type differential equations almost unaffected by the number of independent h.o. coefficients; instead the boundary conditions carry in, in general, problem-dependent computational difficulties increasing with the number of these coefficients. As an application, gradient-symmetric beam models are discussed. A parallel hierarchy of simplified isotropic models with couple stresses is also presented, in which the novel concept of rotational volumetric strain gradient is exploited. A graphical overview on isotropic strain gradient elasticity models is reported. An Appendix is devoted to the concepts of extensional and rotational volumetric strain gradients and to the related pressure-like stresses.