Symmetry conditions in strain gradient elasticity

We study the variational significance of the order-of-differentiation symmetry condition of strain gradient elasticity. This symmetry condition stems from the fact that in strain gradient elasticity, one can interchange the order of differentiation in the components of the second displacement gradient tensor. We demonstrate that this symmetry condition is essential for the validity of free variational formulations commonly employed for deriving the field equations of strain gradient elasticity. We show that relying on this additional symmetry condition, one can restrict consideration to strain gradient constitutive equations with a considerably reduced number of independent material coefficients. We explicitly derive a symmetry unified theory of isotropic strain gradient elasticity with only two independent strain gradient material coefficients. The presented theory has simple stability criteria and its factorized displacement form equations of equilibrium allow for expedient identification of the fundamental solutions operative in specific theoretical and application studies.