Derivation of Mindlin’s first and second strain gradient elastic theory via simple lattice and continuum models

Mindlin, in his celebrated papers of Arch. Rat. Mech. Anal. 16, 51–78, 1964 and Int. J. Solids Struct. 1, 417–438, 1965, proposed two enhanced strain gradient elastic theories to describe linear elastic behavior of isotropic materials with micro-structural effects. Since then, many works dealing with strain gradient elastic theories, derived either from lattice models or homogenization approaches, have appeared in the literature. Although elegant, none of them reproduces entirely the equation of motion as well as the classical and non-classical boundary conditions appearing in Mindlin theory, in terms of the considered lattice or continuum unit cell. Furthermore, no lattice or continuum models that confirm the second gradient elastic theory of Mindlin have been reported in the literature. The present work demonstrates two simple one dimensional models that conclude to first and second strain gradient elastic theories being identical to the corresponding ones proposed by Mindlin. The first is based on the standard continualization of the equation of motion taken for a sequence of mass-spring lattices, while the second one exploits average processes valid in continuum mechanics. Furthermore, Mindlin developed his theory by adding new terms in the expressions of potential and kinetic energy and introducing intrinsic micro-structural parameter without however providing explicit expressions that correlate micro-structure with macro-structure. This is accomplished in the present work where in both models the derived internal length scale parameters are correlated to the size of the considered unit cell.