On a second-order rotation gradient theory for linear elastic continua

A second-order rotation gradient theory for non-classical elastic continua is developed. This theory accounts for the higher-order deformation of the material structure where the material particle inside the elastic domain is idealized as a microvolume having three degrees of freedom, namely, a translation, a micro-rotation, and a higher-order micro-deformation. The associated strain energy density is a function of the infinitesimal strain tensor and the first and second gradients of the rotation tensor. It is demonstrated that for materials in nano-scale applications and because of some defects at the material structure level, a higher-order deformation measure may be needed. The second-strain gradient theory has the merit to account for the higher-order deformation of the material particle. However, this theory has limited applications because it depends on 16 additional material constants for isotropic elastic continua. By discussing some physical concepts relevant to the natures of material structures, crystallinity, and amorphousness, the second-strain gradient theory is reduced to the second-rotation gradient theory for certain types of materials. For isotropic materials, the developed second-rotation gradient theory only depends on three additional material constants instead of 16. A continuum model equipped with an atomic lattice model is then proposed to examine the applicability of the available non-classical theories and the applicability of the proposed theory for different types of materials.