Atomistic-based strain gradient elasticity theory, in the framework of finite deformations

Atomistic-based strain gradient elasticity theory, in the framework of finite deformations, is proposed. As a fundamental quantity of this approach, an objective relative displacement between atomic-pairs is considered. Besides, a deformation energy of an atomic-pair is defined in terms of such a measure in the form of a Lennard-Jones type potential. Thus, the objective relative displacement between atomic-pairs has been Taylor's series expanded up to second order in order to obtain the deformation energy of atomic-pairs in terms of (i) non-linear strain (the Green-Saint-Venant tensor), (ii) its gradient, (iii) the orientation of the atomic-pair and (iv) Lennard-Jones coefficients, including the inter-atomic distance. The strain energy of the continuum is assumed to be the integral over the unit sphere of the previous Lennard-Jones potential and the isotropic case will be obtained by assuming the independence of the Lennard-Jones coefficients with respect to the inter-atomic orientation.