Unconstrained and Cauchy-Born-constrained atomistic systems. Deformational and configurational mechanics

In this contribution, the deformational and configurational mechanics of (elastic) discrete atomistic systems in relation to their continuum counterparts are considered for the quasi-static case. Thereby, we firstly investigate the basic unconstrained case in the sense of lattice statics as a reference. Based on these results, we consider two Cauchy-Born-type constraints that (locally) describe the change of the position of atoms (between the material and the spatial configuration) in terms of either a linear or a quadratic map, respectively. Insertion of these kinematic constraints into the variation of the total potential energy of the unconstrained case renders eventually Cauchy-Born-type definitions for atomistic stresses and hyperstresses, for both deformational and configurational cases. In the continuum limit, these are the relevant continuum stresses and hyperstresses contributing to the local force balances of first- (classical) and second-order (non-classical) gradient continua (here first- and second-order gradient refers to the highest gradient of the deformation map characterizing the kinematics). It is emphasized that the atomistic and the Cauchy-Born-type configurational quantities represent novel and unexpected contributions. Among other things, they may be useful in assessing the singular or non-singular character of deformation fields at crack tips and comparing numerical estimates resulting from atomistic simulations with analytical predictions resulting from solutions of related boundary value problems for gradient continua.