An atomistically-meaningful pseudocontinuum representation for the finite monatomic chain with harmonic nearest-neighbor interactions

An atomistically-meaningful pseudocontinuum representation for the nontrivial lattice dynamics of a finite monatomic chain with linear elastic interactions between nearest neighbor atoms is analytically deduced by mean of a dynamic mechanical analysis extending the memory-dependent pseudocontinuum viewpoint suggested in [M. Charlotte and L. Truskinovsky, Lattice dynamics from a continuum viewpoint, J. Mech. Phys. Solids, 60, pages 1508-1544 (2012)]. For a correct description of the lattice dynamics at its interstice length scale, the pseudocontinuum model integrates both the bulk and boundary inertial (heat-vibration) effects of the atomistic medium through specific modifications of the classical elastodynamic Newton's law model: these modifications involve a generalization of the D'Alembert's principle of inertial forces and Neumann-Robin's boundary conditions, without increasing the number of initial and boundary conditions of the generic mechanical evolution problem, unlike all other generalized continuum models proposed in the literature up to this date. Owing to the spatially local and one-dimensional nature of the discrete and pseudocontinuum models, relationships are thus more clearly pinpointed between the elastodynamic normal stress field of that exact generalized continuum representation and the cohesive (or internal) and inertial forces operating at the lattice sites within the bulk of a finite-size monatomic chain and at its boundary.