Simultaneous spatial and temporal coarse-graining: From atomistic models to continuum elastodynamics

Abstract

This paper aims at providing a new angle to the classical thermodynamic description of the elastodynamics of solids at finite temperature via the simultaneous spatio-temporal coarse-graining of the atomistic Hamiltonian equations of motion. First, we show by means of various numerical examples, that the spectral kinetic energy density of atomistic trajectories as projected in the normal modes of the system, reveals the anticipated coupling between the spatial and temporal scales, as well as the separation between the elastic and thermal material behavior. The ratio between these two scales provides a single parameter ϵ, whose limit to zero, denotes the upscaling to the continuum scale in the limit of infinite length/time scale separation. The coarse-graining procedure for the identified slow (elastic) variables is then performed by combining mathematical results of Bornemann (2006) based on notions of weak convergence, and statistical mechanics results of Berdichevsky (1997). The resulting continuum description and relations, obtained in generalized coordinates, are all in complete agreement with well-known results in the thermodynamics of solids. Yet, the upscaling exercise delivers simple mathematical insight into some of these classical continuum expressions. Specifically, the potential structure of the reversible elastodynamic equations of motion, with the mechanical forces given by the derivatives of free energy at constant temperature, becomes quite apparent. In addition, the non-commutativity of weak limits and nonlinear functions provides a simple mathematical explanation for the temperature dependence of the mechanical properties of materials, or the fact that thermal energy is finite, while thermal vibrations are negligibly small at the continuum scale. The limit ϵ → 0 is also explored numerically for two examples of a beam undergoing elastic vibrations at various temperatures. The first example is an illustrative mass-spring system, where the atoms’ mass, interatomic distance and potentials are suitably scaled with ϵ, while the length of the beam remains constant; and the second example explores an iron beam of increasing length and fixed atomistic description. In both cases, the scaling of all quantities is analyzed in detail, and the coarse-grained description is compared to the behavior obtained via full atomistic simulations.