Abstract
We present a framework for derivation of closed-form continuum equations
governing mesoscale dynamics of large particle systems. Balance
equations for spatial averages such as density, linear momentum, and
energy were previously derived by a number of authors. These equations
are exact, but are not in closed form because the stress and the heat
flux (e.g. stress in the momentum balance equation) cannot be evaluated
without the knowledge of particle positions and velocities. Recently, we
proposed a method for approximating exact fluxes by true constitutive
equations, that is, using non-local operators acting only on the average
density and velocity. In the paper, constitute operators are obtained by
using filtered regularization methods from the theory of ill-posed
problems. We also formulate conditions on fluctuation statistics which
permit approximating these operators by local equations. The performance
of the method is tested numerically using Fermi-Pasta-Ulam particle
chains with two different potentials: the classical Lennard-Jones, and
the purely repulsive potential used in granular materials modeling. The
initial conditions incorporate velocity fluctuations on scales that are
smaller than the size of the averaging window. Simulation results show
good agreement between the exact stress and its closed form
approximation.