Page 20 - The Combined Finite-Discrete Element Method

GENERAL FORMULATION OF DISCONTINUUM PROBLEMS 3

question is, what is the total volume occupied by the particulate after all the particles
have found the state of rest?

This problem is subsequently referred to as container problem. It is self-evident that the
definition of density ρ given by
dm
ρ = (1.2)
dV
and the definition of mass m given by

m = ρdV (1.3)
V

are not valid for the container problem.
The total mass of the system is instead given by

N

m = m i (1.4)
i=1
where N is the total number of particles in the container and m i is the mass of the
individual particles. In other words, the total mass is given as a sum of the masses of
individual particles. It is worth mentioning that the size of the container is not much larger
than the size of the individual particles. The particles pack in the container, and the mass of
particles in the container is a function of the size of the container, the shape of individual
particles, size of individual particles, deposition method, deposition sequence, etc.
Mathematical description of the container problem ought to take into account the shape,
size and mass of individual particles, and also the interaction between the individual par-
ticles and interaction with the walls of the container. The mathematical model describing
the problem has to state the interaction law for each couple of contacting particles. For
each particle, the interaction law is combined with a momentum balance principle, result-
ing in a set of governing equations describing that particle. Sets of differential equations
for different particles are coupled through inter-particle interaction. The resulting global
set of coupled differential equations describes the behaviour of the particulate system as
a whole. The solution of the global set of governing equations results in the final state
of rest for each of the particles. In the case of hypothetical continuum the total number
of governing partial differential equations does not depend on the size of the problem.
In the container problem, each particle has a set of differential equations governing its
motion, and the total number of governing partial differential equations is proportional to
the total number of particles in the container.
The container problem, together with many similar problems pervading science and
engineering, are by nature discontinuous. They are called problems of discontinuum, or
discontinuum problems. Problems for which a hypothetical continuum model is valid
are, in contrast, called problems of continuum, or continuum problems. The mathematical
formulation of problems of continuum involves the constitutive law, balance principles,
boundary conditions and/or initial conditions.
The mathematical formulation of problems of discontinua involves the interaction law
between particles and balance principles. Analytical solutions of these equations are rarely
available, and approximate numerical solutions are sought instead. The most advanced