Page 18 - The Combined Finite-Discrete Element Method

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Introduction

1.1 GENERAL FORMULATION OF CONTINUUM PROBLEMS

The microstructure of engineering materials is discontinuous. However, for a large pro-
portion of engineering problems it is not necessary to take into account the discontinuous
nature of the material. This is because engineering problems take material in quantities
large enough so that the microstructure of the material can be described by averaged
material properties, which are continuous. The continuous nature of such material prop-
erties is best illustrated by mass m, which is defined as a continuous function of volume
through introduction of density ρ such that

dm
ρ = (1.1)
dV
where V is volume. The microscopic discontinuous distribution of mass in space is thus
replaced by hypothetical macroscopic continuous mass distribution. In other words, micro-
scopic discontinuous material is replaced by macroscopic continuum of density ρ.
The continuum hypothesis introduced is valid as long as the characteristic length of
the particular engineering problem is, for instance, much greater than the mean free path
of molecules. For engineering problems the characteristic length is defined by either
the smallest dimension of the problem itself, or the smallest dimension of the part of
the problem of practical interest. The hypothesis of continuum enables the definition of
physical properties of the material as continuous functions of volume. These physical
properties are very often called physical equations,orthe constitutive law, and they are
then combined with balance principles (balance equations). The result is a set of governing
equations. The balance principles are apriori physical principles describing materials
in sufficient bulk so that the effects of discontinuous microstructure can be neglected.
Balance principles include conservation of mass, conservation of energy, preservation of
momentum balance, preservation of moment of momentum balance, etc.
Governing equations are usually given as a set of partial differential equations (strong
formulation) or integral equations (weak or variational formulation). Governing equations,
when coupled with external actions in the form of boundary and initial conditions (such as
loads, supports, initial velocity, etc.), make a boundary value problem or an initial bound-
ary value problem. The solution of a particular boundary value problem is sometimes
expressed in analytical form. More often, approximate numerical methods are employed.

The Combined Finite-Discrete Element Method A. Munjiza
 2004 John Wiley & Sons, Ltd ISBN: 0-470-84199-0