Page 20 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 5

Let now x; be the mapping which associates to any particle P of the
continuum M, at any time f#, a certain position r obviously belonging

to the volume support (configuration) D, 1.e., P “ r. This mapping

is called motion, the equation r = y(P,t) defining the motion of that
particle. Obviously the motion of the whole continuum will be defined
by the ensemble of the motions of all its particles, i.e., by the mapping

x(M,t),x:M + D, which associates to the continuum, at any
moment f, its corresponding configuration.
The motion of a continuum appears then as a sequence of configura-
tions at successive moments, even if the continuum cannot be identified
with its configuration D = x (M,t).
The mapping which defines the motion has some properties which

will be made precise in what follows. But first let us identify the most
useful choices of the independent variables in the study (description) of
the continuum motions. They are the Lagrangian coordinates (material
description) and the Eulerian coordinates (spatial description).
Within the material description, the continuum particles are “‘identi-
fied” with their positions (position vectors) in a suitable reference config-
uration (like, for instance, the configuration at the initial moment to)?

These positions in the reference configuration would provide the “fin-
gerprints” of the continuum particle which at any posterior moment ¢,
will be individualized through this position R belonging to the reference
configuration Do.
Under these circumstances, due to the mentioned identification, the
equation of the motion is

r= x(R,t), (1.1)

the R coordinates ( X* or X; ), together with t, representing the La-

grangian or material coordinates, through which all the other motion
parameters can be expressed. Hence £x(R, t) and 4x (R,t), with R
scanning the points of the domain Dg, will define the velocity field and
the acceleration field respectively at the moment te

The equation of motion, for an R fixed and ¢ variable, defines the
trajectory (path) of the particle P which occupied the position R at the
initial moment.
Finally, from the same equation of motion but for ¢ fixed and R vari-
able in the configuration Dg, we will have that the corresponding r is

"In the theory of elasticity one takes as reference configuration that configuration which
corresponds to the natural (undeformed) state of the medium.
e suppose the existence of these fields and their continuity except, possibly, at a finite
number of points (surfaces) of discontinuity.